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Non-autonomous self-oscillator with hard excitation

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The simplest self-oscillator with hard excitation under external harmonic driving is governed by a following equation:

This is a modification of the Van der Pol - Duffing equation with an addition term in the factor at the first derivative, that of the fourth power. It is controlled by parameter k. In the autonomous system two limit cycles may coexist, one unstable, and another stable. The stable one is of larger diameter, and to reach the associated self-oscillatory regime the system require to get an initial kick of sufficient magnitude. It explains the term 'hard excitation'.

Below the left-hand diagram shows a chart of dynamical regimes for l = 1.0, b=0.34, k=0.1. Observe two distinct sets of synchronization tongues. One is disposed to the right from the main tongue and is arranged along the border corresponding to the saddle-node bifurcation line. The second set of tongues, in the left part of the diagram, has such a peculiarity that the tongues have sharp edges in the top part too. The border of the main tongue in this region represents a bifurcation line of Neimark-Sacker (Andronov-Hopf in the slow-amplitude equations).

The right-hand diagram represents an analogous chart, but obtained in computations in backward time. This trick visualizes unstable periodic regimes (repellers). The main unstable synchronization tongue is seen well (the green one). The stable synchronization regime corresponds on this chart to divergence (orbits running to infiniy) and is shown in gray. vertexs of the stable and unstable tongues are separated along the frequency axis. It is linked with non-isochronous nature of the system, for which the parameter b responses. In the case b=0, the stable and unstable tongues have a common vertex. In the right part of the diagram, one can see a set of tongues of larger periods corresponding to synchronization of regimes associated with an unstable limit cycle.

With increase of k the limit cycles in the autonomous system become closer, then collide at some bifurcation value of the parameter and disappear. It is remarkable that in the non-autonomous system the quasiperiodic regimes do not disappear simultaneously with the limit cycles. This phenomenon was discovered by Mandelstam and Papaleksi yet in 1930-th and called the effect of asynchronous excitation. The figure below illustrates this effect on the parameter plane chart at the parameter value к=0,15, which is larger than the bifurcation one. Observe existence of two notable islands of quasiperiodicity (white), inside which narrow synchronization tongues may be distinguished. In terms of slow-amplitude equations, Andronov-Hopf bifurcation curves are responsible for appearance of the islands, and after their collision the quasiperiodicity disappears ultimately.

Some details of the above picture of synchronization are discussed in the article of A.P.Kuznetsov and S.V.Milovanov: Syncronization in a system with bifurcation of collision of a stable and an unstable limit cycle (in Russian).

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