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On hyperbolic attractors in a modified complex Shimizu-Morioka system
Vyacheslav Kruglov1, Igor Sataev1
1Kotelnikov's Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, Zelenaya 38, Saratov 410019, Russia.
Chaos (Woodbury, N.Y.)
|June 12, 2023
Summary
Researchers modified a complex-valued Shimizu-Morioka system, creating a uniformly hyperbolic attractor. This new system, unlike the original Lorenz attractor, exhibits unique expansion and contraction properties, demonstrating a novel dynamical system behavior.
Area of Science:
- Dynamical Systems
- Chaos Theory
- Nonlinear Dynamics
Background:
- The Shimizu-Morioka system is known for its complex dynamics and Lorenz-like attractor.
- Uniformly hyperbolic attractors are fundamental in understanding chaotic systems, characterized by exponential expansion and contraction rates.
- Previous studies focused on systems with genuine Lorenz attractors, leaving room to explore modified systems.
Purpose of the Study:
- To introduce and analyze a modified complex-valued Shimizu-Morioka system.
- To investigate the nature of the attractor in the modified system, specifically looking for uniform hyperbolicity.
- To compare the dynamics of the modified system with the original Shimizu-Morioka system and the Smale-Williams solenoid.
Main Methods:
- Numerical simulations of the modified complex-valued Shimizu-Morioka system.
- Analysis of the attractor's behavior in the Poincaré cross section.
- Numerical verification of the transversality of tangent subspaces for the flow and its Poincaré map.
Main Results:
- The modified system exhibits a uniformly hyperbolic attractor, distinct from the original Lorenz-type attractor.
- The attractor shows significant expansion in the angular direction and strong contraction in transversal directions, resembling the Smale-Williams solenoid.
- Numerical tests confirmed the transversality of tangent subspaces, a key characteristic of uniformly hyperbolic attractors.
- No genuine Lorenz-like attractors were observed in the modified system.
Conclusions:
- The modified complex-valued Shimizu-Morioka system provides the first example of a system with a uniformly hyperbolic attractor derived from a system with a genuine Lorenz attractor.
- The observed dynamics, characterized by expansion and contraction, align with the properties of uniformly hyperbolic systems.
- This study opens new avenues for exploring modified chaotic systems and their unique attractor structures.
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