Numerical proof for chemostat chaos of Shilnikov's type
Copyright policy )Numerical proof for chemostat chaos of Shilnikov's type
A classical chemostat model is considered that models the cycling of one essential abiotic element or nutrient through a food chain of three trophic levels. The long-time behavior of the model was known to exhibit complex dynamics more than 20 years ago. It is still an open problem to prove the existence of chaos analytically. In this paper, we aim to solve the problem numerically. In our approach, we introduce an artificial singular parameter to the model and construct singular homoclinic orbits of the saddle-focus type which is known for chaos generation. From the configuration of the nullclines of the equations that generates the singular homoclinic orbits, a shooting algorithm is devised to find such Shilnikov saddle-focus homoclinic orbits numerically which in turn imply the existence of chaotic dynamics for the original chemostat model.
- Shanghai Normal University China (People's Republic of)
- National Tsing Hua University Taiwan
- Shanghai Normal University China (People's Republic of)
- University of Nebraska System United States
Numerical Analysis, Computer-Assisted, Homoclinic and heteroclinic solutions to ordinary differential equations, Models, Theoretical, Approximation methods and numerical treatment of dynamical systems, Strange attractors, chaotic dynamics of systems with hyperbolic behavior, Complex behavior and chaotic systems of ordinary differential equations, Nonlinear Dynamics, Predatory Behavior, Qualitative investigation and simulation of ordinary differential equation models, Singular perturbations for ordinary differential equations, Animals, Algorithms
Numerical Analysis, Computer-Assisted, Homoclinic and heteroclinic solutions to ordinary differential equations, Models, Theoretical, Approximation methods and numerical treatment of dynamical systems, Strange attractors, chaotic dynamics of systems with hyperbolic behavior, Complex behavior and chaotic systems of ordinary differential equations, Nonlinear Dynamics, Predatory Behavior, Qualitative investigation and simulation of ordinary differential equation models, Singular perturbations for ordinary differential equations, Animals, Algorithms
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