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Zbl 1105.34031
Lu, Wenlian; Chen, Tianping
New approach to synchronization analysis of linearly coupled ordinary differential systems. (English)
[J] Physica D 213, No. 2, 214-230 (2006). ISSN 0167-2789

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.
[Sergiy Yanchuk (Berlin)]

MSC 2000:: *34D05 Asymptotic stability of ODE
34C15 Nonlinear oscillations of solutions of ODE
34C30 Manifolds of solutions of ODE
34C14 Symmetries, invariants
34C28 Other types of "recurrent" solutions of ODE
34C60 Applications of qualitative theory of ODE
34D20 Lyapunov stability of ODE

Keywords: linearly coupled ordinary differential equations; synchronization manifold; stability

Citations: Zbl 0933.37030

Cited in: Zbl 1220.34075 Zbl 1111.34056

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