Vol. 7, Issue 2, Part B (2025)

Chaotic behavior in nonlinear dynamical systems has long intrigued mathematicians, physicists, and engineers due to its sensitive dependence on initial conditions and long-term unpredictability despite deterministic governing laws. The numerical investigation of chaos not only deepens our understanding of complex dynamical systems but also provides tools for applications in secure communications, climate modeling, biological systems, and control engineering. This study presents a comprehensive numerical exploration of chaos in classical and modern nonlinear systems, including the Lorenz system, Rössler attractor, Duffing oscillator, and Chua's circuit.
We employ various numerical tools—such as phase space reconstruction, bifurcation diagrams, Poincaré sections, and Lyapunov exponent calculations—to identify and quantify chaotic behavior. Time series analysis, numerical simulations using the Runge-Kutta method, and spectrum analysis via Fast Fourier Transform (FFT) provide insights into the route to chaos through period doubling and quasi-periodicity. The role of control parameters and initial conditions is critically examined to understand system bifurcations.
The results confirm that minor variations in parameters can lead to a transition from order to chaos. Lyapunov exponents serve as a quantitative measure for distinguishing between chaotic and periodic regimes. Our study compares several systems numerically and demonstrates how different chaos indicators complement each other in identifying and understanding chaotic regimes.
The article concludes with a discussion of the implications of chaos in real-world systems and outlines emerging research directions, including chaos control and synchronization in higher-dimensional systems. This work contributes to a better understanding of the complex behavior of nonlinear systems and provides robust tools for numerical chaos analysis.