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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 $x_i(t)\in \bbfR^n$, $A=(a_{ij})\in \bbfR^{n\times n}$ is the coupling matrix with $a_{ij}\ge 0$ for $i\ne j$, 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 $S=\{x: x_i=x_j, i\ne j\}$. 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 [{\it L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar} and {\it J. F. Heagy}, Chaos 7, 520--543 (1997; Zbl 0933.37030)]. The paper presents two examples of linearly coupled neural networks.

MSC:
34D05Asymptotic stability of ODE
34C15Nonlinear oscillations, coupled oscillators (ODE)
34C30Manifolds of solutions of ODE (MSC2000)
34C14Symmetries, invariants (ODE)
34C28Complex behavior, chaotic systems (ODE)
34C60Qualitative investigation and simulation of models (ODE)
34D20Stability of ODE
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