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Existence of an energy function for three-dimensional chaotic "sink-source" cascades
M Barinova1, V Grines1, O Pochinka1
1Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod)/Department of Fundamental Mathematics, National Research University Higher School of Economics, Nizhny Novgorod 603155, Russia.
Chaos (Woodbury, N.Y.)
|July 9, 2021
Summary
Researchers constructed an energy function for discrete dynamical systems. This function, a smooth Lyapunov function, applies to specific chaotic systems on 3D manifolds, aiding in stability analysis.
Area of Science:
- Dynamical Systems Theory
- Differential Geometry
- Mathematical Physics
Background:
- Research continues on constructing energy functions for discrete dynamical systems.
- Energy functions, as smooth Lyapunov functions, are crucial for analyzing system stability.
- Previous work focused on defining critical points coinciding with the chain recurrent set.
Purpose of the Study:
- To establish the existence of an energy function for a specific class of discrete dynamical systems.
- To extend the concept of energy functions to systems with complex non-wandering sets.
Main Methods:
- Utilized concepts from differential geometry and dynamical systems theory.
- Focused on A-diffeomorphisms of three-dimensional closed orientable manifolds.
- Analyzed systems where the non-wandering set comprises a chaotic one-dimensional surface attractor and repeller.
Main Results:
- Proved the existence of an energy function for the studied class of systems.
- Demonstrated that this energy function's critical points align with the chain recurrent set.
- The findings apply to systems with a specific chaotic attractor-repeller structure.
Conclusions:
- The existence of energy functions is confirmed for discrete dynamical systems with complex chaotic attractors.
- This research advances the theoretical framework for analyzing stability in such systems.
- The constructed energy functions provide valuable tools for understanding the dynamics of chaotic systems.
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