Simulation of the New Chen Time-Delay Chaotic System

Simulation methodology for the New Chen time-delay chaotic system

Chaotic systems hold significant research value in nonlinear dynamics, with time-delay chaotic systems attracting particular attention due to their complex dynamic characteristics. The New Chen time-delay chaotic system is an enhanced model based on the traditional Chen system, incorporating time-delay components that exhibit richer chaotic behaviors.

For beginners, simulating such systems requires understanding several key aspects. First, one must grasp the fundamental mathematical model, including state equations and the representation of time-delay terms. Second, selecting appropriate numerical integration methods (such as the fourth-order Runge-Kutta method) for solving the equations is crucial, while paying attention to the computational complexity introduced by time-delay components. The implementation typically involves creating a function that handles the delayed state variables using history arrays or specialized delay differential equation (DDE) solvers. Finally, tools like bifurcation diagrams and Lyapunov exponents can be employed to analyze the system's chaotic characteristics. The Lyapunov exponent calculation may involve linearized system equations and orthogonalization procedures to track divergence rates in different directions.

During simulation, it's recommended to start with lower-dimensional simplified models and gradually increase complexity. Visualizing phase portraits and time series plots helps in intuitively observing system behavior. Code implementation often includes plotting commands to generate these visualizations alongside the numerical integration routine, allowing for real-time observation of chaotic dynamics.