Exact analytic solution for a chaotic hybrid dynamical system and its electronic realization.

Area of Science:

• Dynamical Systems
• Chaos Theory
• Nonlinear Dynamics

Background:

• Hybrid dynamical systems combine continuous and discrete dynamics.
• Chaotic dynamics arise in various complex systems.
• Understanding chaotic behavior is crucial for control and prediction.

Purpose of the Study:

• Introduce a novel hybrid dynamical system.
• Analyze its chaotic dynamics.
• Provide an exact analytic solution and practical demonstration.

Main Methods:

• Developed a hybrid system with an unstable filter and asynchronous set-point switching.
• Derived a one-dimensional return map (tent function) from sampled continuous states.
• Employed nonlinear transformation for an exact analytic solution.
• Demonstrated conjugacy to symbolic dynamics and circuit realization.

Main Results:

• The hybrid system exhibits chaotic dynamics.
• An exact analytic solution was derived.
• The system's behavior is conjugate to symbolic dynamics.
• A practical electronic circuit implementation was successful.

Conclusions:

• The novel hybrid system effectively demonstrates chaotic dynamics.
• The derived analytic solution offers deep insights into the system's behavior.
• The successful circuit realization validates the theoretical model and suggests practical applications.