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International Journal of Control Volume 31, 1980 - Issue 6
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Original Articles

A survey on non-linear oscillations

D. P. ATHERTON Electrical Engineering Department, University of New Brunswick, Frcdcricton, N.B, CanadaView further author information
&
H. T. DORRAH Electrical Engineering Department, University of New Brunswick, Frcdcricton, N.B, Canada
Pages 1041-1105 | Received 26 Feb 1979, Published online: 30 Jun 2010

REFERENCES

  • AGGARWAL , J. K. , 1972 , Notes on Nonlinear Systems ( Princeton , N.J. Van Nostrand Reinhold ).
  • AIZERMAN , M. A. , 1958 , Lectures on the Theory of Automatic Regulation ( Moscow Fitzmatgiz ).
  • AIZERMAN , M. A. , 1963 , Theory of Automatic Control ( New York Pergamon ).
  • ANDRONOV , A. A. , and CIIAIKIN , S. E. , 1949 , Theory of Oscillations , English translation ( Princeton University Press ).
  • ANDRONOV , A. A. , VITT , A. A. , and KHAIKIN , S. E. , 1966 , Theory of Oscillators ( Reading , Mass. Addison Wesley ).
  • ATHERTON , D. P. , 1975 , Nonlinear Control Engineering-Describing Function Analysis and Design ( London Van Nostrand Reihnold ).
  • ATHERTON , D. P. , and DORRAH , H. T. , 1976 , A survey on the stability of oscillations . Proc. 19th Midwest Symp. Circuits and Systems , University of Wisconsin-Milwaukee , pp. 354 – 364 .
  • BARNETT , S. ,STOREY, C, 1970 , Matrix Methods in Stability Theory ( London Nelson ).
  • BELLMAN , R. , 1953 , Stability Theory of Differential Equations ( New York McGraw-Hill ).
  • BENNETT , W. R. , 1950 , A general review of linear varying parameter and nonlinear circuit analysis . Proc. Inst. Radio Engrs , 38 , 259 – 263 .
  • BLAQUTERE , A. , 1966 , Nonlinear System Analysis ( New York Academic Press ).
  • BOOOLYUBOV , N. N. , and MITROPOLSKY , Y. A. , 1961 , Asymptotic Methods in the Theory of Nonlinear Oscillations , English translation ( New York Gordon & Breach ).
  • BULGAKOV , B. V. , 1954 , Oscillations , in Russian ( Moscow Gostehizdat ).
  • BUTENIN , N. V. , 1965 , Elements of the Theory of Nonlinear Oscillations , English translation ( Toronto Blaisedell ).
  • CODDINGTON , E. A. , and LEVINSON , N. , 1955 Theory of Ordinary Differential Equations ( New York McGraw-Hill ).
  • CORDUNEANU , C. , 1973 , Integral Equations and Stability of Feedback Systems ( New York Academic Press ).
  • COSGRIFF , R. L. , 1958 , Nonlinear Control Systems ( New York McGraw-Hill ).
  • COUGHANOWR , D. R. , and KOPPEL , L. B. , 1965 , Process Systems Analysis and Control ( New York McGraw-Hill ).
  • CUNNINGHAM , W. J. , 1958 , Introduction to Nonlinear Analysis ( New York McGraw-Hill ).
  • FLOQUET , G. , 1883 , Sur les Equations differentielles lineaires a coefficients periodiques . Annls norm, sup. , 2 , 47 – 88 .
  • FLUGGE-LOTZ , I. , 1968 , Discontinuous and Optimal Control ( New York McGraw-Hill ).
  • FOLLINGER , O. , 1969 , Nichtlineare Regelungen ( Munchen Oldenbourg ).
  • GELB , A. ,VANDER VELDE , W. E. , 1968 , Multiple-input Describing Functions and Nonlinear System Design ( New York McGraw-Hill ).
  • GIBSON , J. F. , 1963 , Nonlinear Automatic Control ( New York McGraw-Hill ).
  • GILLE , J. C. , PELEGRIN , M. J. , and DECAULNE , P. , 1959 , Feedback Control Systems ( New York McGraw-Hill ).
  • GRAHAM , D. , and MCRUER , D. , 1961 , Analysis of Nonlinear Systems ( New York Wiley ).
  • GREENSITE , A. L. , 1970 , Elements of Modern Control Theory ( Washington, D.C. Spartan ).
  • HAAG , J. , 1962 , Oscillatory Motions ( Belmont , California Wadsworth ).
  • HALANAY , A. , 1966 , Differential Equations-Stability, Oscillations, Time Lags ( New York Academic Press ).
  • HALE , J. K. , 1963 , Oscillations in Nonlinear Systems ( New York McGraw-Hill ).
  • HAYASHI , C. , 1964 , Nonlinear Oscillations in Physical Systems ( New York McGraw-Hill ).
  • Hsu , J. C. , and MEYER , A. U. , 1968 , Modern Control Principles and Applications ( New York McGraw-Hill ).
  • KAUDERER , K. , 1958 , Nichtlineare Mechanik ( Berlin Springer-Verlag ).
  • KOVACH , L. D. , 1960 , Life can be so nonlinear . American Scientist , Vol. 48 , No. 2 , June , pp. 218 – 225 .
  • KRYLOV , N. , and BOGOLYUBOV , N. , 1943 , Introduction to Nonlinear Mechanics ( Princeton University Press ).
  • LERNER , A. Y. , 1974 , A survey of soviet contributions to control theory . Control and Dynamic Systems-Advances in Theory and Applications , Vol. 11 , edited by C. T. Leondes ( New York Academic Press ), pp. 491 – 514 .
  • MACMILLAN , R. H. , 1962 , Nonlinear Control Systems Analysis ( Oxford Pergamon ).
  • MEIROVITCH , L. , 1975 , Elements of Vibration Analysis ( New York McGraw-Hill ).
  • MINORSKY , N. , 1947 . Introduction to Nonlinear Mechanics ( Ann Arbor , Michigan Edwards Brothers ).
  • MINORSKY , N. , 1962 , Nonlinear Oscillations ( Princeton , N.J. Van Nostrand ).
  • MINORSKY , N. , 1969 , Theory of Nonlinear Control Systems ( New York McGraw-Hill ).
  • NASLIN , P. , 1965 , The Dynamics of Linear and Nonlinear Systems ( London Blackie ).
  • POPOV , E. P. , 1962 , The Dynamics of Automatic Control Systems ( Reading , Mass. Addison-Wesley ).
  • REISSIG , R. , SANSONE , G. , and CONTI , R. , 1974 , Non-linear Differential Equations of Higher Order ( Leyden Noordhoff ).
  • ROSENBROCK , H. H. , 1974 , Computer-aided Control System Design ( London Academic Press ).
  • ROZENVASSER , E. N. , 1969 , Oscillations of Nonlinear Systems , in Russian ( Moscow Nauka ).
  • SAATY , T. L. , 1967 , Modern Nonlinear Equations ( New York McGraw-Hill ).
  • SANSONE , G. , and CONTI , R. , 1964 , Nonlinear Differential Equations ( New York Pergamon ).
  • SILJAK , D. D. , 1969 , Nonlinear Systems-the Parameter Plane Analysis and Design ( New York Wiley ).
  • STARKERMANN , R. , 1970 , Die harmonische Linearisierung , Vols. 1 and 2 ( Mannheim Bibliographisches Institut ).
  • STOKER , J. J. , 1950 , Nonlinear Vibrations ( New York Interscience ).
  • STRUBLE , R. A. , 1962 , Nonlinear Differential Equations ( New York McGraw-Hill ).
  • TEODORESCU , D. , 1973 , Entwurf nichtlinearer Regelsysteme mittels Abtasl- matrizen ( Heidelberg Alfred Huthig Verlag ).
  • THAXBR , G. J. , and PASTEL , N. P. , 1962 , Analysis and Design of Nonlinear Feedback Control Systems ( New York McGraw-Hill ).
  • THEODORCHIK , K. F. , 1952 , Self-oscillating Systems , in Russian, 3rd edition ( Moscow Gostehizdat ).
  • THOMSON , W. T. , 1972 , Theory of Vibration with Applications ( Englewood Cliffs , N.J. Prentice-Hall ).
  • URABE , M. , 1967 , Nonlinear Autonomous Oscillations-Analytical Theory ( New York Academic Press ).
  • KARMAN , T. , 1940 , The engineer grapples with non linear problems . Bull. Amer. math. Soc , 46 , 615 – 683 .
  • WEST , J. C. , 1960 , Analytical Techniques for Nonlinear Control Systems ( London English University Press ).
  • ZUBOV , V. I. , 1963 , Mathematical Methods for the Study of Automatic Control Systems ( New York Pergamon ).
  • BELEVITCH , V. , 1959 , Theorie des circuits non-lin6aires en regime alternati louvain . Librairie Universitaire Uystpruyst .
  • BENDIXSON , I. , 1901 , Sur Ies courbes definies par des Equations differentielles . Acta math. , 24 , 1 – 88 .
  • KALMAN , R. E. , 1957 , Physical and mathematical mechanisms of instability in nonlinear automatic control systems . J. bos. Engng , 79 , 553 – 563 .
  • KHOKHLOV , R. V. , 1960 , A method of analysis in the theory of sinusoidal self-oscillations . J.R.E. Trans. Circuit Theory , 7 , 398 – 13 .
  • KUMAGAI , S. , and KAWAMOTO , S. , 1960 , Multistable circuits using nonlinear reactances . I.R.E. Trans. Circuit Theory , 7 , 432 – 440 .
  • LAPIDUS , L. , and Luus , R. , 1967 , Optimal Control of Engineering Processes ( Waltham , Mass. Blaisedell ).
  • LE COEBEILLER , P. , 1960 , Two-stroke oscillators . I.R.E. Trans. Circuit Theory , 7 , 387 – 398 .
  • LIENARD , A. , 1928 , Etude des oscillations entretenues . Revue gin. Elect. , 23 , 901 – 946 .
  • MINORSKY , N. , 1960 , Theoretical aspects of nonlinear oscillations . I.R.E. Trans. Circuit Theory , 7 , 368 – 381 .
  • POINCARE , H. , 1892, Les Methodes Nouvelles de la Mecanique Celeste , Vol. 1 ( Paris Gauthier-Villars ).
  • POINCARE , H. , 1928 , Oeuvres , Vol. 1 ( Paris Gauthier-Villars ).
  • RAYLEIGH , LORD , 1929 , Theory of Sound , Vol. 1 ( London Macmillan ).
  • STUMPERS , F. L. H. , 1960 , Balth. Van der Pol's work on nonlinear circuits . I.R.E. Trans. Circuit Theory , 7 , 366 – 367 .
  • VAN DER POL , B. , 1926 , On relaxation oscillations . Phil. Mag. , 2 , 978 – 992 .
  • VAN DER POL , B. , 1927 , Forced oscillations in a circuit with nonlinear resistance (Reception with reactive triode) . Phil. Mag. , 3 , 65 – 80 .
  • ATHERTON , D. P. , 1961 , Stability of a feedback system containing a limited field-of-view error detector . Proc. Instn elect. Engrs , Part C , 108 , 265 – 270 .
  • BICKART , T. A. , 1965 , An extension of Pell's method of phase trajectory construction . I.E.E.E. Trans, autom. Control , 10 , 218 – 219 .
  • CHEN , W. H. , 1965 , A method for studying the behaviour of nonlinear systems near and at infinity of the phase plane . Proc. 3rd Annual Allerton Conf. Circuit and System Theory , University of Illinois , pp. 546 – 562 .
  • EL-GELDAWI , F. , 1970 , On the phase plane analysis of relay control system with dead time . Int. J. Control , 11 , 123 – 131 .
  • FREEMAN , E. A. , 1959 , The effect of speed dependent friction and backlash on the stability of automatic control systems . Trans. Am. Inst, elect. Engrs , Part 2 , 78 , 680 – 691 .
  • FREEMAN , E. A. , 1960 , The stabilization of control systems with backlash using a high frequency on-off loop . Proc. Instn elect. Engrs , Part C , 107 , 150 – 157 .
  • KALMAN , R. E. , 1954 , Phase-plane analysis of automatic control systems with nonlinear gain elements . Trans. Am. Inst, elect. Engrs , Part 2 , 73 , 383 – 390 .
  • PELL , W. H. , 1957 , Graphical solution of single degree of freedom vibration problem with arbitrary damping and restoring forces . Trans. Am. Soc. mech. Engrs , 79 , 311 – 312 .
  • TUSTIN , A. , 1947 , The effects of backlash and of speed dependent friction on the stability of closed-cycle control systems . J. Instn elect. Engrs , Part 2 , 94 , 143 – 151 .
  • VOJTASEK , S. , 1971 , Oscillation systems in the phase plane with piecewise linear characteristics . Elektrotech. Gcus. , 22 , 576 – 590 .
  • ABD-ALI , A. , FRADELLOS , G. , and EVANS , F. J. , 1975 , Structural aspects of stability in non-linear systems-I and II . Int. J. Control , 22 , 481 – 491 , 493 – 516 .
  • BARKHAM , P. G. D. ,SOUDACK, A. C, 1969 , An extension to the method of Kryloff and Bogoliubov . Int. J. Control , 10 , 377 – 392 .
  • BARKHAM , P. G. D. ,SOUDACK, A. C, 1970 , Approximate solutions of nonlinear non-autonomous second-order differentia equations . Int. J. Control , 11 , 101 – 114 .
  • BARKHAM , P. G. D. , and SOUDACK , A. C. , 1974 , Approximate solutions of non-autonomous second-order differential equations with saturating non-linearities . Int. J. Control , 19 , 941 – 946 .
  • BARKHAM , P. G. D. ,SOUDACK, A. C, 1978 , Limit-cycle phenomena in non-linear time delay systems . Int. J. Control , 27 , 407 – 420 .
  • BELEVITCH , V. , 1954 , Perturbations dans les systemes non-lineaires filtres- applications a la th6orie des oscillateurs . Revue H.F. , 2 ( 12 ).
  • DENN , M. M. , and BLACK , J. R. , 1973 , Estimation of nonlinear oscillations by an energy method . Chem. Engng Sci. , 28 , 1515 – 1518 .
  • HALE , J. K. , 1960 , On the method of averaging . I.R.E. Trans. Circuit Theory , 7 , 517 – 519 .
  • HEWIT , J. ,STOREY, C, 1967 , Computer program for obtaining regions of practical stability for second-order autonomous systems . Proc. Instn elect. Engrs , 114 , 1347 – 1350 .
  • Hsu , C. S. , 1975 , Limit cycle oscillations of parametrically excited second-order nonlinear systems . J. appl. Mech. , 42 , 176 – 182 .
  • JUNKINS , J. L. , JACOBSON , I. D. , and BLANTON , J. N. , 1973 , A nonlinear oscillator analog of rigid body dynamics . Celest. Mech. , 7 , 398 – 407 .
  • LEATHRUM , J. F. , JOHNSON , E. F. , and LAPIDUS , L. , 1964 , A new approach to the stability and control of nonlinear processes . A.I.Ch.E. Jl , 10 , 16 – 25 .
  • Luus , R. , and LAPIDUS , L. , 1966 , An averaging technique for stability analysis . Chem. Engng Sci. , 21 , 159 – 181 .
  • MULHOLLAND , R. J. , and HONNELL , P. M. , 1974 , Bounds for a two-parameter nonlinear oscillator . I.E.E.E. Trans. Circuits Syst. , 21 , 96 – 99 .
  • NAYFEH , H. H. , 1968 , Forced oscillations of the Van der Pol oscillator with delayed amplitude limiting . I.E.E.E. Trans. Circuit Theory , 15 , 192 – 200 .
  • POURHIET , A. , LE, and PAQUET , J. G. , 1971 , Jump phenomena in a Van der Pol oscillator . Automatica , 7 , 481 – 487 .
  • PRUSSING , J. E. ,WELEY, J. C, 1976 , An iterative procedure for determining limit cycles using Lagrangian mechanics . AIA Aerospace J. , 14 ( 3 ), 320 – 326 .
  • RAO , G. P. , 1970 , Stability analysis of periodically varying second-order systems . Int. J. Control , 12 , 953 – 960 .
  • SHAKHTARIN , B. I. , 1970 , Estimation of the stability boundaries of system for automatic control of the phase of an oscillator output . Aulomn remote Control , 31 , 1515 – 1519 .
  • SIDOROWICZ , R. S. , 1975 , General nonlinear equation for LMCR oscillators and its nearly sinusoidal solutions . Int. J. Circuit Theory Applic. , 3 , 135 – 148 .
  • SUZUKI , Y. , and IMAI , S. , 1973 , A new method of analyzing the stability and periodic oscillations of second-order nonlinear systems . Electron. Commun. Japan , 56 , 10 – 17 .
  • URABE , M. , 1960 , Periodic solutions of Van der Pol's equation with damping coefficient A = 0~10 . I.R.E. Trans. Circuit Theory , 7 , 382 – 386 .
  • AIZERMAN , M. A. , and GANTMACHER , F. R. , 1964 , Absolute Stability of Regulator Systems ( San Francisco Holden-Day ).
  • ALLWRIQHT , D. J. , 1977 , A global stability criterion for simple control loops . J. math. Biol. , 4 , 363 – 373 .
  • ANDERSON , B. D. O. , 1966 , Stability of control systems with multiple non-linearities . J. Franklin Inst. , 282 , 155 – 160 .
  • ANDERSON , B. D. O. , and MOORE , J. B. , 1968 , Algebraic structure of generalized positive real matrices . SI AM J. Control , 6 ( 4 ), 615 – 624 .
  • ANTOSIEWICZ , H. A. , 1958 , A survey of Liapunov's second method . Contributions to Nonlinear Oscillations , Vol. IV , edited by S. Lefschetz ( Princeton University Press ), pp. 147 – 166 .
  • BAKER , R. A. , and DESOER , C. A. , 1968 , Asymptotic stability in the large of a class of single-loop feedback systems . SI AM J. Control , 6 ( 1 ), 1 – 8 .
  • BERGEN , A. R. , IWENS , R. P. , and RAULT , A. J. , 1966 , On input-output stability of nonlinear feedback systems . I.E.E.E. Trans, autom. Control , 11 , 742 – 744 .
  • BERGEN , A. R. , and SAPERO , M. A. , The parabola test for absolute stability. I.E.E.E. Trans, autom. Control , 12 ( 3 ), 312 – 313 .
  • BOGUSZ , W. , GIERGIEL , J. , KASPRZYK , S. , and ZABAWA , M. , 1973 , Stability problems in lumped systems . Zagad. Drgan Nieliniowych , 12 , 9 – 52 .
  • BOLOTIN , V. V. , 1960 , The Dynamic Stability oj Elastic Systems , English translation ( San Francisco Holden-Day ).
  • BORNE , P. , and GENTINA , J. C. , 1974 , On the stability of large nonlinear systems . Proc. JACC , pp. 70 – 74 .
  • BROCKETT , R. W. , and WHXEMS , J. L. , 1965 , Frequency domain stability criteria . I.E.E.E. Trans, autom. Control , 10 , Pt. I, No. 3, Pt. II, No. 4 .
  • BROCKETT , R. W. , 1966 , The status of stability theory for deterministic systems (a survey paper) . I.E.E.E. Trans, autom. Control , 11 , 596 – 607 .
  • BROCKETT , R. W. , and LEE , H. B. , 1967 , Frequency domain instability criteria for time-varying and nonlinear systems . Proc. Inst, elect, electron. Engrs , 55 , 604 – 619 .
  • CHO , Y. S. , and NARENDRA , K. S. , 1968 , An off-axis circle criterion for the stability of feedback systems with a monotonic nonlinearity . I.E.E.E. Trans, autom. Control , 13 ( 4 ), 413 – 416 .
  • DESOER , C. A. , and VIDYASAGAR , M. , 1975 , Feedback Systems Inputs-Output Properties ( New York Academic Press ).
  • DEWEY , A. G. , 1966 , On the stability of feedback systems with one differentiable nonlinear element . I.E.E.E. Trans, autom. Control , 11 ( 3 ), 485 – 491 .
  • DEWEY , A. G. ,JURY, E. L, 1966 , A stability inequality for a class of nonlinear feedback systems . I.E.E.E. Trans, autom. Control , 11 , 54 – 62 .
  • FALB , P. L. , FREEDMAN , M. I. , and ZAMES , G. , 1969 , Input-output stability- a general viewpoint . Proc. 4th IF AC Congress , Warszawa, Session 41, pp. 3 – 15 .
  • FREEMAN , E. A. . , 1973 , Some control system stability and optimality results obtained via functional analysis . Recent Mathematical Developments in Control , edited by D. J. Bell ( London Academic Press ), pp. 45 – 68 .
  • FREEMAN , E. A. , 1973 , Stability of linear constant multivariate systems- contraction mapping approach . Proc. Instn elect. Engrs , 120 , 379 – 384 .
  • FULLER , A. T. , 1969 , In the large stability of relay and saturating control systems with linear controllers . Int. J. Control , 10 , 457 – 480 .
  • GAKG , D. P. , 1974 , A comparative study of absolute stability results for nonlinear systems . Proc. JACC , pp. 194 – 202 .
  • HADDAD , E. K. , 1972 , New criteria for bounded-input bounded-output and asymptotic stability of nonlinear systems . Proc. 5th IF AC Congress , Paris , Pt. IV, Paper 32.2 .
  • Ho , C. P. , 1974 , A frequency domain stability criterion for nonlinear feedback systems . Radio electron. Engng , 44 , 157 – 165 .
  • HOLTZMAN , J. M. , 1968 , A local bounded-input bounded-output condition for nonlinear feedback systems, I.E.E.E. Trans, autom. Control , 13 , 585 – 587 .
  • HOLTZMAN , J. M. , 1967 , Contraction maps and equivalent linearization . Bell Syst. tech. J. , 46 , 2405 – 2435 . ¦
  • HOLTZMAN , J. M. , 1970 , Nonlinear System. Theory-A Functional Analysis Approach ( Englewood Cliffs , N.J. Prentice-Hall ).
  • IBRAHIM , E. S. , and REKASIUS , Z. V. , 1964 , A stability criterion for nonlinear feedback systems . I.E.E.E. Trans, autom. Control , 9 ( 2 ), 154 – 159 .
  • JOSSELSON , R. , and RAJU , G. V. S. , 1974 , Absolute stability of control systems with many sector and slope-restricted non-linearities . Int. J. Control , 19 , 609 – 614 .
  • JURY , E. I. , and LEE , B. W. , 1965 , The absolute stability of systems with many nonlinearities . Automn remote Control , 26 , 943 – 961 .
  • KALMAN , R. E. , and BERTRAM , J. E. , 1960 , Control system design via the second method of Liapunov-Part I Continuous-time systems ; Part II Discrete time systems . J. bos. Engng , 82 , 371 – 400 .
  • KRASOVSKH , N. N. , 1963 , Stability of Motion ( Stanford University Press ).
  • LASALLE , J. P. , and LEFSCHETZ , S. , 1961 , Stability by Liapunov's Direct Method with Applications ( New York Academic Press ).
  • LEFEVRE , C. , and RICHALET , J. , 1968 , Energy stability criteria . Automn remote Control , 28 , 531 – 539 .
  • LEFSCHETZ , S. , 1965 , Stability of Nonlinear Control Systems ( New York Academic Press ).
  • LETOV , A. M. , 1961 , Stability in Nonlinear Control Systems ( Princeton University Press ).
  • LIAPUNOV , A. M. , 1966 , Stability of Motion ( New York Academic Press ).
  • LURE , A. I. , 1967 , Some Nonlinear Problems in the Theory of Automatic Control ( London H.M.S.O. ).
  • MISHCHENKO , E. F. , and PONTRYAGIN , L. S. , 1960 , Differential equations with a small parameter attached to the higher derivatives and some problems in the theory of oscillation . I.R.E. Trans. Circuit Theory , 7 , 527 – 535 .
  • MULLER , P. C. , 1975 , Some new contributions to Liapunov's stability theory . 7th Int. Conf. Nonlinear Oscillations , Berlin .
  • NARENDRA , K. S. , and GOLDWYN , R. M. , 1964 , A geometrical criterion for the stability of certain nonlinear nonautonomous systems . I.E.E.E. Trans. Circuit Theory , 11 ( 3 ), 406 – 408 .
  • NARENDRA , K. S. , and NEUMAN , C. P. , 1966 , Stability of a class of differential equations with a single monotonic nonlinearity . SI AM J. Control , 4 ( 2 ), 295 – 308 .
  • NARENDRA , K. S. , and CHO , Y. S. , 1967 , Stability of feedback systems containing a single odd monotonic nonlinearity . I.E.E.E. Trans, autom. Control , 12 ( 4 ), 448 – 450 .
  • NARENDRA , K. S. , and TAYLOR , J. H. , 1973 , Frequency Domain Criteria for Absolute Stability ( New York Academic Press ).
  • NOLDUS , E. J. , 1973 , Instability of feedback systems containing several time-varying nonlinear amplifiers . Proc. JACC , pp. 870 – 875 .
  • NOLDUS , E. , GALLE , A. , and JOSSON , L. , 1973 , The computation of stability regions for systems with many singular points . Int. J. Control , 17 , 641 – 652 .
  • NOLDUS , E. , 1973 , Criterion for unbounded motion by positive operator methods . Int. J. Control , 18 , 289 – 296 .
  • O'SHEA , R. P. , 1966 , A combined frequency-time domain stability criterion for autonomous linear systems . I.E.E.E. Trans, autom. Control , 11 ( 3 ), 477 – 484 .
  • O'SHEA , R. P. , 1967 , An improved frequency time domain stability criterion for autonomous continuous systems . I.E.E.E. Trans, autom. Control , 12 ( 6 ), 725 – 731 .
  • POPOV , V. M. , 1962 , Absolute stability of nonlinear control systems of automatic control . Automn. remote Control , 22 , 857 – 858 .
  • RAJU , G. V. S. , and JOSSELSON , R. , 1971 , Stability of reactor control systems in coupled core reactors . I.E.E.E. Trans, nucl. Sci. , 18 , 388 – 394 .
  • SANDBERO , I. W. , 1964 , On the boundedness of solutions of nonlinear functional equations . Bell Syst. tech. J. , 43 , 1581 – 1599 .
  • SANDBERO , I. W. , 1965 , Some results on the theory of physical systems governed by nonlinear functional equations . Bell Syst. tech. J. , 44 , 871 – 898 .
  • SANDBERG , I. W. , 1964 , A frequency domain condition for the stability of feedback systems containing a single time-varying nonlinear element . Bell Syst. tech. J. , 43 ( 4 ), 1601 – 1608 .
  • SANDBERO , I. W. , 1966 , On generalizations and extensions of the Popov criterion . I.E.E.E. Trans. Circuit Theory , 13 ( 1 ).
  • SANDBERO , I. W. , 1965 , A note on the application of the contraction-mapping fixed-point theorem to a class of nonlinear functional equations . SIAM Rev. , 7 , 199 – 204 .
  • SILJAK , D. D. , 1968 , On absolute stability and sensitivity . Proc. 2nd IF AC Symp. System Sensitivity and Adaptivity , Dubrovnik , Yugoslavia .
  • SILJAK , D. D. , 1972 , Stability of large-scale systems . Proc. SthlFAC Congress , Paris , Paper No. C-32 .
  • SRINATH , M. D. , and THATHACHAR , M. A. L. , 1967 , A stability criterion for systems with a single slope-restricted monotone nonlinearity . Proc. Asilomar Conf, Systems and Circuits .
  • SRINATH , M. D. , THATHACHAR , M. A. L. , and RAMAPRIYAN , H. K. , 1968 , Absolute stability of systems with multiple nonlinearities . Int. J. Control , 7 ( 4 ), 365 – 375 .
  • TAKEDA , S. , 1973 , On a frequency condition for local stability of nonlinear feedback systems . Proc. JACC , pp. 876 – 879 .
  • THATHACHAR , M. A. L. , SRINATH , M. D. , and RAMAPRIYAN , H. K. , 1967 , On a modified Lur'e problem . I.E.E.E. Trans, autom. Control , 12 ( 6 ), 731 – 740 .
  • THATHACHAR , M. A. L. , and SRINATH , M. D. , 1970 , An improved stability criterion for a system with a nonmonotonic nonlinearity . Int. J. Control , 12 ( 1 ), 145 – 155 .
  • TSAY , Y. T. , and HAN , K. W. , 1975 , Applications of the stability-equation method to linear systems . J. Franklin Inst. , 300 ( 4 ), 391 – 418 .
  • URONEN , P. , and JUTILA , E. A. A. , 1972 , Stability via the theorem of Gersh- gorin . Int. J. Control , 16 , 1057 – 1061 .
  • WILLEMS , J. L. , 1964 , On the stability of systems containing a single non-linearity . M.Sc. Thesis , M.I.T. , Cambridge .
  • WILLEMS , J. L. , 1969 , Stability, instability, invertibility and casuality . SIAM J. Control , 7 , 645 – 671 .
  • WILLEMS , J. L. , 1970 , Stability Theory of Dynamical Systems ( London Nelson ).
  • WILLEMS , J. L. , 1970 , The Analysis of Feedback Systems , Research Mon. No. 62 ( Cambridge , Mass. M.I.T. Press ).
  • YAKUBOVICH , V. A. , 1970 , Absolute stability of nonlinear control systems, Part I . Avtomatika Telemehk. , 31 ( 12 ) ; 1971, Part II. Ibid., 32 (6) .
  • ZAMES , G. , 1963 , Functional analysis applied to nonlinear feedback systems . I.E.E.E. Trans. Circuit Theory , 10 ( 3 ), 392 – 404 .
  • ZAMES , G. , 1966 , On the input-output stability of time-varying nonlinear feedback systems, Part I . I.E.E.E. Trans, aulom. Control , 11 ( 2 ) ; 1966, Part II. tIbid., 11 (3) .
  • ZAMES , G. , and SHNEYDOE , N. A. , 1977 , Structural stabilization and quenching by dither in nonlinear systems . I.E.E.E. Trans, autom. Control , 22 , 352 – 361 .
  • ZAMES , G. and FALB , P. L. , 1968 , Stability conditions for systems with monotone and slope-restricted nonlinearities . BI AM J. Control , 6 , 89 – 108 .
  • ZAMES , G. and FALB , P. L. , 1967 , On the stability of systems with monotonic and odd monotonic nonlinearities . I.E.E.E. Trans, aulom. Control , 12 ( 2 ), 221 – 223 .
  • BORGES , C. A. , CESARI , L. , and SANCHEZ , D. A. , 1975 , Functional analysis and the method of harmonic balance . Q. appl. Math. , 32 , 457 – 464 .
  • BYRNE , P. C. , and SHEERAN , M. , 1973 , Comment on geometric criterion for the absence of limit cycles in nonlinear control systems . Electronics Lett. , 9 , 576 .
  • CESARI , L. , 1962 , Functional analysis and periodic solutions of nonlinear differential equations . Contributions to Differential Equations , Vol. I ( New York Interscience ).
  • GARBER , E. D. , 1967 , Frequency criteria for the absence of periodic modes . Automn remote Control , 28 , 1776 – 1780 .
  • GREGORY , J. , 1974 , Oscillation theory for second-order integral differential equations . J. math. Analysis Applic , 47 , 69 – 77 .
  • GUSTAFSON , G. B. , 1974 , Bounded oscillations of linear and nonlinear delay-differential equations of even order . J. math. Analysis Applic. , 46 , 175 – 189 .
  • IWENS , R. , 1968 , Bounds on the responses of nonlinear control systems . J. Franklin Inst. , 285 , 261 – 274 .
  • KRASNOSELSKTI , M. A. , and LIFSCHITS , E. A. , 1973 , Some criteria for the existence of periodic oscillations in nonlinear systems . Automn remote Control , 34 , 1374 – 1377 .
  • NOLDUS , E. J. , 1969 , A frequency domain approach to the problem of the existence of periodic motion in autonomous nonlinear feedback systems . Z. angew. Math. Mech. , 3 , 167 – 177 .
  • NOLDUS , E. J. , 1971 , A counterpart of Popov's theorem for the existence of periodic solutions . Int. J. Control , 13 , 705 – 719 .
  • NOLDUS , E. J. , 1974 , Autonomous periodic motion in nonlinear feedback systems . I.E.E.E. Trans, autom. Control , 19 , 381 – 387 .
  • NOLDUS , E. J. , 1975 , Computation of a lower bound of nonlinear oscillation amplitudes . Automatica , 11 , 431 – 435 .
  • NOLDUS , E. J. , 1975 , Oscillation criteria of the Popov type . I.E.E.E. Trans. autom. Control , 20 , 577 – 579 .
  • OHTA , Y. , HANEDA , H. , and MARUHASHI , T. , 1975 , Periodic behavior of nonlinear-periodic differential systems . Electron. Commun. Japan , 58 , 21 – 29 .
  • PLISS , V. A. , 1966 , Nonlocal Problems of the Theory of Oscillations ( New York Academic Press ).
  • PVAUCH , L. L. , 1960 , Oscillation of a third order nonlinear autonomous system . Contributions to the Theory of Nonlinear Oscillations , Vol. I , edited by S. Lefschetz ( Princeton University Press ).
  • ROOTENBERG , J. , and WALK , R. , 1972 , Geometric criterion for the absence of limit cycles in nonlinear control systems . Electronics Lett. , 8 , 497 – 499 .
  • ROOTENBERG , J. , and WALK , R. , 1973 , Frequency criterion for the absence of limit cycles in nonlinear systems . I.E.E.E. Trans, autom. Control , 18 , 64 – 65 .
  • SANDBERG , I. W. , 1964 , On the response of nonlinear control systems to periodic input signals . Bell Syst. tech. J. , 43 , 911 – 926 .
  • SEIFERT , G. , 1958 , On stability in the large for periodic solutions of differentia equations , Annln Math. , 67 , 83 – 89 .
  • WOZNY , M. J. , and SCHCLTZ , D. G. , 1966 , A generalized Zubov formulation for limit cycles in third order nonlinear systems . Proc. 4th Annual Allerton Conf. Circuit and System Theory , University of Illinois , pp. 916 – 924 .
  • ZIEMBA , S. , 1973 , Some remarks on the future of the vibrating theory . Zagad. Drgan Nieliniowych , 14 , 14 – 25 .
  • BEROEN , A. R. , and WILLEMS , I. J. , 1962 , Verification of Aizerman's conjecture for a class of third-order systems . I.R.E. Trans, autom. Control , 7 ( 3 ), 42 – 46 .
  • BERGEN , A. R. and BAKER , R. A. , 1972 , On third-order systems and Aizerman's conjecture . I.E.E.E. Trans, autom. Control , 17 , 220 – 222 .
  • COOK , P. A. , 1973 , Describing function for a sector nonlinearity . Proc. Instn elect. Engrs , 120 , 143 – 144 .
  • DEWEY , A. G. , and JURY , E. I. , 1965 , A note on Aizerman's conjecture . I.E.E.E. Trans, autom. Control , 10 ( 4 ), 482 – 483 .
  • FANNIN , D. R. , and RUSHING , A. J. , 1974 , Verification of the Kalman conjecture based en locus curvature . Proc. Inst, elect, electron. Engrs , 62 , 542 – 543 .
  • FITTS , R. E. , 1966 , Linearization of nonlinear feedback systems . Ph.D. Dissertation , M.I.T. , Cambridge .
  • FITTS , R. E. , 1966 , Two counterexamples to Aizerman's conjecture|at. I.E.E.E. Trans, autom. Control , 11 , 553 – 556 .
  • FUJII , K. , and SHOJI , K. , 1972 , Verification of the Aizerman and/or Kalman conjecture . I.E.E.E. Trans, autom. Control , 17 , 406 – 408 .
  • JACUSIEL , P. , and KING , R. E. , 1971 , On the sector of stability in Aizerman's method . Int. J. Control , 13 , 77 – 81 .
  • LIGHTHILL , M. J. , and MEES , A. I. , 1973 , Stability of nonlinear feedback systems . Recent Mathematical Developments in Control , edited by D. J. Bell ( London Academic Press ), pp. 1 – 20 .
  • LOTTERIO , L. , and MAFFEZZONI , C. 1973 , A limit cycle stability criterion for Lure-type nonlinear systems . Int. J. Control , 17 , 849 – 853 .
  • MUKERJEE . M. R. , 1975 , A note on Aizerman's conjecture for third-order systems , I.E.E.E.- Trans, autom. Control , 20 , 552 – 553 .
  • PLISS , V. A. , 1965 , Certain Problems in the Theory of Stability of Motion in the Whole , English translation, NASA-TT-F-280 , Washington , D.C .
  • RAO , U. M. , 1975 , Some aspects of the stability of nonlinear control systems . M.Sc.E. Thesis , University of New Brunswick , Fredericton .
  • SHIRAKAWA , H. , 1968 , Stability of a certain class of third-order nonlinear differential equation . I.E.E.E. Trans, autom. Control , 13 , 202 – 203 .
  • SINGH , V. , 1975 , A counterexample to harmonic linearization , Proc. Inst, elect. electron. Engrs , 63 , 1610 .
  • VOQT , W. G. , and GEORGE , J. H. , 1967 , On Aizerman's conjecture and bounded-ness . I.E.E.E. Trans, autom. Control , 12 , 338 – 339 .
  • VORONOV , A. A. , 1978 , Systems with differentiable non-decreasing non-linearity absolutely stable in the Hurwitz angle . Proc. 7th IF AC Congress , Helsinki , pp. 1731 – 1736 .
  • ALLWRIGHT , D. J. , 1977 , Harmonic balance and the Hopf bifurcation . Proc. Camb. phil. Soc. math. phys. Sci. , 82 , 453 – 467 .
  • BAILEY , A. R. , 1966 , Stabilisation of control systems by the use of driven limiters . Proc. Instn elect. Engrs , 113 , 169 – 174 .
  • BASS , R. W. , 1961 , Mathematical legitimacy of equivalent linearization by describing functions . Proc. 1st IF AC Congress , London , pp. 895 – 905 .
  • BERGEN , A. R. , and FRANKS , R. L. , 1970 , On periodic responses of nonlinear feedback systems . I.E.E.E. Trans, autom. Control , 15 , 146 – 147 .
  • BERGEN , A. R. , and FRANKS , R. L. , 1971 , Justification of the describing function method . SI AM J. Control , 9 , 568 – 589 .
  • CHAN , W. C. , and DORRAH , H. T. , 1973 , Describing function for servomechanism with backlash and multiple non-linearities . Int. J. Control , 17 , 1251 – 1264 .
  • CHERNIKOV , S. A. , 1974 , Extension of the method of harmonic linearization to nonlinear systems with nonfiltering linear parts . Engng Cybernet. , 12 , 138 – 147 .
  • CHERNOV , V. G. , 1969 , Improving system stability by use of controlled limiters . Engng Cybernet. , 7 , 152 – 158 .
  • DUDKIKAR , K. L. , 1974 , Sustained oscillations for a certain class of nth order system . Proc. 12th Annual Allerton Conf. Circuit and System Theory , University of Illinois , pp. 490 – 494 .
  • DUTILH , J. 1950 , Theoriedesservomechanismsarelais . Ondeelect. , 30 , 438 – 445 .
  • GARG , D. P. , and GEISLER , E. G. , 1973 , Limit cycle analysis for control systems of high order . Int. J. Control , 17 , 1225 – 1232 .
  • GELB , A. , 1961 , The analysis and design of limit cycling adaptive automatic control systems . Sc.D. Dissertation , M.I.T. , Cambridge .
  • GELB , A. ,VANDER VELDE , W. E. , 1963 , On limit cycling control systems . I.E.E.E. Trans, autom. Control , 8 , 142 – 157 .
  • GEORGE , J. H. , GUNDERSON , R. W. , and HAHN , H. , 1975 , Sustained small oscillations in nonlinear control systems . AIA Aerospace J'. , 13 , 1251 – 1253 .
  • GIBSON , J. E. , and PRASANNA-KUMAR , K. S. , 1961 , A new RMS describing function for single-valued nonlinearities . Proc. Inst. Radio Engrs , 49 , 1321 .
  • GOLDFARB , L. C , 1947 , On some nonlinear phenomenon in regulatory system . Avtomatika Telemekh. , 8 , 349 – 383 .
  • GOLDFARB , L. C. , 1956 , On some nonlinear phenomena in regulatory systems . Frequency Response , edited by R. Oldenburger ( New York Macmillan ), pp. 239 – 257 .
  • GRAY , J. O. , and TAYLOR , P. M. , 1974 , Computer aided design using describing function methods . I.E.E. Conj. Publ. , London , No. Ill , pp. 92 – 98 .
  • GRENSTED , P. E. , 1955 , The frequency response analysis of nonlinear systems . Proc. Instn elect. Engrs , Part C , 102 , 244 – 253 .
  • GRUMONDZ , V. T. , 1974 , Justification of and an estimate of the accuracy of a method for investigating nonlinear control systems . Automn remote Control , 35 , 1211 – 1218 .
  • HARRINGTON , H. J. , 1959 , Limit-cycle stability study of a feedback control system by a new describing-function technique . Trans. Am. Inst, elect. Engrs , Part 2 , 79 , 129 – 134 .
  • HOLISTER , F. H. , 1965 , Network analysis and design by parameter plane techniques . Ph.D. Dissertation , U.S. Naval Postgraduate School , Monterey , California .
  • JOHNSON , E. C , 1952 , Sinusoidal analysis of feedback control systems containing nonlinear elements . Trans. Am. Inst, elect. Engrs , Part 2 , 71 , 169 – 181 .
  • JONES , N. B. , 1967 , Simultaneous oscillations at unrelated frequencies in a feedback loop containing only one nonlinearity . Int. J. Control , 5 , 331 – 343 .
  • KOGHENBURGER , R. J. , 1949 , Analysis and synthesis of contactor servo-mechanisms . D.Sc. Dissertation , M.I.T. , Cambridge .
  • KOCHENBURGER , R. J. , 1950 , A frequency response method for analysing contactor servomechanisms . Trans. Am. Inst, elect. Engrs , Part 2 , 69 , 270 – 284 .
  • KODAMA , S. , and SHIRAKAWA , H. , 1968 , Stability of nonlinear feedback systems with backlash . I.E.E.E. Trans, autom. Control , 13 , 392 – 399 .
  • Kou. S. Y. , and HAN , K. W. , 1975 , Limitations of the describing function method . I.E.E.E. Trans, autom. Control , 20 , 291 – 296 .
  • KUDREWICZ , J. , 1967 , Contribution to the theory of describing function . 4th Int. Conf. Nonlinear Oscillations , Prague .
  • KUDREWICZ , J. , 1969 , Theorems on the existence of periodic vibrations based upon the describing function method . Proc. 4th IF AC Congress , Warszawa , Session 41 , pp. 46 – 60 .
  • KUDREWICZ , J. , 1969 , Mathematical basis of describing function method . Rozpr. eleklrotech. , 15 , 587 – 624 .
  • KUDREWICZ , J. , 1973 , The describing function method and free vibrations in nonlinear systems . Int. J. Circuit Theory Applic , 1 , 49 – 57 .
  • MAHALANABIS , A. K. , 1961 , On stabilisation of feedback systems affected by hysteresis nonlinearities . Trans. Am. Inst, elect. Engrs , Part 2 , 80 , 277 – 285 .
  • MEES , A. I. , and BERGEN , A. R. , 1975 , Describing function revisited . I.E.E.E. Trans, autom. Control , 20 , 473 – 78 .
  • MODINA , E. S. , OBABKOV , V. K. , and KHARCHENKO , V. P. , 1975 , Self-oscillating processes in nonlinear automatic control systems . Automn remote Control , 36 , 17 – 23 .
  • OPPELT , W. , 1948 , Locus curve method for regulators with friction . Z. dt. Ingr., Berlin , 90 , 179 – 183 .
  • POPOV , E. P. , 1956 , An approximate investigation of transient processes in nonlinear automatic systems by the method of harmonic linearization . Izv. Alcad. Naulc SSSR , 9 , 3 – 23 .
  • POPOV , E. P. , and PALITOV , I. P. , 1960 , Approximate Methods for Analysis of Nonlinear Automatic Systems ( Moscow State Press for Physics and Mathematical Literature ) (English translation Foreign Technical Division, AFSC, Wright-Patterson AFB, Ohio, Report FTD-TT-62–910) .
  • POPOV , E. P. , and KLYPALO , E. I. , 1969 , The development of harmonic linearization methods (in Russian) . Proc. 4th IF AC Congress , Warszawa , Session 41 , pp. 34 – 45 .
  • RAO , U. M. , and ATHERTON , D. P. , 1977 , Stability analysis of a structurally flexible spacecraft . 20th Midwest Symp. Circuits and Systems , Lubbock , Texas , pp. 652 – 656 .
  • RAPP , P. E. , and MEES , A. I. , 1977 , Spurious predictions of limit cycles in a non-linear feedback system by the describing function method . Int. J. Control , 26 , 821 – 829 .
  • SATYENDRA , K. N. , 1956 , Describing function representing the effect of inertia, backlash and Coulomb friction on the stability of an automatic control system . Trans. Am. Inst, elect. Engrs , Part 2 , 75 , 243 – 249 .
  • SEBASTIAN , L. , 1973 , Method for the determination of self-sustained oscillations for an automatic nonlinear system with a single nonlinearity . Automatisme , 18 , 133 – 137 .
  • SHOOMAN , M. L. , 1964 , Stability analysis of nonlinear systems in the parameter plane . I.E.E.E. Trans, autom. Control , 9 , 267 – 269 .
  • SHULKIND , D. , 1960 , Accuracy requirements for nonlinear compensation of backlash . I.R.E. Trans, autom. Control , 5 , 79 – 84 .
  • SIIJAK , D. D. , 1966 , Generalization of the parameter plane method . I.E.E.E. Trans, autom. Control , 11 , 63 – 70 .
  • SNELSIRE , R. ,CLEMSON, S. C, 1967 , On the use of the describing function as a stability criterion . I.E.E.E. Trans, autom. Control , 12 , 788 – 789 .
  • TAFT , V. A. , 1958 , Stability of periodic conditions in automatic control systems found approximately on the basis of a filter hypothesis . Automn remote Control , 19 , 550 – 555 .
  • TSAY , Y. T. , 1974 , Analytical determination of limit cycle for a class of third-order nonlinear system . Int. J. Control , 19 , 1143 – 1154 .
  • WILLEMS , J. C , 1968 , Nonlinear harmonic analysis . Technical Memo. No. ESL-TM-357 , Electronic Systems Laboratory, M.I.T. , Cambridge .
  • ATHERTON , D. P. , 1974 , The stability of oscillations . Autom. Control Theory Applic , 2 , 47 – 49 .
  • ATHERTON , D. P. , 1975 , Analytical determination of limit cycles for a class of third-order non-linear systems . Int. J. Control , 21 , 1021 – 1023 .
  • BARANYI , A. , and HENK , T. , 1975 , On the stability of sinusoidal oscillations . 7th Int. Conf. Nonlinear Oscillations , Berlin .
  • BONENN , Z. , 1960 , Connection between various methods of investigating absolute stability . I.R.E. Trans, autom. Control , 5 , 330 – 331 .
  • BONENN , Z. , 1963 , Relative stability of oscillations in non-linear control systems . Proc. 2nd IF AC Congress , London , 1 , 35 – 40 .
  • CHOKSY , N. H. , 1963 , A new analytic stability criterion for use with describing functions . Proc. JACC , pp. 481 – 184 .
  • CHOUDHURY , S. K. , 1974 , Stability of oscillations in nonlinear feedback systems . M.Sc.E. Thesis , University of New Brunswick , Fredericton .
  • CHOUDHURY , S. K. , and ATHERTON , D. P. , 1974 , Limit cycles in high-order nonlinear systems . Proc. Instn elect. Engrs , 121 , 717 – 724 .
  • DATTA , K. B. , 1972 , Limit cycle stability of third and higher-order systems . Int. J. Control , 15 , 889 – 895 .
  • DEMCHENKO , V. V. , and EL-SIRAGY , N. M. , 1973 , On stability of electromagnetic oscillations in media with cubic nonlinearity . Physica , 67 , 336 – 346 .
  • DORRAH , H. T. , and ATHERTON , D. P. , 1976 , Stability of oscillatory feedback systems with frequency-dependent nonlinearities . Int. J. Electron. , 41 , 137 – 146 .
  • DOUCE , J. L. , and KING , R. E. , 1959 , Instability of a nonlinear conditionally stable system subjected to a sinusoidal input . Trans. Am. Inst, elect. Engrs , 78 , 665 – 670 .
  • FILATOV , I. V. , 1972 , Application of stability criteria to the analysis of nonlinear systems . Engng Cybernet. , 10 , 351 – 357 .
  • FREEMAN , E. A. , 1968 , On the stability of nonlinear oscillations predicted by the harmonic-linearisation technique . Proc. Instn elect. Engrs , 115 , 445 – 52 .
  • GRAYSON , P. , 1965 , Stability and oscillations in systems containing one or more single-valued symmetric nonlinearities . J. Franklin Inst. , 280 , 52 – 64 .
  • JOHNSON , R. A. , and LEACH , B. W. , 1972 , Stability of oscillations in low-order nonlinear systems . I.E.E.E. Trans, autom. Control , 17 , 672 – 675 .
  • LEACH , B. W. , 1972 , An evaluation of estimates of stability of oscillations in nonlinear systems . Ph.D. Dissertation , University of Manitoba , Winnipeg .
  • LEACH , B. W. , 1973 , An evaluation of estimates of stability of oscillations in nonlinear systems . Proc. Canadian Conf. Automatic Control , University of New Brunswick , Fredericton , Paper No. 4.2 .
  • LOEB , J. M. ; 1956 , Recent advances in nonlinear servo theory . Frequency Response , edited by R. Oldenburger ( New York Macmillan ), pp. 260 – 268 .
  • MEES , A. I. , 1973 , Limit cycle stability . J. Inst. Maths. Applic , 11 , 281 – 295 .
  • MORGAN , A. T. , 1971 , A general stability criterion of the oscillations in nonlinear systems . Proc. Inst, elect, electron. Engrs , 59 , 78 – 79 .
  • SEN , A. K. , 1968 , A simpler stability criterion for a class of nonlinear systems . I.E.E.E. Trans, autom. Control , 13 , 451 – 152 .
  • TSAY , Y. T. , WANG, B. O, and HAN , K. W. , 1974 , Stability analysis of nonlinear control systems with characteristic equations having complex coefficients . J. Franklin Inst. , 297 , 179 – 200 .
  • TSAY , Y. T. , and HAN , K. W. , 1975 , A gradient approach to the determination of stability of limit cycles . J. Franklin Inst. , 300 , 391 – 418 .
  • WEISS , H. , and SIDAR , M. , 1973 , Relationships between stability test methods of nonlinear control systems . Int. J. Control , 17 , 1009 – 1019 .
  • WILLEMS , J. C , 1966 , Perturbation theory for the analysis of instability in nonlinear feedback systems . Proc. 4th Annual Allerton Conf. Circuit and System Theory , University of Illinois , pp. 836 – 848 .
  • WILLEMS , J. L. , 1968 , The stability of oscillations in nonlinear networks . I.E.E.E. Trans. Circuit Theory , 15 , 284 – 286 .
  • BASUTHAKUK , S- N. , and Joa , S. T. , 1971 , The effect of parameter variation on the periodic oscillations in a non-linear system . Int. J. Control , 13 , 793 – 799 .
  • DAVISON , E. J. , 1968 , Application of the describing function technique in a single-loop feedback system with two nonlinearities . I.E.E.E. Trans. autom. Control , 13 , 168 – 170 .
  • DAVISON , E. J. , and CONSTANTINESCU , D. , 1971 , A describing function technique for multiple nonlinearities in a single-loop feedback system . I.E.E.E. Trans, autom. Control , 16 , 56 – 60 .
  • ENGELS , D. A. , 1967 , The describing function for a constrained range integration process with bang-bang input and dead zone . I.E.E.E. Trans, autom.. Control , 12 , 582 – 585 .
  • FREEMAN , E. A. , 1962 , Stability analysis of control systems having two nonlinear elements with calculations for saturation and backlash . Proc. Instn elect. Engrs , Part C , 109 , 665 – 675 .
  • FREEMAN , E. A. ,BARNEY, G. C, 1963 , Divergent oscillations and their excitation in control systems with two saturation type nonlinear elements . Proc. Instn elect. Engrs , 110 , 1096 – 1106 .
  • GAN , R. , and RIMER , M. , 1965 , Stability analysis of systems with multiple nonlinearities . I.E.E.E. Trans, autom. Control , 10 , 94 – 97 .
  • GARO , D. P. , and ALEMAN , R. , 1975 , A limit cycle analysis of control systems with multiple non-linearities . Int. J. Control , 16 , 689 – 697 .
  • GRAY , J. O. , SAWIDES , L. S. , and TAYLOR , P. M. , 1975 , Interactive graphics design using sinuosidal describing function methods . I.S.A. Trans. , 14 , 68 – 77 .
  • HOPKINS , W. G. , 1973 , On the prediction of limit cycles in systems containing multiple nonlinearities . I.E.E.E. Southeast Conf. , Kentucky , Paper M-2 .
  • JELONEK , I. J. , POMELLA , P. L. V. , and KARUNARATNE , N. S. , 1964 , Oscillations in feedback systems subjected to periodic interference and stability of a self-optimising control system . Proc. Instn elect. Engrs , 111 , 1881 – 1893 .
  • JUD , H. G. , 1964 , Limit cycle determination for parallel linear and nonlinear elements . I.E.E.E. Trans, autom. Control , 9 , 183 – 184 .
  • LEONHARD , A. , 1963 , Extension of the describing function method to the investigation of parametric oscillations . Proc. IFAC , Basel , Switzerland , pp. 187 – 197 .
  • MEES , A. I. , 1972 , The describing function matrix . J. Inst. Math. Applic , 10 , 49 – 67 .
  • MEES , A. I. , 1973 , Oscillations in nonlinear feedback systems . Ph.D. Dissertation , Cambridge , England .
  • PELLEGRINI , C. A. , YOCKEY , H. M. , and THALER , G. J. , 1966 , System performances of control systems with two nonlinearities . Proc. 4th Annual Allerton Conf. Circuit and System Theory , University of Illinois , pp. 877 – 886 .
  • PENFIELD , P. , JR., 1966 , Circuit theory of periodically driven nonlinear systems . Proc. Inst, elect, electron. Engrs , 54 , 266 – 280 .
  • PRUSTY , S. , 1970 , A new approach for the study of limit-cycle behaviour for a class of third-order nonlinear control systems . Int. J. Control , 12 , 257 – 263 .
  • RANQACHARYULU , M. A. V. , and SRINIVASAN , P. , 1973 , A note on the period of oscillation of non-linear systems . J. Sound Vibr. , 29 , 207 – 213 .
  • SEBASTIAN , L. , and Voicu , M. , 1974 , Stability of boundary oscillations in control systems with two nonlinearities . Regelungstechnik , 22 , 317 – 319 .
  • SELTZER , S. M. , 1971 , Analysis of a control system containing two nonlinearities . Int. J. Control , 13 , 1057 – 1064 .
  • SMETANIN , N. A. , 1965 , Harmonic linearization of a component with variable structure . Automn remote Control , 26 , 998 – 1001 .
  • SOMLO , J. , 1973 , On the computer aided design of control systems . Probl. Control Inf. Theory , 2 , 41 – 65 .
  • TEODORESCU , D. , 1970 , Describing function series-A new means for nonlinear system analysis . Proc. Instn elect. Engrs , 117 , 2170 – 2180 .
  • WILLIAMSON , D. , 1973 , Oscillations in nonlinear feedback systems . Proc. JACG , pp. 752 – 758 .
  • WILLIAMSON , D. , 1975 , Periodic motion in nonlinear systems . I.E.E.E. Trans, autom. Control , 20 , 479 – 86 .
  • YEUNQ , K. S. , 1975 , Improvement of harmonic balance by structural transformations . Regelungstechnik , 23 , 312 – 317 .
  • ATHEBTON , D. P. , and CHOUDHURY , S. K. , 1975 , An exact solution for oscillations in relay systems with complex pole . Int. J. Control , 21 , 1015 – 1020 .
  • ATHERTON , D. P. , and RAMANI , N. , 1975 , A method for the evaluation of limit cycles . Inl. J. Control , 21 , 375 – 384 .
  • BERGEN , A. R. , 1962 , A note on Tsypkin's locus . I.R.E. Trans, autom. Control , 7 , 78 – 80 .
  • BOHN , E. V. , 1961 , Stability margins and steady-state oscillations in on-off feedback systems . I.R.E. Trans. Circuit Theory , 8 , 127 – 130 .
  • BROOKS , B. E. , and JOHNSON , R. A. , 1969 , Limit cycles in time-delay relay systems . Int. J. Control , 9 , 387 – 398 .
  • CHUNG , J. K.-C , and ATHEBTON , D. P. , 1966 , The determination of periodic modes in relay systems using the state space approach . Int. J. Control , 4 , 105 – 126 .
  • GILLE , J. C , and PAQUET , J. G. , 1962 , Subharmonic oscillations in on-off control systems . Trans. Am. Inst, elect. Engrs , Part 2 , 81 , 210 – 216 .
  • GILLE , J. C , PAQUET , J. G. , and LEMAITRE , J. F. , 1968 , Synchronisation des asservissements par plus-ou-moins . C. r. hebd. Seanc. Acad. Sci., Paris , 267 , 238 – 240 .
  • GILLE , J. C. , PAQUET , J. G. , and LEMAITRE , J. P. , 1968 , Oscillations libres des systemes asservis par plus-ou-moins . C. r. hebd. Seanc. Acad. Sci., Paris , 267 , 335 – 336 .
  • JUDD , F. F. , and CHIRLAIN , F. M. , 1974 , Graphical analysis and design of limit cycles in autonomous relay control systems . Int. J. Control , 20 , 321 – 334 .
  • JUDD , F. F. , 1975 , Relationships between Tsypkin, Hamel and approximate limit cycle analyses . Int. J. Control , 21 , 641 – 653 .
  • JUDD , F. F. , 1977 , Error bounds for approximate analysis of self-oscillations in autonomous relay control systems . Int. J. Control , 25 , 557 – 574 .
  • JUMABIE , G. , 1969 , Generalization of the theory of Ya. Z. Tsypkin for relay automatic control system . Automn remote Control , 12 , 1929 – 1937 .
  • LEMAITRE , J. F. , PAQUET , J. G. ,GILLE, J. C, 1970 , A general approach for on-off control system oscillations . Automatica , 6 , 609 – 613 .
  • LEMAITRE , J. F. , CHAMPENOIS , A. , PAQUET , J. G. ,GILLE, J. C, 1968 , Oscillations combinees des systemes par plous-ou-moins . Annls Telecom- mun. , 23 .
  • MICHAELS , L. H. , 1967 , The relationship between periodic solutions and stability in third-order relay control systems . Ph.D. Dissertation , Rensselaer Polytechnic Institute , Troy , New York .
  • MICHAELS , L. H. , and FREDERICK , D. K. , 1968 , The relationship between periodic solutions and stability in a class of third-order relay control systems . I.E.E.E. Trans, autom. Control , 13 , 400 – 407 .
  • MOHAMMED , A. , 1965 , Steady state oscillations and stability of on-off feedback systems . Ph.D. Dissertation , University of British Columbia , Vancouver .
  • PAQUET , J. G. , LEMAITRE , J. F. ,GILLE, J. C, 1966 , La methode de Hamel Cypkin et son application aux fonctions de transfert particulieres . Auto matisme , 11 , 3 – 5 .
  • PAQUET , J. G. , and LEMAITRE , J. F. , 1970 , Mhodes Pratiques d'tftude des Oscillations Nonlineaires ( Paris Gordon & Breach ).
  • RAO , U. M. ,ATHERTON,. D. P., 1978 , Multi-pulse oscillations in relay systems . Proc. 7th IFAC Congress , Helsinki , Vol. 3 , Paper No. 42.4 , pp. 1747 – 1754 .
  • SAKAWA , Y . , 1960, Subharmonic oscillations in relay control systems. Proc. IF AC , Moscow, pp. 404 – 409 .
  • TSYPKTN , Y. Z. , 1955 , Theory of Relay Control Systems , in Russian ( Moscow State Press for Technical and Theoretical Literature ).
  • TTJRNBULL , G. F. , ATHERTON , D. P. , and TOWNSEND , J. M. , 1965 , Method for the theoretical analysis of relay feedback systems . Proc. Instn elect. Engrs , 112 , 1039 – 1055 .
  • APRILLE , T. J. , JR., and TRICK , T. N. , 1971 , An algorithm to obtain the steady-state response of nonautonomous systems . Proc. 5th Asilomar Conf. Circuits and Systems .
  • APRILLE , T. J. , JR., and TRICK , T. N. , 1971 , Steady state analysis of nonlinear circuits that have a periodic response . Report No. R515, Coordinated Sciences Laboratory , University of Illinois .
  • APRILLE , T. J. , JR., and TRICK , T. N. , 1972 , A computer algorithm to determine the steady-state response of nonlinear oscillators . I.E.E.E. Trans. Circuit Theory , 19 , 354 – 360 .
  • APRILLE , T. J. , JR., and TRICK , T. N. , 1972 , Steady state analysis of nonlinear circuits with periodic inputs . Proc. Inst, elect, electron. Engrs , 60 , 108 – 114 .
  • AROYRIS , J. H. , DUNNE, P. C, and ANGELOPOULOS , T. , 1973 , Nonlinear oscillations using the finite element technique Compul. Meth. appl. Mech. Engng , 2 , 203 – 250 .
  • ARMANAZI , A. N. , 1974 , Determination of steady state oscillations in piecewise linear systems . I.E.E.E. Trans. Circuits Syst. , 21 , 703 – 704 .
  • BAILY , E. M. , 1968 , Steady state harmonic analysis of nonlinear networks . Ph.D. Dissertation , Stanford University .
  • BAILY , E. M. , 1969 , Steady-state harmonic analysis of nonlinear networks-a computer-oriented approach . Proc. 1st Annual Houston Conf. Circuits, Systems and Computers , Houston , pp. 390 – 399 .
  • BELYTSCHKO , T. , CHIAPETTA , R. L. , and BARTEL , H. D. , 1976 , Efficient large scale nonlinear transient analysis by finite elements . Int. J. Numer. Meth. Engng , 10 , 579 – 596 .
  • CHANDRA , J. ,DAVIS, P. W- , 1973 , On the approximation of periodic solutions of equations with bounded nonlinearities . Numer. Math. , 21 , 96 – 108 .
  • CHUA , L. O. , and GREEN , D. N. , 1973 , Synthesis of nonlinear systems having prescribed periodic solutions . Proc. I.E.E.E. Int. Symp. Circuit Theory , Toronto , pp. 352 – 354 .
  • CHUA , L. O. , and GREEN , D. N. , 1974 , Synthesis of nonlinear periodic systems . I.E.E.E. Trans. Circuits Syst. , 21 , 286 – 294 .
  • COLON , F. R. , and TRICK , T. N. , 1973 , Fast periodic steady-state analysis for large-signal electronic circuits . I.E.E.E. Jl Solid-St. Circuits , 8 , 260 – 269 .
  • DIRECTOR , S. W. , BRODERSEN , A. J. , and WAYNE , D. A. , 1971 , A method for quick determination of the periodic steady-state in nonlinear networks . Proc. 9th Annual Allerlon Conf. Circuit and System Theory , University of Illinois , pp. 131 – 139 .
  • GRIEPENTROO , E. , 1975 , Numerical treatment of oscillation problems in electrical networks . 7th Int. Conf. Nonlinear Oscillations , Berlin .
  • JONNADA , R. K. , 1969 , Limit cycles of higher order nonlinear autonomous systems . Ph.D. Dissertation , University of Pennsylvania .
  • JONNADA , R. K. , and WEYGANDT , C. N. , 1971 , A numerical method for the location of limit cycles of higher-order nonlinear autonomous systems . J. Franklin Inst. , 292 , 333 – 355 .
  • JONNADA , R. K. , and WEYGANDT , C. N. , 1971 , Limit cycles of higher order nonlinear autonomous systems . J. Franklin Inst. , 291 , 199 – 210 .
  • LAUTERBORN , W. , 1976 , Numerical investigation of nonlinear oscillations of gas bubbles in liquids . J. acoust. Soc. Am. , 59 , 283 – 293 .
  • LINDENLAUB , J. C. , 1969 , An approach for finding the sinusoidal steady state response of nonlinear systems . Proc. 7th Annual Allerton Conf. Circuit and System Theory , University of Illinois , 323 – 329 .
  • NAKHLA , M. S. , and BEANIN , F. H. , JR., 1974 , A gradient method for finding the periodic response of nonlinear systems . Proc. 12th Annual Allerton Conj. Circuits and System Theory , University of Illinois , pp. 703 – 704 .
  • NAKHLA , M. S. , 1975 , Steady state analysis of nonlinear periodic systems . Ph.D. Thesis , University of Waterloo .
  • NAKHLA , M. S. , and VLACH , J. , 1975 , Piecewise steady state harmonic analysis of nonlinear periodic systems . Proc. Conf. on Information Sciences and Systems , John Hopkins University .
  • NAKHLA , M. S. , and VLACH , J. , 1976 , Piecewise harmonic balance technique for determination of periodic response of nonlinear systems . I.E.E.E. Trans. Circuits Syst. , 23 , 85 – 91 .
  • NIBTZSCH , J. , 1975 , Analytical and numerical treatment of nonlinear oscillation problems . 7th Int. Conf. Nonlinear Oscillations , Berlin .
  • TRICK , T. N. , 1973 , Computation of the periodic response of nonlinear communication circuits . Proc. 7th Asilomar Conf. Circuits, Systems and Computers , pp. 633 – 636 .
  • BALASUBRAMANIAN , R. , and ATHERTON , D. P. , 1967 , Quasi-linear analysis of feedback systems containing a nonlinearity with output dependent upon both the input and its derivative . Proc. 5th Int. Analogue Computation Meeting , Lausanne , Session B2, No. 6 , pp. 432 – 441 .
  • BALASUBRAMANIAN , R. , and ATHERTON , D. P. , 1968 , Prediction of jump resonance in systems containing certain multidimensional nonlinearities . Proc. Instn elect. Engrs , 115 , 1369 – 1372 .
  • BESHAI , M. E. , and DOKAINISH , M. A. , 1975 , Transient response of a forced non-linear system . J. Sound Vibr. , 41 , 53 – 62 .
  • CARTWRIGHT , M. L. , 1950 , Forced oscillations in nonlinear systems . Contributions to the Theory of Nonlinear Oscillations , Vol. I , edited by S. Lefschetz ( Princeton University Press ), pp. 149 – 241 .
  • DOUCE , J. L. , and KING , R. E. , 1959 , Instability in a nonlinear conditionally stable system subjected to a sinusoidal input . Trans. Am. Inst, elect. Engrs , Part 2 , 78 , 665 – 670 .
  • DREW , J. H. , 1974 , Stability of certain periodic solutions of a forced system with hysteresis . Int. J. non-linear Mech. , 9 , 121 – 125 .
  • FUKUMA , A. , and MATSUBARA , M. , 1966 , Jump resonance criteria of nonlinear control systems . I.E.E.E. Trans, autom. Control , 11 , 699 – 706 .
  • HATANAKA , H. , 1963 , The frequency responses and jump resonance phenomena of nonlinear feedback control systems . J. has. Engng , 85 , 236 – 242 .
  • HISAYUKI , K. , 1972 , Forced oscillations in system of two degrees of freedom with nonlinear restoring force . Electron. Commun. Japan , 55 , 8 – 15 .
  • JAY AW ANT , B. V. , and TAHA , M. F. E. , 1971 , Analysis of f err o-resonant jump phenomenon in mutually coupled non-linear LCR circuits . Int. J. Control , 13 , 1009 – 1029 .
  • LAMBA , S. S. , 1968 , Partial avoidance of jump resonance in feedback systems . Int. J. Control , 8 , 537 – 542 .
  • LAMBA , S. S. , and KAVANAGH , R. J. , 1969 , Jump resonance criteria for systems containing double-valued and frequency dependent nonlinearities . Proc. Instn elect. Engrs , 116 , 1225 – 1228 .
  • LAMBA , S. S. , and KAVANAGH , R. J. , 1971 , The phenomenon of isolated jump resonance and its applications . Proc. Instn elect. Engrs , 118 , 1047 – 1050 .
  • LAMBA , S. S. , and KAVANAGH , R. J. , 1973 , Synthesis of certain nonlinear systems for multimodal operation . Proc. Instn elect. Engrs , 120 , 919 – 920 .
  • LEVTNSON , E. , 1953 , Some saturation phenomena in servomechanisms with emphasis on the tachometer stabilized system . Trans. Am. Inst, elect. JSngrs , Part 2 , 72 , 1 – 9 .
  • LOPES , O. , 1975 , Forced oscillations in nonlinear neutral differentia equations . SI AM J. appl. Math. , 29 , 196 – 207 .
  • MAEZAWA , S. 1961 , Steady, forced vibration of unsymmetrical piece-wise linear system (1st report Explanation of analytical procedure) . Bull. J.S.M.E. , 4 , 201 – 212 .
  • MEES , A. I. , 1973 , Periodic waveforms in forced nonlinear systems . Int. J. Control , 18 , 1169 – 1188 .
  • MOORE , J. , and SILJAK , D. , 1966 , Parameter plane analysis of forced nonlinear oscillations . Proc. 4th Annual Allerton Gonj. Circuit and System, Theory , University of Illinois , pp. 887 – 902 .
  • NAOY , F. L. , URAZ , A. , and TAKAHASHI , S. , 1973 , Compensation method for nonlinear systems having jump and hysteresis properties . Int. J. Control , 18 , 801 – 816 .
  • PADMANABHAN , T. R. , 1976 , Investigations on some aspects of the stability of forced oscillations in nonlinear systems . Ph.D. Dissertation , I.I.T. , Kharagpur , India .
  • SHIRAI , K. , KATAOXA , F. , and IINO , Y. , 1973 , Forced oscillation of nonlinear systems having two degrees of freedom . Electl Engng Japan , 93 , 23 – 30 .
  • SINGH , Y. P. , 1968 , Some aspects of forced oscillations in nonlinear systems . Ph.D. Dissertation , I.I.T. , Kharagpur , India .
  • SINGH , Y. P. , 1965 , Graphical method for finding the closed loop frequency response of nonlinear feedback control systems . Proc. Instn elect. Engrs , 112 , 2167 – 2170 .
  • STEIN , W. A. , and THALER , G. J. , 1958 , Obtaining the frequency response characteristics of a nonlinear servomechanism from an amplitude and frequency-sensitive describing function . Trans. Am. Inst, elect. Engrs , Part 2 , 77 , 91 – 96 .
  • WEST, J. C- , and DOUCE , J. L. , 1954 , The frequency response of a certain class of nonlinear feedback systems . Br. J. appl. Phys. , 5 , 204 – 210 .
  • AMSLER , B. E. , and GOROZDOS , R. E. , 1959 , On the analysis of bi-stable control systems . I.R.E. Trans, autom. Control , 4 , 46 – 58 .
  • BONENN , Z. , 1958 , Stability of forced oscillations in nonlinear feedback systems . I.R.E. Trans, autom. Control , 3 , 109 – 111 .
  • DEWAN , E. M. , 1972 , Harmonic entrainment of Van der Pol oscillations Phaselocking and asynchronous quenching . I.E.E.E. Trans, autom. Control , 17 , 655 – 663 .
  • FEIGUTH , W. , 1968 , Multi-input quasi-linearization . Ph.D. Dissertation , University of New Brunswick , Fredericton .
  • Fu , C. C , 1974 , Stable harmonic and subharmonic oscillations of a system with piecewise linear restoring forces . J. appl. Mech. , 41 , 273 – 277 .
  • HAYASHI , C , SHIBAYAMA , H. , and NISHIXAWA , Y. , 1960 , Frequency entrainment in a self-oscillatory system with external force . I.R.E. Trans. Circuit Theory , 7 , 413 – 422 .
  • HELMHOLTZ , H. , 1895 , Sensation of Tone ( London Longmans, Green & Co. ).
  • HUEY , R. M. , PAWLOFF, O, and GLUCHAROFF , T. , 1960 , Extension of the dual input describing function technique to systems containing reactive non-linearity . J. Instn elect. Engrs , Part C , 107 , 334 – 341 .
  • JAMES , D. J. G. , 1971 , Stability and subharmonics in a sinusoidal perturbation hill-climbing system . Int. J. Control , 13 , 165 – 178 .
  • KULIK , A. D. , MADATOV , G. L. , and SHYCHANIN , V. M. , 1973 , Subharmonic oscillations in relay control systems . Avtomatyka , 6 , 19 – 24 .
  • LUDEKE , C. A. , 1953 , The generation and extinction of subharmonics . Proc. Nonlinear Circuit Analysis Symp. , New York , Vol. 2 , pp. 215 – 233 .
  • MAEZAWA , S. , and FURUKAWA , S. , 1973 , Superharmonic resonance in piecewise linear system (Effect of damping and stability problem) . Bull. J.S.M.E. , 16 , 931 – 941 .
  • MANDELSTAM , L. , and PAPALEXI , N. , 1973 , Z. Phys. , 73 .
  • MCCABDLE , J. C , 1968 , Forced oscillations in relay control systems . M.Sc.E. Thesis , University of New Brunswick , Fredericton .
  • MOOK , D. T. , MARSHALL , L. R. , and NAYFEH , A. H. , 1974 , Subharmonic and superharmonic resonances in the pitch and roll modes of ship motions . J. Hydronaut. , 8 , 32 – 10 .
  • O'DAY , J. , and HYDE , J. P. , 1972 , Subharmonics and jump resonance in a third-order nonlinear system . l.E.E.E. Trans, autom. Control , 17 , 721 – 722 .
  • OLDENBURGER , R. , and NICHOLLS , R. , 1963 , Stability of subharmonic oscillations in nonlinear systems . Proc. JACC , pp. 675 – 680 .
  • OLDENBURGER , R. , and NICHOLLS , R. E. , 1964 , Stability of subharmonic oscillations in nonlinear systems . J. bas. Engng , 86 , 116 – 120 .
  • RAJAOOPALAN , P. K. , and SINOH , Y. P. , 1969 , Analysis of harmonic and almost periodic oscillations in forced self oscillating systems . Proc. 4th IF AC Congress , Warsaw , Technical Session No. 41 , pp. 80 – 122 .
  • REIF , Z. F. , 1973 , Subharmonic resonance of order 1/2 with an asymmetrical restoring force . Aeronaut. J. , 77 , 98 – 101 .
  • SAKAWA , Y. , 1960 , Subharmonic oscillations in relay-control systems . Proc. 1st IF AC Congress , Moscow , Vol. 1 , pp. 404 – 409 .
  • WEST , J. C , and DOUCE , J. L. , 1954 , The mechanism of subharmonic generation in a feedback system . Proc. Instn elect. Engrs , Part B , 102 , 569 – 574 .
  • YAMAMOTO , T. , and YASUDA , K. , 1974 , Occurrence of the summed and differential harmonic oscillations in a nonlinear multidegree-of-freedom vibrating system . J. appl. Mech. , 41 , 781 – 786 .
  • ATHERTON , D. P. , 1978 , Applications of high frequency signal injection in nonlinear systems . Int. J. Control , 28 , 161 – 164 .
  • ATHERTON , D. P. , 1962 , The evaluation of the response of single-valued non-linearities to several inputs . Proc. Instn elect. Engrs , Part C , 109 , 146 – 157 .
  • BALASUBRAMANIAN , R. , 1968 , Analysis of systems containing a multidimensional nonlinearity . Ph.D. Dissertation , University of New Brunswick , Fredericton .
  • BALASUBRAMANIAN , R. , and ATHERTON , D. P. , 1968 , Response of multidimensional nonlinearities to inputs which are separable processes . Proc. Instn elect. Engrs , 115 , 581 – 590 .
  • GIBSON , J. E. , and SRIDHAR , R. , 1963 , A new dual-input describing function and an application to the stability of forced oscillations . Trans. Am. Inst. elect. Engrs , Part 2 , 82 , 65 – 70 .
  • MAHALANABIS , A. K. , and NATH , A. K. , 1966 , On the DIDFs of a nonlinear element with memory . l.E.E.E. Trans, autom. Control , 11 , 138 – 139 .
  • OCHIAI , S. , and OLDENBURGER , R. , 1966 , The second describing function and removal of jump resonance by signal stabilization . Proc. JACC , pp 569 – 578 .
  • OLDENBURGER , R. , 1957 , Signal stabilisation of a control system . J. bas. Engng , 79 , 1869 – 1872 .
  • OLDENBURGER , R. , and NAKADA , T. , 1961 , Signal stabilisation of self-oscillatory systems . I.R.E. Trans, autom. Control , 6 , 313 – 325 .
  • OLDENBURGER , R. ,BOYER, R. C, 1962 , Effects of extra sinusoidal inputs to nonlinear systems . J. bas. Engng , 84 , 559 – 570 .
  • PAP , L. E. , 1975 , Some remarks on synchronized oscillators in presence of interfering carrier . 7th Int. Conf. Nonlinear Oscillations , Berlin .
  • DORRAH , H. T. , and HAMZA , M. H. , 1974 , On the asymmetrical oscillatory transient describing function . Int. J. Control , 19 , 385 – 399 .
  • RAJAGOPALAN , P. K. , and SINGH , Y. P. , 1972 , The complete dual input response of nonlinear elements . Int. J. Control , 15 , 1009 – 1026 .
  • RAO , U. M. , and ATHERTON , D. P. , 1975 , Bounds of the 2-input describing function . Electronics Lett. , 11 , 273 .
  • SEA , G. R. , and VACROUX , G. A. , 1969 , Steady state analysis of nonlinear systems and multiple input describing functions/MIDF . Automatica , 5 , 763 – 772 .
  • SOMBRVILLE , M. J. , and ATHERTON , D. P. , 1958 , Multigain representation for a single-valued nonlinearity with several inputs and the evaluation of their equivalent gains by a Cursor method . Proc. Instn elect. Engrs , Part C , 105 , 537 – 549 .
  • TEODORBSCU , D. , 1972 , Dual-input describing series and their use for forced non-linear system analysis . Int. J. Control , 15 , 693 – 704 .
  • WEST , J. C. , DOUCE , J. L. , and LIVESLEY , R. K. , 1956 , The dual-input describing function and its use in the analysis of non-linear feedback systems . Proc. Instn elect. Engrs , Part B , 103 , 463 – 474 .
  • Cox , C. S. , 1965 , Transient analysis of nonlinear control systems . M.Sc. Thesis , University of Newcastle-upon-Tyne , England .
  • DORRAH , H. T. , 1972 , Prediction of transient oscillations in nonlinear control systems . M.Sc. Thesis , University of Calgary , Calgary , Alberta .
  • DORRAH , H. T. , and HAMZA , M. H. , 1972 , Analysis of transient oscillations in nonlinear control systems . Autom. Control Theory Applic , 1 , 1 – 8 .
  • DORRAH , H. T. , and HAMZA , M. H. , 1974 , On the asymmetrical oscillatory describing function . Int. J. Control , 19 , 385 – 399 .
  • FINNIGAN , R. E. , 1957 , Transient analysis of nonlinear servomechanism using describing functions with root locus technique . Ph.D. Dissertation , University of Illinois , Urbana .
  • FREEMAN , E. A. , 1967 , Characterization of nonlinearity for transient processes- Its evaluation and application . l.E.E.E. Trans, autom. Control , 12 , 491 – 501 .
  • FREEMAN , E. A. , and Cox , C. S. , 1967 , Dynamic analysis of nonlinear control systems . Proc. Instn elect. Engrs , 114 , 129 – 138 .
  • GRENSTED , P. E. , 1958 , Analysis of transient response of nonlinear control systems . J. bos. Engng , 80 , 427 – 32 .
  • SCHLICHT , D. L. , and THALER , G. J. , 1967 , Transient response of nonlinear systems . Proc. 5th Annual Allerton Conf. Circuit and System Theory , University of Illinois , pp. 586 – 595 .
  • SO.UDACK , A. C , and BARKHAM , P. G. D. , 1971 , On the transient solution of the unforced Duffing equation with large damping . Int. J. Control , 13 , 767 – 769 .
  • BLIGHT , J. D. , and MCCLAMROCH , N. H. , 1975 , Graphical stability criteria for large-scale nonlinear multiloop systems . Proc. IF AC 6th Triennial World Congress , Boston , Paper 44.5 .
  • COOK , P. A. , 1973 , Modified multivariate circle theorems . Recent Mathematical Developments In Control Theory , edited byD. J. Bell ( London Academic Press ), pp. 367 – 372 .
  • COOK , P. A. , 1974 , Nyquist-plot methods of investigating multivariable feedback systems . Proc. IE AC Symp. on Multivariable Technological Systems , Manchester , Paper S-28 .
  • COOK , P. A. 1976 , Conditions for the absence of limit cycles . l.E.E.E. Trans. autom. Control , 21 , 339 – 345 .
  • DEWEY , A. G. , 1967 , Frequency domain stability criteria for nonlinear multi-variable systems . Int. J. Control , 5 , 77 – 84 .
  • EL-SAKKARY , A. K. , 1978 , Stability of and oscillations in nonlinear multi-variable control systems . M.Sc.E. Thesis , University of New Brunswick , Fredericton .
  • HTTSEYIN , O. , 1973 , On the circle criterion . Int. J. Control , 18 , 9 – 16 .
  • HUSEYIN , O. , 1973 , On the Popov criterion . Int. J. Control , 17 , 1137 – 1142 .
  • MCCLAMROCH , N. H. , and IANCULESCU , G. D. , 1975 , Global stability of two linearly interconnected nonlinear systems . I.E.E.E. Trans, autom. Control , 20 , 678 – 682 .
  • MEES , A. I. , and RAPP , P. E. , 1977 , Stability criteria for multiple-loop nonlinear feedback systems . Proc. IF AC Symp. on Multivariable Technological Systems , Fredericton , pp. 183 – 188 .
  • MOORE , J. B. , and ANDERSON , B. D. O. , 1967 , Applications of the multivariable Popov criterion . Int. J. Control , 5 , 345 – 353 .
  • MOORE , J. B. , and ANDERSON , B. D. O. , 1968 , A generalization of the Popov criterion . J. Franklin Inst. , 285 , 488 – 492 .
  • MCGEE , R. W. , and MACLELLAN , G. D. S. , 1968 , Stability of nonlinear multivariable control systems . Proc. Insin elect. Engrs , 115 , 590 – 591 .
  • PABTOVI , S. , and NAHI , E. H. , 1969 , Absolute stability of dynamic system containing nonlinear functions of several state variables . Automatica , 5 , 465 – 473 .
  • RAE , W. G. , and MACLELLAN , G. D. S. , 1966 , Stability of two classes of nonlinear multivariable control systems . Proc. 3rd IF AC Congress , London , Paper No. 12-c .
  • RAMANI , N. , and ATHERTON , D. P. , 1975 , A note on the stability and eigenvalues of G(s) . Int. J. Control , 22 , 701 – 704 .
  • ROOTENBERG , J. , and Oso , J. B. , 1971 , On cross coupling and stability in nonlinear control systems . I.E.E.E. Trans, autom. Control , 16 , 73 – 75 .
  • ROSENBROCK , H. H. , 1972 , Multivariable circle theorems . IMA Conf. on Recent Developments in Control , Bath .
  • ROSENBROCK , H. H. , and COOK , P. A. , 1975 , Stability and the eigenvalues of G(s) . Int. J. Control , 21 , 99 – 104 .
  • SHANKAR , S. , 1976 , Stability analysis of nonlinear control systems . M.Sc.E. Thesis , University of New Brunswick , Fredericton .
  • SHANKAR , S. , and ATHERTON , D. P. , 1977 , Graphical stability analysis of nonlinear multivariable control systems . Int. J. Control , 25 , 375 – 388 .
  • TOKUMARU , H. , and SAITO , N. , 1965 , On the absolute stability of automatic control systems with many non-linear characteristics . Mem. Fac. Engng Kyoto Univ. , 27 , 347 – 379 .
  • ANAND , G. V. , 1969 , Large-amplitude damped free vibration of a stretched string . J. acoust. Soc. Am. , 45 ( 5 ), 1089 – 1096 .
  • ANAND , G. V. , 1969 , Stability of nonlinear oscillations of stretched strings . J. acoust. Soc. Am. , 46 ( 3 ), 667 – 677 .
  • DATARDINA , S. P. , and LINKENS , D. A. , 1978 , Multimode oscillations in mutually coupled van der Pol type oscillators with fifth-power nonlinear characteristics . I.E.E.E. Trans. Circuits Sysi. , 25 , 308 – 315 .
  • ENDO , T. , and MORI , S. , 1976 , Mode analysis of a multimode ladder oscillator . I.E.E.E. Trans. Circuits Syst. , 23 , 100 – 113 .
  • ENDO , T. , and MORI , S. , 1976 , Mode analysis of a two-dimensional low-pass multimode oscillator . I.E.E.E. Trans. Circuits Syst. , 23 , 517 – 530 .
  • GELB , A. , 1963 , Limit cycles in symmetric multiple nonlinearity systems . I.E.E.E. Trans, autom. Control , 8 , 177 – 178 .
  • GRAY , J. O. , and TAYLOR , P. M. , 1976 , The computer aided design of multi-variable nonlinear control systems using frequency domain techniques . 1st IFACjIFIP Symp. on Software for Computer Control , Paper R.ll , pp. 213 – 219 .
  • GRAY , J. O. , and TAYLOR , P. M. , 1977 , Frequency response methods in the design of multivariable non linear feedback systems . Proc. IF AC Symp. on Multivariable Technological Systems , Fredericton , pp. 225 – 232 .
  • GRAY , J. O. , and AL-JANABI , T. H. , 1977 , The numerical design of multivariable nonlinear feedback systems . Proc. IFAC Syrtvp. on Multivariable Technological Systems , Fredericton , pp. 233 – 238 .
  • HIRAI , K. , and KTTREMATSU , Y. , 1974 , A graphical and computational stability criterion for nonlinear multivariable control systems . Mem. Fac. Engng Kobe Univ. , No. 2 , pp. 93 – 98 .
  • LINDGREN , A. S. , 1964 , Limit cycles in symmetric multivariable systems . I.E.E.E. Trans, autom. Control , 9 , 119 – 120 .
  • LINDGREN , A. G. , and PINKOS , R. F. , 1966 , Stability of symmetric nonlinear multivariable systems . J. Franklin Inst. , 282 , 92 – 101 .
  • LINKENS , D. A. , 1974 , Analytical solution of large numbers of mutually coupled nearly sinusoidal oscillators . I.E.E.E. Trans. Circuits Syst. , 21 , 294 – 300 .
  • LINKENS , D. A. , 1976 , Stability of entrainment conditions for a particular form of mutually coupled Van der Pol oscillators . I.E.E.E. Trans. Circuits Syst. , 23 , 113 – 121 .
  • MEES , A. I. , 1973 , Describing functions, circle criteria and multiple-loop feedback systems . Proc. Instn elect. Engrs , 120 , 126 – 130 .
  • NIKIFORUK , P. N. , and WINTONYK , B. L. M. , 1968 , Frequency response analysis of two-dimensional nonlinear symmetrical and non-symmetrical control systems . Int. J. Control , 7 , 49 – 62 .
  • RAMANI , N. , and ATHERTON , D. P. , 1975 , A describing function method for the approximate stability of nonlinear multivariable systems . University of New Brunswick, Electrical Engineering Department , Report SDC-1 .
  • RAMANI , N. , ATHERTON , D. P. , and MEES , A. I. , 1973 , Describing functions circle criterion and multiple-loop feedback systems . Proc. Instn- elect. Engrs , 120 , 814 .
  • RAMANI , N. , and ATHERTON , D. P. , 1973 , Frequency response methods for nonlinear multivariable systems . Proc. Canadian Con]. AiUomatic Control , University of New Brunswick , Fredericton , Paper No. 9.2 .
  • RAMANI , N. , and ATHERTON , D. P. , 1974 , Stability of nonlinear multivariable systems . Proc. IFAC Symp. Multivariable Technological Systems , University of Manchester , Paper No. S10 .
  • RAMANI , N. , 1975 , Some aspects of the analysis and design of multivariable systems . Ph.D. Dissertation , University of New Brunswick, Fredericton .
  • RASIS , Y. , and BONEN , Z. , 1971 , Oscillations in a two-dimensional nonlinear system . Int. J. Control , 14 , 129 – 139 .
  • TAYLOR , J. H. , 1977 , A new algorithmic limit cycle analysis method for multivariable systems . TASC Report , 28 pp.
  • TAYLOR , J. H. el al., 1976 , High angle of attack stability and control . Office of Naval Research Report , ONR-CR215-237-1 .
  • TAYLOR , J. H. , 1977 , A new algorithmic limit cycle analysis method for multi-variable systems . TIM-612–2 , TASC , Reading , Massachusetts .
  • TAYLOR , J. H. , 1975 , An algorithmic state-space/describing function technique for limit cycle analysis . TIM-612–1 , TASC , Reading , Massachusetts .
  • VISWANADHAM , N. , G. URURAJA , H. V. , and DEEKSHATULU , B. L. , 1966 , Design of nonlinear multivariable control systems . Int. J. Control , 4 , 153 – 166 .
  • VISWANADHAM , N. and DEEKSHATULU , B. L. , 1966 , Stability analysis of nonlinear multivariable systems . Int. J. Control , 5 , 369 – 375 .
  • WOON , S. K. , and NICHOLSON , H. , 1977 , Describing functions, limit cycles and reduced models in nonlinear systems . Electronics Lett. , 13 , 185 – 187 .
  • ATHERTON , D. P. , 1966 , Conditions for periodicity in control systems containing several relays . Proc. 3rd IFAC Congr. , London , Paper No. 28-E .
  • CHUNG , J. K.-C , and ATHERTON , D. P. , 1968 , Analytical design of multi-variable relay control systems for non-interaction in the mean . Proc. IFAC Symp. on Multivariable Control Systems , Dusseldorf .
  • JORGL , H. P. , 1976 , On the analysis of two-variable relay control systems with a linear-asymmetric plant . Int. J. Control , 23 , 703 – 712 .
  • NUGENT , S. T. , 1967 , Steady state oscillations in multivariable relay control systems . Ph.D. Dissertation , University of New Brunswick , Fredericton .
  • NUGENT , S. T. , and KAVANAGH , R. J. , 1969 , Self and forced oscillations in multivariable relay control systems . Automatica , 5 , 519 – 527 .
  • BONENN , Z. , 1971 , Frequency response of feedback relay amplifiers . Proc. Instn elect. Engrs , Part C , 108 , 287 – 295 .
  • CLARK , R. N. , 1969 , Analysis of oscillations in pulse-modulated satellite attitude control systems . Ph.D. Dissertation , Stanford University , California .
  • CLARK , R. N. , DUMAS , P. ,FOSTH, D. C, 1975 , Scheme to improve limit cycle performance of an attitude control system . J. Spacecr. Rockets , 12 , 253 – 254 .
  • DE , G. , and MANDAL , A. K. , 1973 , Incremental describing function analysis of subharmonic oscillations in control systems with thyristor converters . I.E.E.E. Trans, ind. Electron. Control Instrumn , 20 , 229 – 235 .
  • DE , G. , and MANDAL , A. K. , 1973 , Impulse analysis of subharmonic oscillations in control systems with thyristor converters . Proc. Instn elect. Engrs , 120 , 1030 – 1037 .
  • HAN , K. W. , and THALER , G. J. , 1970 , Limit cycles in instrument servos . Proc. 8th Annual Allerton Conf. Circuit and System Theory , University of Illinois , pp. 552 – 561 .
  • HARVEY , C. A. , 1972 , Limit-cycle bounds of a satellite attitude-control system . I.E.E.E. Trans, autom. Control , 17 , 526 – 528 .
  • LANDA , P. S. , and STRELKOV , S. P. , 1961 , On the stability of an aileron control system disturbed by turbulence . Automn remote Control , 22 , 953 – 961 .
  • MILLAR , R. A. , and VIGNERON , F. A. , 1976 , Attitude stability of flexible spacecraft which use dual time constant feedback lag network pseudorate control . AIAAjCASI 6th Communications Satellite Conf. , Montreal , No. 76 – 266 .
  • ROOTS , W. K. , 1969 , Temperature control in industrial processes . I.E.E.E. Trans, ind. Electron. Control Instrumn. , 16 , 147 – 156 .
  • ROOTS , W. K. , and GONENC , G. , 1969 , Indirect electroheat regulation . Proc. Instn elect. Engrs , 116 , 1471 – 1474 .
  • ROOTS , W. K. , 1969 , Fundamentals of Temperature Control ( New York Academic Press ).
  • ROOTS , W. K. , and WALKER , F. , 1967 , Dynamic equilibrium cycling in dis- continuously controlled electric process heating . I.E.E.E. Trans. Industry gen. Applic , 3 , 131 – 140 .
  • ROOTS , W. K. , and WALKER , F. , 1967 , Secondary feedback in on-off electric heating processes . I.E.E.E. Trans. Industry gen. Applic. , 3 , 401 – 420 .
  • ROOTS , W. K. , WOODS , J. T. , WELLS , F. M. , and TULL , R. H. , 1967 , Mode dependent time constants in electric heating processes . I.E.E.E. Trans. Industry gen. Applic , 3 , 315 – 322 .
  • SHUCK , O. H. , 1959 , Honeywell's history and philosophy in the adaptive control field . Wright Air Development Center, Technical Report , Wright Patterson AFB , Ohio .
  • TURNBULL , G. F. , and TOWNSEND , J. M. , 1965 , A feedback pulse with modulated audio amplifier . Wireless World , 71 , 160 – 168 .
  • AGGARWAL , J. K. , and RICHIE , C G. , 1965 , On coupled oscillators . Proc. 3rd Annual Allerton Conf. Circuit and System Theory , University of Illinois , pp. 514 – 523 .
  • BOWLER , O E. J. , EWART , D. N. ,CONCORDIA, C, 1973 , Self excited torsional frequency oscillations with series capacitors . I.E.E.E. Winter Power Meeting , New York , Paper No . T 73218 – 5 .
  • CANDY , J. C , 1974 , A use of limit cycle oscillations to obtain robust analog-to-digital converters . I.E.E.E. Trans. Commun. , 22 , 298 – 305 .
  • FALLSIDE , F. , and FARMER , A. R. , 1967 , Ripple instability in closed-loop systems with thyristor amplifiers . Proc. Instn elect. Engrs , 114 , 139 – 152 .
  • FALLSIDE , F. , 1968 , Ripple instability in closed-loop pulse-modulation systems including inverter drives . Proc. Instn elect. Engrs , 115 , 218 – 228 .
  • FOUAD , A. A. , 1973 , Trends in stability analysis of power systems . Proc. 11th Annual Allerton Conf. Circuit and System Theory , University of Illinois , pp. 54 – 63 .
  • GRAHAM , J. W. ,EHRAM, L. (editors) , 1973 , Nonlinear system modelling and analysis with applications to communications receivers . Technical Report No. RADC-TR-73–178 , Rome Air Development Center , Griffiss AFB , New York .
  • INIOO , R. M. , 1975 , Piecewise linear approximation of nonlinear oscillators . Int. J. Circuit Theory Applic. , 3 , 301 – 306 .
  • JACKSON , R. D., 1969 , Oscillations in single converter drives . Proc. Instn. elect. Engrs , 116 , 633 – 638 .
  • KAVANAGH , R. J. , and SINCLAIR , D. M. , 1974 , A novel frequency meter using jump resonance . I.E.E.E. Trans. Instrumn Measmt , 23 , 178 – 179 .
  • KOSTYUK , A. M. , 1973 , The method of small oscillations and static stability of power systems-A criticism of modern stability theory . Avtomatyka , 6 , 80 – 90 .
  • MIDDLEBROOK , R. D. , 1973 , Describing function properties of a magnetic pulsewidth modulator . I.E.E.E. Trans. Aerospace Electron. Syst. , 9 , 386 – 398 .
  • MORARU , A. , 1973 , Nonlinear oscillations of electromagnetic motor armature . Zagad. Drgan Nieliniowych , 14 , 623 – 638 .
  • PRUSTY , S. , and SANYAL , S. K. , 1976 , New approach for study of ferroreso- nance . Proc. Instn elect. Engrs , 123 , 916 – 918 .
  • WEST , J. C , JAYAWANT , B. V. , and REA , D. P. , 1967 , Transition characteristics of the jump phenomenon in nonlinear resonant circuits . Proc. Instn elect. Engrs , 114 , 381 – 393 .
  • YEH , S. N. , and SUN , Y. , 1974 , Nonlinear analysis of harmonic oscillators . Proc. 7th I.E.E.E. Int. Symp. Circuits and Systems , San Francisco , pp. 147 – 151 .
  • DESHMUKH , J. , and ATHERTON , D. P. , 1976 , Hybrid simulation of an impact hammer . 8th AICA Congr. , Delft , The Netherlands , Paper No. 3.8.1, pp. 855 – 864 .
  • EGLE , D. M. , 1967 , An investigation of an impact vibration absorber . J. Engng Industry , 89 , 653 – 661 .
  • HAIOHT , E. C , and KING , W. W. , 1972 , Stability of nonlinear oscillations of an elastic rod . J. acoust. Soc. Am. , 52 , 899 – 911 .
  • Kuo , B. C , SINGH , G. , and SELTZER , S. M. , 1975 , Limit cycle analysis of large space telescope with CMG nonlinearity . J. Spacecr. Rockets , 12 , 472 – 76 .
  • MARKEEV , A. P. , 1975 , Stability of plane oscillations and rotations of a satellite on a circular orbit . Kosm. Issled,. , 13 , 322 – 336 .
  • MASRI , S. F. , and CAUGHEY , T. K. , 1966 , On the stability of the impact damper . J. appl. Mech. , 33 , 586 – 592 .
  • MASRI , S. F. , 1968 , Stability boundaries of the impact damper . J. appl. Mech. , 99 , 416 – 417 .
  • SARAVANJA-FABRIS , N. , 1972 , Nonlinear stability analysis of chatter in metal cutting . M.Sc. Thesis , Illinois Institute of Technology , Chicago .
  • SARAVANJA-FABRIS , N. , and D'SODZA , A. F. , 1973 , Nonlinear stability analysis of chatter in.metal cutting . Trans. Am. Soc. mech. Engrs , Paper No. 73-DET-33 .
  • STEPHEN , A. C. , 1967 , Vibration of a single degree of freedom system under |combined impulse and sinusoidal excitation. National Engineering Laboratory Report No. 287|ot.
  • TSAPLIN , S. A. , 1953 , Vibratory Impact Mechanism for Road and Bridge Construction ( in Russian ) ( Moscow N.E.L. , Avtotransizdat ).
  • YAMAMOTO , T. , 1960 , Response curves at the critical speeds of sub-harmonic and summed and differential harmonic oscillations . Bull. J.S.M.E. , 3 , 397 – 403 .
  • Ai , C. H. , and HAN , K. W. , 1975 , Stability analysis of a nuclear reactor control system with multiple transport-lags and an asymmetrical nonlinearity . I.E.E.E. Trans, nucl. Sci. , 22 , 2103 – 2112 .
  • ASAHI , Y. , and AKCASU , A. Z. , 1973 , An investigation of nonlinear xenon oscillation by method of averaging . J. nucl. Energy , 27 , 839 – 855 .
  • KEENER , J. P. , and COHN , D. S. , 1975 , Nonlinear oscillations in a reactor with two temperature coefficients . Nucl. Sci. Engng , 56 , 354 – 359 .
  • BREARLEY , J. R. ,SOUDACK, A. C, 1978 , Approximate solutions to the Lotka-Vol terra competition equations . Int. J. Control , 27 , 933 – 941 .
  • CHANCE , B. , PYE , E. K. , GHOSH , A. K. and HESS , B. , 1973 , Biological and Biochemical Oscillators ( New York Academic Press ).
  • CRONIN , J. , 1975 , Stable periodic solutions of some nonlinear systems . 7th Int. Conj. Nonlinear Oscillations , Berlin .
  • GOODWIN , B. C. , 1963 , Temporal Organization in Cells-A Dynamic Theory of Cellular Control Processes ( New York Academic Press ).
  • HAUSNER , A. , 1971 , Analog and Analog/Hybrid Computer Programming ( Englewood Cliffs Prentice-Hall ), p. 74 .
  • MEES , A. I. , and RAPP , P. E. , 1978 , Periodic metabolic systems. Oscillations in multiple-loop negative feedback biochemical control networks . J math Biol , 5 , 99 – 114 .
  • MTLSUM , J. H. , 1966 , Biological Control System Analysis ( New York McGraw-Hill ).
  • PAVLIDIS , T. , 1973 , Biological Oscillators Their Mathematical Analysis ( New York Academic Press ).
  • RAPP , P. , 1976 , Nonlinear oscillations in biochemical processes . Colloquium on Stability Theory and Nonlinear Oscillations , Savoy Place , England .
  • RAPP , P. , 1975 , Biochemical oscillators a search procedure . Math. Biosci 23 , 289 – 303 .
  • RAPP , P. , 1975 , A theoretical investigation of a large class of biochemical oscillators . Math. Biosci. , 25 , 165 – 188 .
  • RAPP , P. , 1975 , Analytic procedures for large dimension nonlinear biochemical oscillators . Biosystems , 7 , 92 – 100 .
  • RAPP , P. , 1976 , Mathematical techniques for the study of oscillations in biochemical control loops . Bull. Inst. Math. Applic , 12 , 11 – 21 .
  • RAPP , P. , 1976 , Analysis of biochemical phase shift oscillators by a harmonic balancing technique . J. math. Biol. , 3 , 203 – 224 .
  • SCHWAN , H. P. , 1969 , Biological Engineering ( New York McGraw-Hill ).
  • STUCKI , J. W. , 1978 , Stability analysis of biochemical systems-a practical guide . Prog. Biophys. molec. Biol. , 33 , 99 – 187 .
  • TYSON , J. J. , 1975 , On the existence of oscillatory solutions in negative feedback cellular control processes . J. math. Biol. , 1 , 311 – 315 .
  • WALTER , O , 1969 , Oscillations in controlled biochemical systems . Biophys J 9 , 863 – 872 .
  • WALTER , C , 1969 , Stability of controlled biological systems . J. theor. Biol 23 , 23 – 28 .
  • WEED , H. R. , 1972 , Survey paper on biological receptors and their technical models . Proc. 5th IF AC Congr. , Paris , Paper No. S-6 .

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