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Smale-Williams solenoids in autonomous system with saddle equilibrium.

S P Kuznetsov1, V P Kruglov1, I R Sataev1

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Summary

Researchers created a new chaotic system using complex variables, revealing a Smale-Williams type attractor. This model, resembling wave dynamics, offers insights into complex systems and chaos theory.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Nonlinear Dynamics
  • Mathematical Physics

Background:

  • Self-oscillating systems with homoclinic loops are fundamental in understanding complex dynamics.
  • Saddle equilibria with specific eigenvalue properties are crucial for chaotic behavior.
  • The Smale-Williams attractor is a canonical example of a uniformly hyperbolic chaotic attractor.

Purpose of the Study:

  • To construct a novel low-dimensional autonomous system exhibiting chaotic dynamics.
  • To investigate the formation of a Smale-Williams type chaotic attractor using complex-valued variables.
  • To analyze the role of saddle equilibria and Bernoulli maps in generating chaos.

Main Methods:

  • Replacement of real-valued variables with complex-valued variables in a self-oscillating system.
  • Analytical and numerical investigations of the system's phase space dynamics.
  • Examination of trajectory behavior near saddle equilibria and in Poincaré cross sections.

Main Results:

  • Construction of a four-dimensional system with a uniformly hyperbolic chaotic attractor of Smale-Williams type.
  • Demonstration that complex variable arguments undergo a Bernoulli map near the saddle equilibrium.
  • Identification of distinct regions in the phase space contributing to the attractor's formation.

Conclusions:

  • The proposed model successfully generates a Smale-Williams type chaotic attractor.
  • The use of complex variables offers a new perspective on constructing chaotic systems.
  • The model's resemblance to complex amplitude equations suggests broad applicability in wave dynamics and related fields.