##
**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.

\[ H(-\Phi_ 0)=0=H(\Phi_{N+1}). \] 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.