Duffing oscillator is a classic example used to demonstrate chaotic behavior and dynamics. It is represented in mathematical form as follows (the details of the form and symbols may vary depending on the documents, but overall form would be similar to this).

The Duffing oscillator is a well-known example of a system used to demonstrate chaotic behavior and dynamics. It is represented mathematically by a second-order differential equation that incorporates terms for displacement, velocity, and nonlinear restoring forces. The general form of the equation is:

The meaning of each terms and coefficients are as follows :

x = displacement

dx/dt = velocity

d²x/dt² = acceleration

δ = damping coefficient

ω0 = natural frequency of oscillation

β = nonlinearity coefficient

γ = forcing amplitude

ω = forcing frequency

φ = phase shift of forcing function

Here, each term and coefficient has a specific meaning. The variable x represents displacement, dx/dt represents velocity, and d²x/dt² represents acceleration. The parameter δ is the damping coefficient, which quantifies the system's resistance to motion. The terms β and ω₀ describe the nonlinear and natural frequency of oscillation, respectively. The forcing term γ cos(ωt + φ) accounts for an external periodic force, where γ is the forcing amplitude, ω is the forcing frequency, and φ is the phase shift of the forcing function.

The characteristics of the Duffing oscillator arise from the nonlinearity introduced by the x³ term. This nonlinearity enables the system to exhibit chaotic behavior under certain conditions. Depending on the parameter ranges, the oscillator may display bounded, periodic oscillations or chaotic dynamics. Varying the forcing frequency ω relative to the natural frequency ω₀ has a significant impact on the behavior of the system. As parameters are adjusted, the oscillations may switch between regular periodic motion and chaotic trajectories.

An important feature of the Duffing oscillator is its sensitivity to initial conditions. Even slight changes in the starting values can result in vastly different trajectories over time. The phase shift φ of the external forcing function plays a crucial role in determining when chaotic regimes emerge.

To explore the behavior of the Duffing oscillator, tools like phase portraits and Poincaré sections can be used. These visualizations reveal the underlying attractors of the system and help illustrate its transition between order and chaos. By numerically solving the equation and experimenting with different parameters and initial conditions, one can observe how the system evolves and how its dynamics change in response to external influences.

The characteristics of this equation can be summarized as follows :

• The x3 term introduces nonlinearity which enables chaotic dynamics.
• The system exhibits chaotic, bounded oscillations for certain parameter ranges.
• Varying the forcing frequency ω relative to the natural frequency ω0 impacts the behavior.
• The oscillations switch between periodic and chaotic as parameters change.
• Slight changes in initial conditions yield dramatically different trajectories over time.
• The phase shift φ of the external forcing alters when chaotic regimes emerge.
• Tools like phase portraits and Poincar sections reveal the underlying attractor.

You can plot out the solution of this equation using a simple script below. Play with parameters and initial conditions and see how the plot changes.