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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.

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
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