A strange attractor in the unfolding of an orbit-flip homoclinic orbit

References

• Bonckaert , P. 1993 . On the continuous dependence of the smooth change of coordinates in parametrized normal forms theorems . Journal of Differential Equations , 106 : 107 – 120 .
   Web of Science ®Google Scholar
• Bonckaert , P. 1997 . Conjugacy of vector fields respecting additional properties . Journal of Dynamical and Control Systems , 3 : 419 – 432 .
   Google Scholar
• Chow , S. N. , Deng , B. and Fiedler , B. 1990 . Homoclinic bifurcation at resonant eigenvalues . Journal of Dynamics and Differential Equations , 2 : 177 – 244 .
   Google Scholar
• Coullet , P. and Tresser , C. 1978 . Iterations d'endomorphismes et groupe de renormalisation . Journal de Physique , C5 : 25
   Google Scholar
• Deng , B. 1993 . Homoclinic twisting bifurcation and cusp horseshoe maps . Journal of Dynamics and Differential Equations , S : 417 – 467 .
   Google Scholar
• Evans , J. W. , Fenichel , N. and Feroe , J A. 1982 . Double impulse solutions in nerve axon equations . SIAM Journal on Applied Mathematics , 42 : 219 – 234 .
   Web of Science ®Google Scholar
• Feigenbaum , M. J. 1978 . Quantitative universality for a class of nonlinear transformations . Journal of Statistical Physics , 19 : 669 – 706 .
   Web of Science ®Google Scholar
• Gambaudo , J.-M. 1987 . Ordre, desordre, et frontière des systeme Morse-Smale , l'Université de Nice . Thesis
   Google Scholar
• Hirsch , M. , Pugh , C. and Shub , M. 1977 . Invariant Manifolds, Lecture Notes in Mathematics , Vol. 583 , Berlin : Springer-Verlag .
   Google Scholar
• Homburg , A. J. 1996 . Global Aspects of Homoclinic Bifurcations of Vector Fields , Vol. 578 , Memoirs AMS .
   Google Scholar
• Homburg , A. J. , Kokubu , H. and Krupa , M. 1994 . The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit . Ergodic Theory and Dynamical Systems , 14 : 667 – 693 .
   Web of Science ®Google Scholar
• Homburg , A. J. , Kokubu , H. and Naudot , V. 2001 . Homoclinic doubling cascade . Archive for Rational Mechanics , 160 : 195 – 213 .
   Web of Science ®Google Scholar
• Homburg , A. J. and Krauskopf , B. 2000 . Resonant homoclinic flip bifurcations . Journal of Dynamics and Differential Equations , 12 (4) : 807 – 850 .
   Google Scholar
• Homburg , A. J. and Young , T. 2001 . Universal scaling in homoclinic doubling cascades . Communications in Mathematical Physics , 222 : 269 – 292 .
   Web of Science ®Google Scholar
• Kiriki , S. and Yongluo , C. 2001 . Global dynamics of some Henon maps . Bulletin of China , (in press)
   Google Scholar
• Kisaka , M. , Hokubu , H. and Oka , K. 1993 . Bifurcations to N-homoclinic orbits and N-periodic orbits in vector fields . Journal of Dynamics and Differential Equations , S : 305 – 357 .
   Google Scholar
• Kokubu , H. and Naudot , V. 1997 . The existence of infinitely many homoclinic doubling bifurcations from some codimension 3 homoclinic orbits . Journal of Dynamics and Differential Equations , 9 (3) : 445 – 462 .
   Google Scholar
• Koper , M. 1995 . Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der PolDuffing model with a cross shaped phase diagram . Physica D , 80 : 72 – 94 .
   Google Scholar
• Krupa , M. , Sandstede , B. and Szmolyan , P. 1997 . Fast and slow waves in the FitzHugh-Nagumo equation . Journal of Differential Equations , 133 : 49 – 97 .
   Web of Science ®Google Scholar
• Mora , L. and Viana , M. 1993 . Abundance of strange attractors . Acta Mathematica , 171 : 1 – 71 .
   Web of Science ®Google Scholar
• audot , V. 1996a . Strange attractor in the unfolding of an inclination-flip homoclinic orbit . Ergodic Theory and Dynamical Systems , 16 : 1071 – 1086 .
   Web of Science ®Google Scholar
• Naudot , V. 1996b . Bifurcations homoclines des champs de vecteurs en dimension trois , Dijon : l'Université de Bourgogne . Thèse
   Google Scholar
• Newhouse , S. 1974 . Diffeomorphisms with infinitely many sinks . Topology , 13 : 9 – 18 .
   Google Scholar
• Nii . 1996 . N-homoclinic orbits bifurcations for homoclinic orbits changing their twisting . Journal of Dynamics and Differential Equations , 2 (2) : 549 – 572 .
   Google Scholar
• Oldman , B. E. , Krauskopf , B. and Champneys , R. 2000 . Death of period-doubling cascades: locating the homoclinic doubling cascade . Physica D , 126 : 100 – 120 .
   Google Scholar
• Oldman , B. E. , Krauskopf , B. and Champneys , R. 2001 . Numerical unfoldings: codimension-three resonant homoclinic flip bifurcations . Non-linearity , 14 (3) : 597 – 621 .
   Google Scholar
• Palis , J. and Tokens , F. 1993 . “ Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations ” . In Fractal Dimensions and Infinitely Many Attractors , Cambridge : Cambridge University Press .
   Google Scholar
• Rychlik , M. R. 1990 . Lorenz attractors through Shil'nikov-type bifurcation . Part I. Ergodic Theory and Dynamical Systems , 10 : 793 – 821 .
   Web of Science ®Google Scholar
• Sandstede , B. 1993 . Verzweigungstheorie homokliner Verdopplungen , University of Stuttgart . Thesis
   Google Scholar
• Shil'nikov , L P. 1965 . A case of the existence of a denumerable set of periodic motions . Soviet Math. Doklady , 8 : 163 – 166 .
   Google Scholar
• Shil'nikov , A. 1993 . On bifurcation of the Lorentz attractor in the Shimizu-Morioka model . Physica D , 62 (Special Issue)
   Google Scholar
• Yanagida , E. 1987 . Branching of double pulse solutions from single pulse solutions in nerve axon equations . Journal of Differential Equations , 66 : 243 – 262 .
   Web of Science ®Google Scholar