Coupled Van der Pol oscillator | Theory of Hybrid Systems

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Model description

Two Van der Pol oscillators can be coupled directly [1]. The composed model is described by the following 4-dimensional ODE.

$\left\{ \begin{array}{lcl} \dot{x}_1 & = & y_1 \\ \dot{y}_1 & = & (1 - x_1^2)\cdot y_1 - x_1 + (x_2 - x_1) \\ \dot{x}_2 & = & y_2 \\ \dot{y}_2 & = & (1 - x_2^2)\cdot y_2 - x_2 + (x_1 - x_2) \\ \end{array} \right.$

Reachability settings

We consider the initial set $x_1\in [1.25,1.55]$ , $y_1\in [2.25,2.35]$ , $x_2\in [1.25,1.55]$ and $y_2\in [2.25,2.35]$ .

Results

The following figure shows an overapproximation computed by Flow* for the time horizon $[0,7]$ :

References

[1] R. H. Rand and P. J. Holmes. Bifurcation of periodic motions in two weakly coupled Van der Pol oscillators. Volume 15 of International Journal of Non-Linear Mechanics, pages 387–399, Pergamon Press Ltd., 1980.