Reduction Methods and Chaos for Quadratic Systems of Differential Equations

Sergey Vakulenko, Dima Grigoriev und Andreas Weber
In: Studies in Applied Mathematics (2015)
 

Abstract

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 inline image. 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.

Bibtex

@ARTICLE{VakulenkoGrigorievWeber2015a,
    author = {Vakulenko, Sergey and Grigoriev, Dima and Weber, Andreas},
     title = {Reduction Methods and Chaos for Quadratic Systems of Differential Equations},
   journal = {Studies in Applied Mathematics},
      year = {2015},
  abstract = {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 inline image. 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.},
       doi = {10.1111/sapm.12083}
}