In this paper we study the bifurcation phenomena of quasi–periodic states of a model of the human circadian rhythm, which is described by a system of coupled van der Pol equations with a periodic external forcing term. In the system a periodic or quasi–periodic solution corresponds to a synchronized or desynchronized state of the circadian rhythm, respectively. By using a stroboscopic mapping, called a Poincar mapping, the periodic or quasi–periodic solution is reduced to a fixed point or an invariant closed curve (ab. ICC). Hence we can discuss the bifurcations for the periodic and quasi–periodic solutions by considering that of the fixed point and ICC of the mapping. At first, the geometrical behavior of the 3 generic bifurcations, i.e., tangent, Hopf and period doublig bifurcations, of the periodic solutions is given, Then, we use a qualitative approach to bring out the similar behavior for the bifurcations of the periodic and quasi–periodic solutions in the phase space and in the Poincarsection respectively. At last, we show bifurcation diagrams concerning both periodic and quasi–periodic solutions, in different parameter planes. For the ICC, we concentrate our attention on the period doubling cascade route to chaos, the folding of the parameter plane, the windows in the chaos and the occurrence of the type I intermittency.

We consider a ring of n Rayleigh oscillators coupled hybridly. Using the symmetrical property of the system we demonstrate the degeneracy of the Hopf bifurcation of the equilibrium at the origin. The degeneracy implies the exstence and stability of the n-phase oscillation. We discuss some consequences of the perturbation of the symmetry. Then we study the case n = 3. We show the bifurcation diagram of the equilibria and of hte periodic solutions. Especially, we analyze the mechanism for the symmetry breaking bifurcation of the fully symmetric solution. We report and explain the occurrence of both chaotic attractors and repellors and show two types of symmetry recovering crisis they undergo.

In this paper we study the properties induced by the symmetrical properties of a system of hybridly coupled oscillators of the Rayleigh type on the bifurcations of its equilibria. We first discuss the symmetrical properties of the system. Then we classify the equilibria according to their symmetrical properties. Demonstrating the structural degeneracy of the system, we give the complete stability analysis of the equilibria.

In this paper we study the bifurcations of the periodic solutions induced by the symmetrical properties of a system of hybridly coupled oscillators of the Rayleigh type. By analogy with the results concerning with the equilibria, we classify the periodic solutions according to their spatial and temporal symmetries. We discuss the possible bifurcations of each type of periodic solution. Finally we analyze the phase portraits of the system when the parameters vary.