Properties of the Dominant Behaviour of Quadratic Systems

References

• Andriopoulos , K , Leach , P G L and Nucci , M C . 2003 . The Ladder Problem: Painlevé integrability and explicit solution . Journal of Physics A: Mathematical and General , 36 : 11257 – 11265 .
   Google Scholar
• Andriopoulos , K and Leach , P G L . 2005 . Decomposition of the scalar autonomous Riccati equation, preprint: Research Group in Mathematical Physics , Karlovassi 83 200 , Greece : Department of Communication and Systems Engineering, School of Sciences, University of the Aegean .
   Google Scholar
• Cairó , L , Feix , M R and Goedert , J . 1989 . Invariants for models of interacting populations . Physics Letters A , 140 : 421 – 427 .
   Web of Science ®Google Scholar
• Cairó , L and Feix , M R . 1992 . Families of invariants of the motion for the Lotka-Volterra equations: The linear polynomials family . Journal of Mathematical Physics , 33 : 2440 – 2455 .
   Web of Science ®Google Scholar
• Contopoulos , G , Grammaticos , B and Ramani , A . 1993 . Painlevé analysis for the mixmaster universe model . Journal of Physics A: Mathematical and General , 25 : 5795 – 5799 .
   Google Scholar
• Contopoulos , G , Grammaticos , B and Ramani , A . 1994 . The Mixmaster Universe revisited . Journal of Physics A: Mathematical and General , 27 : 5357 – 5362 .
   Google Scholar
• Cotsakis , S and Leach , P G L . 1994 . Painlevé analysis of the Mixmaster universe . Journal of Physics A: Mathematical and General , 27 : 1625 – 1631 .
   Google Scholar
• Ermakov , V . 1880 . Second-order differential equations. Conditions of complete integrability . Universita Izvestia Kiev Series III , 9 : 1 – 25 . (trans Harin AO)
   Google Scholar
• Feix , M R , Geronimi , C , Cairó , L , Leach , P G L , Lemmer , R L and Bouquet , S É . 1997 . Right and left Painlevé series for ordinary differential equations invariant under time translation and rescaling . Journal of Physics A: Mathematical and General , 30 : 7437 – 7461 .
   Google Scholar
• Hua , D D , Cairó , L , Feix , M R , Govinder , K S and Leach , P G L . 1996 . Connection between the existence of first integrals and the Painlevé Property in Lotka-Volterra and Quadratic Systems . Proceedings of the Royal Society of London , 452 : 859 – 880 .
   Google Scholar
• Govinder , K S and Leach , P G L . 1996 . The nature and uses of symmetries of ordinary differential equations . South African Journal of Science , 92 : 23 – 28 .
   Web of Science ®Google Scholar
• Golubchik , I Z and Sokolov , V V . 1997 . On some generalisations of the factorisation method . Theoretical and Mathematical Physics , 110 : 267 – 276 .
   Web of Science ®Google Scholar
• Golubchik , I Z and Sokolov , V V . 2000 . Operator Yang-Baxter equations, integrable odes and nonassociative algebras . Journal of Nonlinear Mathematical Physics , 7 : 184 – 197 .
   Web of Science ®Google Scholar
• Head , A K . 1993 . LIE, a PC program for Lie analysis of differential equations . Computer Physics Communications , 77 : 241 – 248 .
   Web of Science ®Google Scholar
• Karasu , A and Leach , P G L . 2005 . Nonlocal symmetries and integrable systems . Journal of Physics A: Mathematical and General , (Kalkanlı) (submitted)
   PubMedGoogle Scholar
• Leach , P G L , Cotsakis , S and Flessas , G P . 2000 . Symmetry, singularity and integrability in complex dynamics: I The reduction problem . Journal of Nonlinear Mathematical Physics , 7 : 445 – 479 .
   Web of Science ®Google Scholar
• Lemmer , R L and Leach , P G L . 1993 . The Painlevé test, hidden symmetries and the equation . y^″ + yy^′ + ky³ = 0 Journal of Physics A: Mathematical and General , 26 : 5017 – 5024 .
   Google Scholar
• Lie , S . 1967 . Differentialgleichungen , New York : Chelsea .
   Google Scholar
• Lotka , A J . 1956 . Elements of Mathematical Biology , New York : Dover .
   Google Scholar
• Malthus , T R . 1970 . An Essay on the Principal of Population , Harmondsworth : Penguin .
   Google Scholar
• Morozov , V V . 1958 . Classification of six-dimensional nilpotent Lie algebras . Izvestia Vysshikh Uchebn Zavendeniĭ Matematika , 5 : 161 – 171 .
   Google Scholar
• Mubarakzyanov , G M . 1963 . On solvable Lie algebras . Izvestia Vysshikh Uchebn Zavendeniĭ Matematika , 32 : 114 – 123 .
   Google Scholar
• Mubarakzyanov , G M . 1963 . Classification of real structures of five-dimensional Lie algebras . Izvestia Vysshikh Uchebn Zavendeniĭ Matematika , 34 : 99 – 106 .
   Google Scholar
• Mubarakzyanov , G M . 1963 . Classification of solvable six-dimensional Lie algebras with one nilpotent base element . Izvestia Vysshikh Uchebn Zavendeniĭ Matematika , 35 : 104 – 116 .
   Google Scholar
• Murray , J D . 2002 . Mathematical Biology, I: An Introduction , Third , New York : Springer-Verlag .
   Google Scholar
• Nucci , M C and Leach , P G L . 2005 . Jacobi’s last multiplier and the complete symmetry group of the Ermakov-Pinney equation . Journal of Nonlinear Mathematical Physics , 12 : 305 – 320 .
   Web of Science ®Google Scholar
• Pinney , E . 1950 . The nonlinear differential equation . y^″(x) + p(x)y + cy⁻³ = (0), Proceedings of the American Mathematical Society , 1 : 681
   Web of Science ®Google Scholar
• Ramani , A , Grammaticos , B and Bountis , T . 1989 . The Painlevé property and singularity analysis of integrable and nonintegrable systems . Physics Reports , 180 : 159 – 245 .
   Web of Science ®Google Scholar
• Sherring , J , Head , A K and Prince , G E . 1997 . Dimsym and LIE: symmetry determining packages . Mathematical and Computer Modelling , 25 : 153 – 164 .
   Web of Science ®Google Scholar
• Rajasekar , S . 2004 . Painlevé analysis and integrability of two-coupled non-linear oscillators . Pramana – journal of physics , 62 : 1 – 12 .
   Web of Science ®Google Scholar
• Tabor , M . 1989 . Chaos and Integrability in Nonlinear Dynamics , New York : John Wiley .
   Google Scholar
• Verhulst , P-F . 1838 . Notice sur le loi que la population suit dans son accroissement . Correspondences des Mathématiques et Physiques , 10 : 113 – 121 .
   Google Scholar
• Volterra , V . 1926 . Variazione fluttuazione del numero d’individui in specie animali conviventi . Memoiri della Academia Linceinsis , 2 : 31 – 113 .
   Google Scholar