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Frequency plateaus in a chain of weakly coupled oscillators. I. (English) Zbl 0558.34033
This paper studies equations of the form $(1)\quad X'_ k=F(X_ k)+\epsilon R_ k(X_ k,\epsilon)-\epsilon D(X_{k+1}-2\gamma X_ k+X_{k-1})\quad X_ 0=0=X_{N+2}$ where $$X_ k\in R_ m$$, $$F: R^ m\to R^ m$$, D is an (m$$\times m)$$ matrix, $$\gamma =0$$ or 1, $$k=1,...,n+1$$, and $$\epsilon \ll 1$$. $$X'_ k=F(X_ k)$$ is assumed to have a stable limit cycle. It is shown that (1) has a stable invariant torus of dimension $$n+1$$, on which the equations have the form $\theta '_ 1=\omega_ 1+\epsilon H(\Phi_ 1)+O(\epsilon^ 2)\quad \Phi '_ k=\epsilon [\Delta_ k+H(\Phi_{k+1})+H(-\Phi_ k)-H(\Phi_ k)-H(- \Phi_{k-1})]+O(\epsilon^ 2)$ $H(-\Phi_ 0)=0=H(\Phi_{N+1}).$ Here H is $$2\pi$$-periodic, $$\omega_ 1$$ is the frequency of the first oscillator, $$\theta_ k$$ is the phase of the kth oscillator, $$\Phi_ k(resp$$. $$\Delta_ k)$$ is the phase difference (resp. the frequency difference) between the kth and $$(k+1)th$$ oscillators. For $$\epsilon$$ small, phaselocking exists if there is a stable critical point of an almost invariant n-dimensional system parameterized by $$\Phi_ 1,...,\Phi_ N$$. Most of the paper deals with the special case $$H=\sin \Phi$$ and $$\Delta_ k\equiv \Delta$$ (a linear frequency gradient). It is shown that if $$\Delta$$ is increased until phaselocking is no longer possible, there emerges a large-scale invariant circle in this N- dimensional system, which corresponds to the existence of a pair of ”plateaus” on which the frequency of the coupled oscillators is independent of k, and whose homotopy class within the N-torus corresponds to the position of the frequency jump. Phaselocking properties of equations of the form (1), with general H, (rather than $$H=\sin \Phi)$$, have recently been studied (the authors, ”Symmetry and phaselocking in chains of coupled oscillators”, to appear). It is shown that the anti- symmetric symmetry of sin $$\Phi$$ implies that, if $$H=\sin \Phi$$, equations (2) lose phaselocking for a smaller frequency gradient than if H lacks this symmetry. The frequency at which the oscillators phaselock (in the presence of a small enough gradient to allow this) is also different.

##### MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
##### Keywords:
oscillator; stable critical point; Phaselocking properties
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