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Maximum performance at minimum cost in network synchronization. (English) Zbl 1117.34048

The subject of the paper is the network of coupled identical oscillators

x ˙ i =F(x i )+σ j=1 n A ij [H(x j )-H(x i )],i=1,,n,

where σ is the parameter which controls the overall coupling, x i m is the phase variable of i-th oscillator, A ij 0, is the relative strength of the coupling, and H: m m is the output function.

The following optimization problems are considered: 1. Maximization of synchronizability, i.e. the maximization of the region of parameter σ, for which stable synchronization occurs. 2. Minimization of the synchronization cost, which is defined as the sum of the total input strength of all nodes at the lower synchronization threshold σ min : C=σ min i,j=1 n A ij .

The main result shows that the solution sets for these two optimization problems coincide and are characterized by the following condition on the Laplacian (i.e. the matrix (A ij )) eigenvalues:

0=λ 1 <λ 2 ==λ n ·

The authors study such optimal networks and drive some useful conclusions about their structure.

34D05Asymptotic stability of ODE
34C15Nonlinear oscillations, coupled oscillators (ODE)
34C30Manifolds of solutions of ODE (MSC2000)
34C28Complex behavior, chaotic systems (ODE)
34D08Characteristic and Lyapunov exponents
49N99Miscellaneous topics in calculus of variations