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Complexity in spatio-temporal dynamics

Published online by Cambridge University Press:  14 November 2011

D. L. Rod  and
B. D. Sleeman
D. L. Rod
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada
B. D. Sleeman
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K
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Extract

Complex and chaotic structures in certain dynamical systems in biology arise as a consequence of noncomplete integrability of two-degree-of-freedom Hamiltonian systems. A study of this problem is made using Ziglin theory and implemented with the aid of the Kovacic algorithm.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 5 , 1995 , pp. 959 - 974
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Baider, A., Churchill, R. C., Rod, D. L. and Singer, M. F.. On the infinitesimal geometry of integrable systems (Preprint FI 92-DS07, The Fields Institute for Research in Mathematical Sciences, December 9, 1992, to appear).Google Scholar
2Churchill, R. C., Pecelli, G. and Rod, D. L.. A survey of the Henon-Heiles Hamiltonian with applications to related examples. In Como Conference Proceedings on Stochastic Behaviour in Classical and Quantum Hamiltonian Systems, eds Casati, G. and Ford, J., Springer Lecture Notes in Physics 93 73–136, (New York: Springer, 1979).Google Scholar
3Churchill, R. C. and Rod, D. L.. Pathology in dynamical systems (III): Analytic Hamiltonians, J. Differential Equations 37 (1980), 23–38.Google Scholar
4Churchill, R. C. and Rod, D. L.. On the determination of Ziglin monodromy groups. SIAM J. Math. Anal. 22 (1991), 1790–802.Google Scholar
5Churchill, R. C., Rod, D. L. and Singer, M. F.. Group-theoretic obstructions to integrability (Preprint FI 93-DS04, The Fields Institute for Research in Mathematical Sciences, March 15, 1993, to appear).Google Scholar
6Duval, A.. The Kovacic algorithm with applications to special functions, in Differential Equations and Computer Algebra, ed. Singer, M. (San Diego: Academic Press, 1991).Google Scholar
7Duval, A. and Loday-Richard, M.. A propos de l'algorithme de Kovacic (Preprint 89–12, Universite de Paris-Sud, Paris, 1989).Google Scholar
8Kodaira, K.. Complex Manifolds and Deformation of Complex Structures, Grundlehren der mathematischen Wissenschaften 238 (New York: Springer, (1986).CrossRefGoogle Scholar
9Kot, M.. Discrete-time travelling waves: Ecological examples. J. Math. Biol. 30 (1992), 413–36.Google Scholar
10Kovacic, J.. An algorithm for solving second order linear homogeneous differential equations. J. Symbolic Comput. 2 (1986), 3–43.CrossRefGoogle Scholar
11Massey, W. S.. Algebraic Topology: An Introduction (New York: Springer, 1989).Google Scholar
12Moser, J.. Stable and Random Motions in Dynamical Systems, Annals of Mathematical Studies (Princeton, NJ: Princeton University Press, 1973).Google Scholar
13Rod, D. L.. On a theory of Ziglin in Hamiltonian dynamics. In Hamiltonian Dynamical Systems, eds Meyer, K. and Saari, D. E., Contemporary Mathematics 81, 259–70 (Providence, RI: American Mathematical Society, 1988).Google Scholar
14Sleeman, B. D.. Complexity in biological systems and Hamiltonian dynamics. Proc. Roy. Soc. London Ser. A 425(1989), 17–47.Google Scholar
15Sleeman, B. D.. The dynamics of reversible systems and spatio-temporal complexity in biology. Proceedings of the Conference on Conservative Systems and Quantum Chaos, Fields Institute.Google Scholar
16Wiggins, S.. Global Bifurcations and Chaos, Applied Mathematical Sciences 73 (Heidelberg: Springer, 1988).Google Scholar
17Ziglin, S. L.. Branching of solutions and non-existence of first integrals in Hamiltonian mechanics (I). Functional Anal. Appl. 16 (1982), 181–9.Google Scholar