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Diffusive coupling, dissipation, and synchronization. (English) Zbl 1091.34532
The author investigates the phenomenon of synchronization in differential equation models with diffusive coupling. The concept of synchronization can be described through the following example: Consider $$n$$ subsystems of differential equations $$\dot z_j= f_j(z_j), z_j\in \mathbb R^N, 1\leq j \leq n$$. The vector fields $$f_j$$, $$1\leq j \leq n$$, could be close to one another or not. Now suppose that these subsystems are coupled to obtain the system $$\dot z = A(k)z + f(z)$$ where $$z=(z_1, \dots , z_n)^\top$$, $$f(z)=(f_1(z_1), \dots , f_n(z_n))^\top$$ and the coupling matrix $$A(k)$$ is a real and symmetric $$n\times n$$-matrix depending on the coupling parameter $$k=(k_1,\dots ,k_p)$$. Roughly speaking, synchronization occurs (for some value of $$k$$) if as time evolves the solutions of the coupled system (we are only interested in bounded solutions) approach the diagonal $$\{z\in \mathbb R^{nN}:\;z_1=\cdots = z_n\}$$. When for some $$l<n$$ and index set $$\{j_1,\dots , j_ l\}\subset \{1,2,\dots ,n\}$$ the solutions of the coupled system approach the set $$\{z\in \mathbb R^{nN}: z_{j_1}= \cdots = z_{j_ l}\},$$ we say that a partial synchronization occurs. The coupling parameter $$k$$ acts as a control and given a coupled system the goal is to find values of $$k$$ for which synchronization is observed.
Here, the author aims to give a unified framework to treat the problem of synchronization which comprises many of the previous efforts and includes examples of ordinary, partial and functional-differential equations. The problem of finding sharp estimates on the values of the parameter for which synchronization is observed is not of concern here. The paper has many interesting examples in ordinary, partial and functional-differential equations. The author also surveys most of the literature in the subject.

##### MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 34K12 Growth, boundedness, comparison of solutions to functional-differential equations 35K57 Reaction-diffusion equations
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