Self-adaptive attractor-shaping for oscillators networks

We consider a network of N coupled limit cycle oscillators, each having a set of control parameters Λk, 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 Λc. This situation therefore strongly differs from classical synchronization problems where the Λk are kept constant. The deformations of the Λk towards the consensual Λc are “plastic” — once Λc is reached, it remains permanent even in absence of interactions. Again, this has to be contrasted with classical synchronization which does not affect the Λk (in the absence of interactions, individual behaviors are restored). The resulting consensual Λc can be analytically characterized. In general, the set of initial conditions from which Λc 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.