Stabilization of discrete breathers using continuous feedback control

Abstract

Localized oscillations in 1- and 2-dimensional nonlinear lattices are by now recognized as a widely occurring phenomenon with applications to many problems of physical and biological interest. When the spatial distribution of the amplitudes of these oscillations possesses a single extremum they are called discrete breathers and are known to be stable for a sizable range of the inter-particle coupling parameter α⩾0. However, when the amplitudes possess more than one (local) extremum they are called multibreathers and are generally unstable for all α>0. In this Letter, we demonstrate that it is possible to stabilize breathers and multibreathers of a 1-dimensional chain with quartic on-site potential, using a continuous feedback control (CFC) method, originally proposed by Pyragas. CFC is called conservative if it preserves the Hamiltonian nature of the system and dissipative if it introduces loss terms proportional to the velocity of the particles. As is well-known from low-dimensional examples CFC works by inducing pitchfork, period-doubling or transcritical bifurcations to the unstable periodic orbits under study. Here, we demonstrate, by computing the eigenvalues of the corresponding monodromy matrix and following the phase space oscillations of the particles that CFC stabilizes our breathers and multibreathers also via low-dimensional bifurcations.