Exactly solvable chaos in an electromechanical oscillator

Area of Science:

• Physics
• Engineering
• Dynamical Systems

Background:

• Chaotic oscillators are crucial in various scientific fields.
• Developing systems with exact analytical solutions remains a significant challenge.
• Hybrid dynamical systems offer complex behaviors but are often difficult to analyze.

Purpose of the Study:

• To introduce a novel electromechanical chaotic oscillator.
• To derive an exact analytic solution for its chaotic oscillations.
• To characterize the system's chaotic behavior and compare it to established models.

Main Methods:

• Modeling a hybrid dynamical system with a linear ODE and discrete switching.
• Deriving exact analytical solutions using linear convolution.
• Analyzing chaotic properties through waveform return maps.

Main Results:

• The oscillator admits an exact analytic solution.
• The system generates chaotic oscillations topologically similar to Lorenz or Rössler attractors.
• Analytical solutions closely match physical oscillations.
• Return maps confirm chaotic dynamics, resembling shift or tent maps.

Conclusions:

• The novel electromechanical oscillator provides a rare example of a chaotic system with an exact solution.
• The findings validate the analytical model against physical behavior.
• This work offers new insights into the analysis of hybrid chaotic systems.