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Numerical test for hyperbolicity of chaotic dynamics in time-delay systems
Pavel V Kuptsov1, Sergey P Kuznetsov2,3
1Institute of Electronics and Mechanical Engineering, Yuri Gagarin State Technical University of Saratov, Politekhnicheskaya 77, Saratov 410054, Russia.
Physical Review. E
|August 31, 2016
Summary
We created a numerical test to assess the hyperbolicity of chaotic dynamics in time-delay systems. This method confirms hyperbolicity in two systems and identifies a new nonhyperbolic chaotic system.
Area of Science:
- Dynamical Systems and Chaos Theory
- Nonlinear Dynamics
- Computational Physics
Background:
- Hyperbolicity is a key property for understanding the stability and predictability of chaotic systems.
- Time-delay systems exhibit complex dynamics, but testing their hyperbolicity remains challenging.
- Existing methods for analyzing chaotic dynamics may not be directly applicable to systems with time delays.
Purpose of the Study:
- To develop and validate a numerical test for assessing the hyperbolicity of chaotic dynamics in time-delay systems.
- To investigate the prevalence of hyperbolic chaos in different types of time-delay systems.
- To provide a computational tool for distinguishing between hyperbolic and nonhyperbolic chaos in these systems.
Main Methods:
- Development of a numerical test based on the angle criterion.
- Computation of angle distributions between expanding, contracting, and neutral manifolds of trajectories.
- Application of the test to three distinct time-delay system examples.
Main Results:
- The developed numerical test successfully confirmed hyperbolicity in two previously studied time-delay systems.
- The test identified a novel example of a time-delay system exhibiting nonhyperbolic chaos.
- Angle distributions provided clear indicators for classifying the hyperbolicity of the tested systems.
Conclusions:
- The numerical angle criterion test is effective for assessing hyperbolicity in chaotic time-delay systems.
- Nonhyperbolic chaos can occur in time-delay systems, highlighting the need for specialized analysis.
- This work offers a valuable tool for the quantitative study of chaotic dynamics in systems with time delays.
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