Design of Complex Oscillator Network with Multiple Limit Cycles

Abstract

Rhythmic behaviors are widely observed in animal motions. A fundamental control mechanism for producing and regulating rhythmic movements is based on the Central Pattern Generator (CPG), which is a distributed network of neuronal oscillators. Mathematical models of CPGs can be useful as a basic component in feedback control designs to achieve oscillations. This paper develops a CPG model as a network of nonlinear oscillators described by ordinary differential equations with complex variables. The use of complex state variables simplifies the CPG design and makes it transparent how the network connectivity relates to the resulting oscillation pattern. We will provide a method for designing CPGs to achieve oscillations of prescribed frequency, amplitude, and phase as a stable limit cycle with guaranteed local convergence. Moreover, we show how the network can be designed to embed multiple limit cycles in the state space.