Inducing Chaos by Resonant Perturbations: Theory and Experiment

We propose a scheme to induce chaos in nonlinear oscillators that either are by themselves incapable of exhibiting chaos or are far away from parameter regions of chaotic behaviors. Our idea is to make use of small, judiciously chosen perturbations in the form of weak periodic signals with time-varying frequency and phase, and to drive the system into a hierarchy of nonlinear resonant states and eventually into chaos. We demonstrate this method by using numerical examples and a laboratory experiment with a Duffing type of electronic circuit driven by a phase-locked loop. The phase-locked loop can track the instantaneous frequency and phase of the Duffing circuit and deliver resonant perturbations to generate robust chaos.

Figure 1

For the periodically forced Duffing oscillator described by Eq. (1) that exhibits a period-1 attractor, (a) change in the frequency in the external resonant perturbation and three representative attractors [in the $(x,dx/dt)$ plane] during the induction of chaos. (b) Evolution of the energy of the oscillator. Here time $n$ denotes the number of cycles of the periodic forcing.

Figure 2

Young-Silva circuit implementation of the Duffing oscillator. A differential amplifier (U5) was added to generate a single output proportional to the voltage across (R7) in order to visualize the phase plot of the voltage at Y1 versus that at Y2 on an oscilloscope.

Figure 3

Experimental bifurcation diagram for the Duffing circuit. The vertical arrow indicates the initial period-1 state of the system that is to be brought into chaos using a resonant perturbation.

Figure 4

(a) Phase-space plot of a periodic attractor from the Duffing circuit, (b) induced chaotic attractor under resonant driving, (c) histogram of the largest Lyapunov exponent from the induced chaotic attractor, and (d) attractor due to noisy driving, which is nonchaotic. The units of $Y_1$ and $Y_2$ in (a), (b), and (d) are in volts.