Duffing Equation




After downloading duffing.jar, please execute it by double-clicking, or typing "java -jar duffing.jar".

If the above application does not start, please install Java from www.java.com.

The points which show the states of pendulums are plotted in the right field every period. If you wait about 10 minutes, you can see the ordered figure.


Here we treat the Duffing equation written as

d2x /dt2 = - δ dx/dt + x - x3 + γ cos(ω t),

where δ=0.20, γ=0.30, ω=1.0. This simulator solves this equation numerically and visualize it as oscillator's motion. Two independent oscillators with different initial conditions are drawn in the same field. Because of the chaotic properties of the derived solution, two oscillators' motions are separated from each other.
Embedding this solution to the three dimensional space (x, dx/dt, t) and cutting the trajectory with a plane, you can see the periodical motion of chaos attractor as shown in "Duffing equation" animation.
The data (x(nT),dx(nT)/dt) (n=0,1,2,...) sampled every T=2 π/ω are shown in the right field of this simulator. After a while, a strange attractor will be observed.

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Introduction to Chaos and Nonlinear Dynamics