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Partial synchronization in linearly and symmetrically coupled ordinary differential systems. (English) Zbl 1167.37019

The paper studies the following general model of a coupled system composed from linearly and symmetrically coupled ordinary differential equations:

$\frac{d{x}^{i}\left(t\right)}{dt}=f\left({x}^{i}\left(t\right),t\right)+\epsilon \sum _{j=1}^{m}{a}_{ij}{\Gamma }{x}^{j}\left(t\right),\phantom{\rule{1.em}{0ex}}i=1,2,\cdots ,m,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $m>1$ is the network size, ${x}^{i}\in {ℝ}^{n}$ is the state variable of the $i$-th oscillator, $t\in \left[0,+\infty \right)$ is a continuous time, $f:{ℝ}^{n}×\left[0,+\infty \right)\to {ℝ}^{n}$ is a continuous map, $A=\left({a}_{ij}\right)\in {ℝ}^{m×m}$ is a coupling matrix with ${a}_{ij}={a}_{ji}$ and $-1\le {a}_{ij}\le 1$, for all $i,j=1,\cdots ,m$, which is determined by the topological structure of the network, $\epsilon >0$ is the coupling strength, and ${\Gamma }=\text{diag}\phantom{\rule{4.pt}{0ex}}\left\{{\gamma }_{1},{\gamma }_{2},\cdots ,{\gamma }_{n}\right\}$ with ${\gamma }_{i}\ge 0$ for all $i=1,2,\cdots ,n$, and ${\sum }_{i=1}^{n}{\gamma }_{i}>0$. The synchronization phenomena in system (1) are investigated via invariant synchronization manifolds. By means of decomposing the whole space into a direct sum of the synchronization manifold and the transverse space, several criteria for the global asymptotic attractiveness of the invariant synchronization manifold are given. Combining these criteria with some numerical examples, it is shown how topological structure affects partial synchronization. A valuable discussion about the possibility of partial synchronization with increasing coupling strength $\epsilon$ is presented. The results on simulations of several numerical examples (coupled 3-D neural networks, coupled Chua circuits, and coupled Lorenz oscillators) are given.

##### MSC:
 37D10 Invariant manifold theory 34D05 Asymptotic stability of ODE