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
_c
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.