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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References:

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