Self-Adaptive Attractor-Shaping for Oscillators Networks

We consider a network of N coupled limit cycle oscillators, each having a set of control parameters Λ_k, k = 1, . . . , N, that controls the frequency and the geometry of the limit cycle. We implement a self-adaptive mechanism that drives the local systems to share a common set of parameters Λ_c. This situation therefore strongly differs from classical synchronization problems where the Λ_k are kept constant. The deformations of the Λ_k towards the consensual Λ_k are “plastic” - once Λ_c is reached, it remains permanent even in absence of interactions. Again, this has to be contrasted with classical synchronization which does not affect the Λ_k (in the absence of interactions, individual behaviors are restored). The resulting consensual Λ_c can be analytically characterized. In general, the set of initial conditions from which Λ_c is reached depends on the network topology. The class of models discussed here unveil the basic features necessary to construct a wider class of dynamical system sharing self-adaptive attractor-shaping capability. Finally, we present numerical simulations that corroborate our theoretical assertions.