This paper proposes a piecewise linear non-autonomous chaos generator that includes a switched inductor with time delay. The dynamics can be grasped by using piecewise exact solutions and one-dimensional return map can be derived rigorously. Using the return map, we formulate bifurcation equations and clarify tangent bifurcation route to chaos. Rough global bifurcation sets are given. Some of chaotic attractors are verified in the laboratory.

In this study, we show how changing a frequency in one of N chaotic circuits coupled by a resistor effects our system by means of both circuit experiment and computer calculation. In these N chaotic circuits, N-1 circuits are completely identical, and the remaining one has altered the value of the oscillation frequency. It is found that for the case of N = 3 when a value of a coupling resistor is gradually increased, only one circuit with different frequency exhibits bifurcation phenomena including inverse period-doubling bifurcation, and for larger value of coupling resistor, the chaotic circuit with different frequency suddenly stops oscillating and the remaining two chaotic circuits exhibit completely anti-phase synchronization.

This paper presents the several bifurcation phenomena of harmonic oscillations occurred in nonlinear three-phase circuit. The circuit consists of delta-connected nonlinear inductors, capacitors and three-phase symmetrical voltage sources. We analyze the bifurcations of the oscillations by the homotopy method. Additionally, we confirm the bifurcation phenomena by real experiments. Furthermore, we reveal the effect of nonlinear couplings of inductors by the comparison of harmonic oscillations in a single-phase circuit.

In this paper, we investigate bifurcation phenomena ovserved from two autonomous three-dimensional chaotic circuits coupled by an inductor. Two types of synchronization modes are ovserved in this coupled system, i.e., in-phase synchronization and anti-phase synchronization. For the purpose of detailed analysis, we consider the case that the diodes in the subcircuits are assumed to operate as ideal switches. In this case Poincare map is derived as a three-dimensional map, and Lyapunov exponents can be calculated by using exact solutions. Various bifurcation phenomena related with chaos synchronization are clarified. We confirm that various bifurcation phenomena are observed from circuit experiments.

We found a novel connecting orbit in the averaged Duffing-Rayleigh equation. The orbit starts from an unstable manifold of a saddle type equilibrium point and reaches to a stable manifold of a node type equilibrium. Although the connecting orbit is structurally stable in terms of the conventional definition of structural stability, it is structually unstable since a one-deimensional manifold into which the connecting orbit flows is unstable. We can consider the orbit is one of global bifurcations governing the differentiability of the closed orbit.

Synchronization and chaos of the oscillator circuit that is composed of two Duffing-Rayleigh oscillators coupled by resistor are investigated. The characteristic feature of this system is that the cubic nonlinear restoring force of each oscillator. The restoring force causes the Neimark-Sacker bifurcation with various synchronizations in the parameter plane. We clarify the bifurcation structure related with this nonlinear phenomenon, and study the chaotic state and its bifurcation process. Especially, we deals with the case that the symmetrical property is broken by changing system parameters.

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.

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.

Bifurcation Phenomena observed in a circuit containing two Josephson junctions coupled by a resistor are investigated. This circuit model has a mechanical analogue: Two damped pendula linked by a clutch exchanging kinetic energy of each pendulum. In this paper, firstly we study equilibria of the system. Bifurcations and topological properties of the equilibria are clarified. Secondly we analyze periodic solutions in the system by using suitable Poincare mapping and obtain a bifurcation diagram. There are two types of limit cycles distinguished by whether the motion is in S1R3 or T2R2, since at most two cyclic coordinates are included in the state space. There ia a typical structure of tangent bifurcation for 2-periodic solutions with a cusp point. We found chaotic orbits via the period-doubling cascade, and a long-period stepwise orbit.

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.

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.

The pendulum equation with a periodic impulsive force is investigated. This model described by a second order differential equation is also derived from dynamics of the stepping motor. In this paper, firstly, we analyze bifurcation phenomena of periodic solutions observed in a generalized pendulum equation with a periodic impulsive force. There exist two topologically different kinds of solution which can be chaotic by changing system parameters. We try to stabilize an unstable periodic orbit embedded in the chaotic attractor by small perturbations for the parameters. Secondly, we investigate the intermittent drive characteristics of two-phase hybrid stepping motor. We suggest that the unstable operations called pull-out are caused by bifurcations. Finally, we proposed a control method to avoid the pull-out by changing the repetitive frequency and stepping rate.

It is important to analyze a tracking or synchronizing process in Spread Spectrum (SS) receiving system. The most common SS tracking system considered here consists of pseudorandom (PN) generator, Lowpass Filter (LPE) and Voltage Controlled Oscillator (VCO). The SS receiver is to track or synchronize its local PN generator to the received PN waveform by VCO. The fundamental equation of the system is known by a second order nonlinear differential equation in terms of phase difference between local PN generator and received PN waveform. The differential equation is nonautonoumous due to PN function of time t with period T. Picking up the gain of VCO as the main parameter in the system we show that the system has bifurcation from the normal oscillation through subharmonic oscillation to finally chaos. In the final case, chaos is confirmed by investigating maximum Liapunov number and both stable and unstable manifolds.

In this paper, we discuss computational methods for obtaining the bifurcation points and the branch directions at branching points of solution curves for the nonlinear resistive circuits. There are many kinds of the bifurcation points such as limit point, branch point and isolated point. At these points, the Jacobian matrix of circuit equation becomes singular so that we cannot directly apply the usual numerical techniques such as Newton-Raphson method. Therefore, we propose a simple modification technique such that the Newton-Raphson method can be also applied to the modified equations. On the other hand, a curve tracing algorithm can continuously trace the solution curves having the limit points and/or branching points. In this case, we can see whether the curve has passed through a bifurcation point or not by checking the sign of determinant of the Jacobian matrix. We also propose two different methods for calculating the directions of branches at branching point. Combining these algorithms, complicated solution curves will be easily traced by the curve tracing method. We show the example of a Hopfield network in Sect.5.

To express period-doubling, intermittency and period-adding chaos observed in the thyristor, we propose a simple model which describes a first return map based on experimental data. This model can express whole the aspects observed in the thyristor through changing a couple of parameters in the map function. Simulated bifurcation diagram reproduced experimental results well in its qualitative nature.

In this paper, bifurcation and chaotic behavior which occur in simple looped MOS inverters with high speed operation are described. The most important point in this work is to change a nonlinear transfer characteristic of a MOS inverter to the nonlinearity generating a chaos. Three types of circuits which include four, three and one MOS inverters, respectively, are proposed. A switched capacitor (SC) circuit to operate sampling holding is added in the loop in each of the circuits. The bifurcation and chaotic behavior have been found along with a variation of an external input, and/or a sampling clock frequency. The bifurcation and chaotic behavior of the proposed simple looped MOS inverters are verified by employing SPICE circuit simulator as well as the experiments. For the first type of four looped CMOS inverters, Lyapunov exponent λ which has the positive regions for the chaotic behavior can be calculated by use of the fitting nonlinear function synthesized from two sigmoid functions. For the second type of three looped CMOS inverters and the third type of one looped MOS inverter, the nonlinear charge/discharge characteristics of the hold capacitor in the SC circuit is utilized efficiently for forming the nonlinearity generating the bifurcation and chaotic behavior. Their bifurcation can be generated by the sampling clock frequency parameter which is controlled easily.

A systematic method for locating and computing branches of connecting orbits developed by the authors is outlined. The method is applied to the sine–Gordon and Hodgkin–Huxley equations.

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.

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.

Some qualitative properties of an inductively coupled circuit containing two Josephson junction elements with a dc source are investigated. The system is described by a four–dimensional autonomous differential equation. However, the phase space can be regarded as S1×R3 because the system has a periodicity for the invariant transformation. In this paper, we study the properties of periodic solutions winding around S1 as a bifurcation problem. Firstly, we analyze equilibria in this system. The bifurcation diagram of equilibria and its topological classification are given. Secondly, the bifurcation diagram of the periodic solutions winding around S1 are calculated by using a suitable Poincar mapping, and some properties of periodic solutions are discussed. From these analyses, we clarify that a periodic solution so–called "caterpillar solution" is observed when the two Josephson junction circuits are weakly coupled.