We propose an equivalent circuit model described by the Rössler equation. Then we can consider a coupled Rössler system with a physical meaning on the connection. We consider an oscillatory circuit such that two identical Rössler circuits are coupled by a resistor. We have studied three routes to entirely and almost synchronized chaotic attractors from phase-locked periodic oscillations. Moreover, to simplify understanding of synchronization phenomena in the coupled Rössler system, we investigate a mutually coupled map that shows analogous locking properties to the coupled Rössler System.

We investigate bifurcations of burst oscillations with rectangular waveform observed in a modified Bonhöffer-van der Pol equation, which is considered as a circuit model for neurons of a feeding rhythm generator. In particular, we clarify a mechanism of properties in a one-parameter graph on the period of oscillations, showing a staircase with hysteresis jumps, by studying a successive bifurcation process including a chain of homoclinic bifurcations. The occurrence of homoclinic bifurcations is confirmed by using the linking number of limit cycles related with the stable manifold through an equilibrium.

Bifurcations of quasi–periodic responses in an oscillator described by conductively coupled van der Pol equations with a sinusoidal forcing term are investigated. According to the variation of three base frequencies, i.e., two natural frequencies of oscillators and the forcing frequency, various nonlinear phenomena such as harmonic or subharmonic synchronization, almost synchronization and complete desynchronization are ovserved. The most characteristic phenomenon observed in the four–dimensional nonautonomous system is the occurrence of a double Hopf bifurcation of periodic solutions. A quasi–periodic solution with three base spectra, which is generated by the double Hopf bifurcation, is studied through an investigation of properties of limit cycles observed in an averaged system for the original nonautonomous equations. The oscillatory circuit is particularly motivated by analysis of human circadian rhythms. The transition from an external desynchronization to a complete desynchronization in human rest–activity can be referred to a mechanism of the bifurcation of quasi–periodic solutions with two and three base spectra.

In a nonlinear dynamical circuit with sinusoidal external source, we frequently encounter various bifurcation phenomena of steady states such as jump and hysteresis phenomenon, frequency entrainment, etc. The steady state corresponds to a periodic solution of the circuit equations described by nonlinear ordinary differential equations. The generic bifurcations of the periodic solution are known as codimension one bifurcations: tangent bifurcation, period doubling bifurcation and the Hopf bifurcation. At a bifurcation value of parameters, if a periodic solution satisfies two bifurcation conditions, then the bifurcation refers as a codimension two bifurcation. This type of bifurcation may be observed in high dimensional systems with several parameters. In Ref.(1), we have classified codimension two bifurcations and proposed a numerical method for obtaining the bifurcation parameters. To illustrate the occurrences of some types of codimension two bifurcations, we analyzed a circuit described by 3-dimensional differential equation. For 3-dimensional system, however, two types of bifurcations never occur. In this paper, we shall treat 4-dimensional system as an illustrating example. In this example, we shall see all types of codimension two bifurcations defined in this paper. For a global property of bifurcation set of parameters, it is found that a type of codimension two bifurcation occurs successively together with the period doubling cascade and the Hopf bifurcations. This bifurcation sequence may cause a new route to the generation of chaotic oscillations.

A numerical method is presented for calculating transverse and non-transverse (or tangent) types of homoclinic points of a two-dimensional noninvertible map having an invariant set that reduces to a one-dimensional noninvertible map. To illustrate bifurcation diagrams of homoclinic points and transitions of chaotic states near the bifurcation parameter values, three systems including coupled chaotic maps are studied.

We investigate bifurcations of the periodic solution observed in a phase converter circuit. The system equations can be considered as a nonlinear coupled system with Duffing's equation and an equation describing a parametric excitation circuit. In this system there are two types of solutions. One is with x = y = 0 which is the same as the solution of Duffing's equation (correspond to uncoupled case), another solution is with xy0. We obtain bifurcation sets of both solutions and discuss how does the coupling change the bifurcation structure. From numerical analysis we obtain a codimension two bifurcation which is intersection of double period-doubling bifurcations. Pericdic solutions generated by these bifurcations become chaotic states through a cascade of codimension three bifurcations which are intersections of D-type of branchings and period-doubling bifurcations.

This letter describes a new computational method to obtain the bifurcation parameter value of a limit cycle in nonlinear autonomous systems. The method can calculate a parameter value at which local bifurcations; tangent, period-doubling and Neimark-Sacker bifurcations are occurred by using properties of the characteristic equation for a fixed point of the Poincare mapping. Conventionally a period of the limit cycle is not used explicitly since the Poincare mapping needs only whether the orbit reaches a cross-section or not. In our method, the period is treated as an independent variable for Newton's method, so an accurate location of the fixed point, its period and the bifurcation parameter value can be calculated simultaneously. Although the number of variables increases, the Jacobian matrix becomes simple and the recurrence procedure converges rapidly compared with conventional methods.

Bifurcations of phase-locked modes for diffusively coupled van der Pol equations are investigated. It is known that in-phase and out-of-phase modes are typically observed in the system if two oscillatory equations are identical. There have been many studies on the behavior of diffusively coupled equations of van der Pol type. Many of these, however, persist in the limits of weakly nonlinearity and weak coupling. In this paper we study global feature of bifurcation sets of the modes under relatively wide range of variation of system parameters: coefficient of nonlinear term, parameter related to the frequency of the uncoupled equations, diffusion coefficient and so on.