Smale-Williams solenoids in autonomous system with saddle equilibrium

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.