A study of type I intermittency of a circular differential equation under a discontinuous right-hand side ✩

In this paper we study a circular differential equation under a discontinuous periodic input, developing a quadratic differential equations system on S1 and a linear differential equations system in the Minkowski space M3. The symmetry groups of these two systems are, respectively, PSOo(2, 1) and SOo(2, 1). The Poincaré circle map is constructed exactly, and a critical value αc of the parameter is identified. Depending on α of the input amplitude the equation may exhibit periodic, subharmonic or quasiperiodic motions.When α varies from α >αc to α <αc, there undergoes an inverse tangent bifurcation; consequently, the resultant Poincaré circle map offers one route to the quasiperiodicity via a type I intermittency. In the parameter range of α <αc the orbit generated by the Poincaré circle map is either m-periodic or quasiperiodic when n/m is a rational or an irrational number. © 2006 Elsevier Inc. All rights reserved.