Properties of the dominant behaviour of quadratic systems. (English) Zbl 1121.34043
This well-written paper examines a class of first-order quadratic systems of two equations with constant coefficients from both the symmetry and singularity analyses points of view. These equations of Lotka-Volterra-type arise in mathematical modeling and are thus of interest. The relationship between an increase in the number of singularities and the possession of the Painlevé property is pointed out. Moreover, the authors observe that for special values of the parameters of the system the possession of the Painlevé property is characterized by a left Painlevé series rather than the standard right Painlevé series.
Reviewer: F. M. Mahomed (Johannesburg)
MSC:
| 34C14 | Symmetries, invariants of ordinary differential equations |
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
| 92D25 | Population dynamics (general) |
References:
| [1] | Andriopoulos, K.; Leach, PGL; Nucci, MC, The Ladder Problem: Painlevé integrability and explicit solution, Journal of Physics A: Mathematical and General, 36, 11257-11265, 2003 · Zbl 1040.37039 |
| [2] | Andriopoulos K & Leach P G L, Decomposition of the scalar autonomous Riccati equation, preprint: Research Group in Mathematical Physics, Department of Communication and Systems Engineering, School of Sciences, University of the Aegean, Karlovassi 83 200, Greece, 2005. |
| [3] | Cairó, L.; Feix, MR; Goedert, J., Invariants for models of interacting populations, Physics Letters A, 140, 421-427, 1989 |
| [4] | Cairó, L.; Feix, MR, Families of invariants of the motion for the Lotka-Volterra equations: The linear polynomials family, Journal of Mathematical Physics, 33, 2440-2455, 1992 · Zbl 0767.92021 |
| [5] | Contopoulos, G.; Grammaticos, B.; Ramani, A., Painlevé analysis for the mixmaster universe model, Journal of Physics A: Mathematical and General, 25, 5795-5799, 1993 · Zbl 0811.53086 |
| [6] | Contopoulos, G.; Grammaticos, B.; Ramani, A., The Mixmaster Universe revisited, Journal of Physics A: Mathematical and General, 27, 5357-5362, 1994 · Zbl 0844.58120 |
| [7] | Cotsakis S & Leach P G L, Painlevé analysis of the Mixmaster universe, Journal of Physics A: Mathematical and General, 27 (1994), 1625-1631. · Zbl 0843.58144 |
| [8] | Ermakov V, Second-order differential equations. Conditions of complete integrability, Universita Izvestia Kiev Series III, 9, (1880) 1-25 (trans Harin AO). |
| [9] | Feix, MR; Geronimi, C.; Cairó, L.; Leach, PGL; Lemmer, RL; Bouquet, SÉ, 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, 1997 · Zbl 0927.34006 |
| [10] | Hua, DD; Cairó, L.; Feix, MR; Govinder, KS; Leach, PGL, 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, 1996 · Zbl 0873.34005 |
| [11] | Govinder K S & Leach P G L, The nature and uses of symmetries of ordinary differential equations South African Journal of Science, 92 (1996), 23-28. |
| [12] | Golubchik, IZ; Sokolov, VV, On some generalisations of the factorisation method, Theoretical and Mathematical Physics, 110, 267-276, 1997 · Zbl 0922.58034 |
| [13] | Golubchik, IZ; Sokolov, VV, Operator Yang-Baxter equations, integrable odes and nonassociative algebras, Journal of Nonlinear Mathematical Physics, 7, 184-197, 2000 · Zbl 1119.37318 |
| [14] | Head, AK, LIE, a PC program for Lie analysis of differential equations, Computer Physics Communications, 77, 241-248, 1993 · Zbl 0854.65055 |
| [15] | Karasu (Kalkanlı) A & Leach P G L, Nonlocal symmetries and integrable systems, Journal of Physics A: Mathematical and General, (2005) (submitted) |
| [16] | Leach, PGL; Cotsakis, S.; Flessas, GP, Symmetry, singularity and integrability in complex dynamics: I The reduction problem, Journal of Nonlinear Mathematical Physics, 7, 445-479, 2000 · Zbl 0989.34074 |
| [17] | Lemmer, RL; Leach, PGL, The Painlevé test hidden symmetries and the equation y” + yy’ + ky^3 = 0, Journal of Physics A: Mathematical and General, 26, 5017-5024, 1993 · Zbl 0806.34008 |
| [18] | Lie, S., Differentialgleichungen, 1967, New York: Chelsea, New York |
| [19] | Lotka, AJ, Elements of Mathematical Biology, 1956, New York: Dover, New York · Zbl 0074.14404 |
| [20] | Malthus, TR, An Essay on the Principal of Population, 1970, Harmondsworth: Penguin, Harmondsworth |
| [21] | Morozov, VV, Classification of six-dimensional nilpotent Lie algebras, Izvestia Vysshikh Uchebn Zavendeniĭ Matematika, 5, 161-171, 1958 · Zbl 0198.05501 |
| [22] | Mubarakzyanov, GM, On solvable Lie algebras, Izvestia Vysshikh Uchebn Zavendeniĭ Matematika, 32, 114-123, 1963 · Zbl 0166.04104 |
| [23] | Mubarakzyanov, GM, Classification of real structures of five-dimensional Lie algebras, Izvestia Vysshikh Uchebn Zavendeniĭ Matematika, 34, 99-106, 1963 · Zbl 0166.04201 |
| [24] | Mubarakzyanov, GM, Classification of solvable six-dimensional Lie algebras with one nilpotent base element, Izvestia Vysshikh Uchebn Zavendeniĭ Matematika, 35, 104-116, 1963 · Zbl 0166.04202 |
| [25] | Murray, JD, Mathematical Biology, I: An Introduction, 2002, New York: Third edition, Springer-Verlag, New York · Zbl 1006.92001 |
| [26] | Nucci M C & Leach P G L (2005) Jacobi’s last multiplier and the complete symmetry group of the Ermakov-Pinney equation Journal of Nonlinear Mathematical Physics12 305-320 · Zbl 1079.34026 |
| [27] | Pinney E, The nonlinear differential equation y” (x) + p(x)y + cy^-3 = 0, Proceedings of the American Mathematical Society, 1, (1950) 681. · Zbl 0038.24303 |
| [28] | Ramani, A.; Grammaticos, B.; Bountis, T., The Painlevé property and singularity analysis of integrable and nonintegrable systems, Physics Reports, 180, 159-245, 1989 |
| [29] | Sherring, J.; Head, AK; Prince, GE, Dimsym and LIE: symmetry determining packages, Mathematical and Computer Modelling, 25, 153-164, 1997 · Zbl 0918.34007 |
| [30] | Rajasekar S (2004) Painlevé analysis and integrability of two-coupled non-linear oscillators Pramana – journal of physics62 1-12 |
| [31] | Tabor, M., Chaos and Integrability in Nonlinear Dynamics, 1989, New York: John Wiley, New York · Zbl 0682.58003 |
| [32] | Verhulst, P-F, Notice sur le loi que la population suit dans son accroissement, Correspondences des Mathématiques et Physiques, 10, 113-121, 1838 |
| [33] | Volterra, V., Variazione fluttuazione del numero d’individui in specie animali conviventi, Memoiri della Academia Linceinsis, 2, 31-113, 1926 · JFM 52.0450.06 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
