L 2-invariants and morse-smale flows on manifolds

Sharko, V. V.

doi:10.1007/s11253-007-0038-8

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Hyperbolic Sets and Chaos1

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2015

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“…Note that numerical approximations are correct if our system is structurally stable. Some topological methods for the Morse-Smale systems can be found in [38].…”

Section: Hyperbolic Sets and Chaosmentioning

confidence: 99%

Reduction Methods and Chaos for Quadratic Systems of Differential Equations

Vakulenko¹,

Grigoriev

Weber

2015

Stud Appl Math

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We consider systems of differential equations with quadratic nonlinearities having applications for biochemistry and population dynamics, which may have a large dimension n. Due to the complexity of these systems, reduction algorithms play a crucial role in study of their large time behavior. Our approach aims to reduce a large system to a smaller one consisting of m differential equations, where m << n. Under some restrictions (that allow us to separate slow and fast variables in the system) we obtain a new system of differential equations, involving slow variables only. This reduction is feasible from a computational point of view for large n that allows us to investigate sensitivity of dynamics with respect to random variations of parameters. We show that the quadratic systems are capable to generate all kinds of structurally stable dynamics including chaos.

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“…Note that numerical approximations are correct if our system is structurally stable. Some topological methods for the Morse-Smale systems can be found in [38].…”

Section: Hyperbolic Sets and Chaosmentioning

confidence: 99%