## New approach to synchronization analysis of linearly coupled ordinary differential systems.(English)Zbl 1105.34031

The authors consider the following linearly coupled system of ordinary differential equations $\frac{dx_i(t)}{dt} = f(x_i(t),t)+\sum_{j=1}^{n}a_{ij}\Gamma x_j(t), \quad i=1,\dots,n,$ where $$x_i(t)\in \mathbb{R}^n$$, $$A=(a_{ij})\in \mathbb{R}^{n\times n}$$ is the coupling matrix with $$a_{ij}\geq 0$$ for $$i\neq j$$, and $$\Gamma = \mathrm{diag}\{ \gamma_1,\gamma_2,\dots,\gamma_n\}$$ with $$\gamma_i>0$$. The main result of the paper is a number of criteria for synchronization of the above system, i.e., for the stability of the invariant subspace $$S=\{x: x_i=x_j, i\neq j\}$$. Using Lyapunov function approach, conditions for global synchronization are given. Presented criteria for local synchronization are mathematically rigorous reformulation of the so-called master stability function approach [L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar and J. F. Heagy, Chaos 7, 520–543 (1997; Zbl 0933.37030)].
The paper presents two examples of linearly coupled neural networks.

### MSC:

 34D05 Asymptotic properties of solutions to ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34C30 Manifolds of solutions of ODE (MSC2000) 34C14 Symmetries, invariants of ordinary differential equations 34C28 Complex behavior and chaotic systems of ordinary differential equations 34C60 Qualitative investigation and simulation of ordinary differential equation models 34D20 Stability of solutions to ordinary differential equations

Zbl 0933.37030
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