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Synchronization of oscillators coupled through an environment. (English) Zbl 1184.34060

The author considers synchronized solutions in a system of \(n\) identical oscillators \(x_k\in\mathbb{R}^d\) coupled through an additional vector of environment \(y\in\mathbb{R}^p\) as follows
\[ \dot{x}_k(t)=f(x_k(t),y(t)),\quad k=1,\dots,n,\qquad \dot{y}(t)=g(y(t))+\frac{\beta}{n}\sum_{j=1}^n h(x_j(t),y(t)), \tag \(*\) \]
where \(f\), \(g\) and \(h\) are smooth functions. By a synchronized solution of the system \((*)\) one means a solution periodic with period \(T>0\)
\[ x_k(t+T)=x_k(t),\quad k=1,\dots,n,\qquad y(t+T)=y(t) \]
such that all the \(x_k\) are identical
\[ x_1(t)=x_2(t)=\dots=x_n(t). \]
Following the traditional scheme of linear stability analysis and exploiting the symmetry of the system with respect to permutation of the oscillators, a criterion for stability of the synchronized solution is established. Besides, the author obtains sufficient conditions for the existence and stability/instability of a synchronized solution to system \((*)\) with weak coupling to environment (i.e. small \(|\beta|\)), in the case when the equation \(\dot{y}=g(y)\) has a stable steady state \(\bar{y}\), and the equation \(\dot{x}=f(x,\bar{y})\) has a non-degenerate periodic solution. The results are illustrated by their applications to a model of coupled neuron oscillators, and to indirectly weakly coupled \(\lambda-\omega\) oscillators.

MSC:

34D06 Synchronization of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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