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Van der Pol oscillator is a basic model of oscillation theory and nonlinear dynamics, which describes self-oscillations and the simplest version of the Andronov - Hopf bifurcation. In presence of external periodic (harmonic) force this system is governed by a following equation:
Здесь l is a control parameter, w – is a frequency of the external driving, b is its amplitude. At l=0 in the autonomous system the Andronov - Hopf bifurcation takes place: at negative l the unique fixed point at the origin is stable, and for positive l it is unstable and enclosed by a stable limit cycle.
The phenomenon of synchronization in the driven Van der Pol oscillator attracts attention of researches starting from pioneering works of the first half of XX century (Appleton, Van der Pol, Andronov, Vitt, Mandelstam, Papaleksi et al.) till the resent time.
The chart of dynamical regimes for the harmonically driven Van der Pol oscillator at l=3 is shown here on the plane of frequency and amplitude of the external force.
It manifests a typical picture of synchronization tongues (Arnold's tongues). Note that internal arrangement of these tongues differs from that for the circle sine map.
Model of Van der Pol - Duffing with external driving is governed by the equation
In this equation an additional term with cubic nonlinearity is included by analogy with the Duffing equation. This nonlinearity is characterized by parameter b and is responsible for a novel effect in the autonomous system, presence of amplitude dependence of the oscillation period, the oscillations are non-isochronous. A principal significance of the Van der Pol - Duffing model is that in the slow amplitude approach it gives rise to a general normal form of the Andronov - Hopf bifurcation. In the equations, the temporal derivative of phase contains a term quadratic in amplitude with coefficient b. So, b may be called phase nonlinearity parameter.
The following diagram shows a chart of regimes of the non-autonomous Van der Pol - Duffing oscillator for l=1 at a sufficiently large b=2.5. Observe that in comparison with the previous chart the edges of synchronization tongues are shifted notably along the horizontal axis, that reveals the non-isochronous nature of the system. The set of tongues between the regions of period 1 and 2 manifests a pronounced internal structure with period-doubling transition to chaos inside the tongues. It looks similar to that in the circle map.
It is remarkable that the model of Van der Pol - Duffing continues to attract interest of researchers (for example, details of the bifurcation picture were discovered in the very end of the XX century).
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