Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit.

Area of Science:

• Nonlinear Dynamics
• Electronic Circuit Analysis
• Chaos Theory

Background:

• Chaotic phenomena in electronic circuits are increasingly important for various applications.
• Robust chaos generation is crucial but difficult to prove, often relying on numerical simulations.
• Existing studies lack rigorous proofs of chaos robustness in simple circuits.

Purpose of the Study:

• To rigorously prove the robustness of chaos in a specific electronic switching circuit.
• To determine the precise parameter ranges that yield chaotic dynamics.
• To provide a theoretical foundation for reliable chaos generation in electronic systems.

Main Methods:

• Modeling the electronic switching circuit as a two-dimensional piecewise smooth map.
• Associating the system with a one-dimensional piecewise linear map for analysis.
• Analytical determination of bifurcations in the parameter space.

Main Results:

• Identification of parameter ranges for robust chaotic dynamics.
• Analytical characterization of border collision, degenerate flip, and homoclinic bifurcations.
• Demonstration of chaos arising from cyclical chaotic regions.

Conclusions:

• The electronic switching circuit can reliably generate robust chaos.
• Analytical methods provide rigorous proof of chaos robustness.
• The findings contribute to the theoretical understanding and practical application of chaotic electronic circuits.