Book Chapter

Self-shaping attractors for coupled limit cycle oscillators

We consider a complex network of N diffusively coupled stable limit cycle oscillators. Each individual system has its own set of local parameters Λ, characterizing its frequencies and the shape of limit cycle. The Λ are allowed, thanks to appropriate interactions, to self-adapt. The self-adaptive mechanism ultimately drive all oscillators to a consensual dynamical state where all local systems share a common and constant consensual set of parameters Λ. Interactions are implemented via a coupling matrix whose spectral properties characterize the convergence conditions leading to a consensual state. Convergence, which is due to the dissipative character of the dynamics, gives rise to the "plastic" deformations of Λ towards Λ. Once reached, the consensual oscillatory state remains invariant even if interactions are removed (i.e. plasticity). This situation therefore strongly differs from classical synchronization problems where the Λ are kept constant (in the absence of interactions, individual behaviors are restored). The resulting Λ is analytically calculated and, in our class of models, their values do not depend on the networks topology. However, the network's connectivity, characterized by the Fiedler number, affects the convergence rate. Finally, we present numerical simulations that corroborate our theoretical assertions.


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