Global view on a nonlinear oscillator subject to time-delayed feedback control
In this paper the authors present a comprehensive bifurcation analysis of a generic nonlinear oscillator model subjected to time-delayed feedback control. In particular, they have demonstrated how the increasing complexity of the solution structure is organized by saddle-node bifurcations, giving rise to an increasing number coexisting control modes. While such a bifurcation analysis is not able to capture, a priori, global stability properties like basins of attraction, they have obtained a global overview of the local linear stability properties of all the states involved in time-delayed feedback control. Thus, an appropriate adiabatic shift of control parameters yields successful control of the desired target state whenever a periodic output signal has been generated. At the upper boundary of the control domain a stable quasiperiodic torus solution is generated. On further increase of the control amplitude such a state is destabilised in a torus or a period-doubling bifurcation, depending on the control phase. No phase locking occurs in this model due to the rotational symmetry. For large values of the control amplitude no stability region is found, neither for periodic nor for quasiperiodic solutions. The sequence of Hopf bifurcations for the Pyragas mode and the first control mode indicate a higher-dimensional unstable manifold and a complex dynamical behaviour is expected to occur. From the plain theoretical point of view the model shows quite interesting bifurcations of higher codimension, subjected to a symmetry which forces the genericity of the transcritical bifurcation. Applying the same type of analysis of the model of time-delayed feedback control can uncover the yet unknown global properties of the control scheme. But we expect that our bifurcation diagrams exhibit a substantial degree of structural stability. In particular, the identification of large stability regions with regard to control parameters which are directly accessible in applications, and the transitions involved at the stability boundary and their physical manifestation can significantly facilitate experimental confirmations.
scientific article; zbMATH DE number 5608111