Van der Pol model in two-delay differential equation representation.

Area of Science:

• Nonlinear Dynamics and Control Systems
• Electrical Engineering and Circuit Theory
• Mathematical Modeling and Simulation

Background:

• The Van der Pol equation, a second-order ODE with cubic nonlinearity, is a fundamental model in nonlinear dynamics.
• Incorporating time delays into the Van der Pol model has been an active area of research, expanding its applicability.
• Resistor-inductor-capacitor (RLC) circuits are essential components in electrical engineering, often modeled by differential equations.

Purpose of the Study:

• To deduce the Van der Pol model and RLC circuit as a delay differential equation (DDE).
• To explore the behavior of a Van der Pol delay model with two delays.
• To investigate the potential application of this model in Parkinson's disease modification.

Main Methods:

• Derivation of delay differential equations for the Van der Pol model and RLC circuit.
• Application of Taylor series expansion to approximate DDEs with ordinary differential equations for small delays.
• Numerical simulations using MATLAB to analyze the behavior of the delay differential equations.

Main Results:

• Successfully formulated the Van der Pol equation and RLC circuit as delay differential equations.
• Demonstrated that the proposed Van der Pol delay model with two delays can be re-used in various applications.
• Numerical simulations visualized diverse dynamic behaviors exhibited by the delay differential equations.

Conclusions:

• The study successfully established a Van der Pol delay model derived from RLC circuits.
• The model's flexibility with two delays opens avenues for diverse applications, including potential relevance to Parkinson's disease.
• Numerical simulations confirm the rich dynamics and potential of these delay differential equations.