Visualizing Chaos and Critical Chaos in the Duffing Chaotic System

In this article, we investigate the chaotic and critical chaotic behaviors of the Duffing system. The Duffing oscillator represents a vibration system described by a first-order nonlinear ordinary differential equation, incorporating both a linear dissipative term and a nonlinear restoring force. This system serves as a fundamental example of chaotic systems, capable of modeling various natural phenomena such as spring-mass oscillators and electronic circuits. Chaotic phenomena refer to unpredictable, random motion patterns arising from minute variations in initial conditions within seemingly stable systems. Critical chaos describes chaotic behavior emerging at critical system states, exhibiting significant similarities to phase transition phenomena. From an implementation perspective, the Duffing equation can be numerically solved using methods like the fourth-order Runge-Kutta algorithm. Key parameters include damping coefficient δ, nonlinear stiffness β, and driving force amplitude γ. Phase portraits and Poincaré sections serve as primary visualization tools for analyzing chaotic trajectories. Critical chaos analysis typically involves bifurcation diagrams to observe system behavior transitions near critical points. Understanding chaotic and critical chaotic patterns in the Duffing system provides crucial insights for both chaos theory and phase transition research, particularly in characterizing sensitivity to initial conditions and identifying regime transitions through computational simulations.