In this paper, we examine oscillatory modes generated by the Hopf bifurcations of equilibrium points except for the origin in a system of coupled four oscillators. (The bifurcation analyses of the origin for many coupled oscillators were already done.) The Hopf bifurcations of the equilibrium points with strong symmetrical property and the generated oscillatory modes are classified. We observe four-phase, in-phase and a pair of anti-phase synchronized states. Even in a system of four coupled oscillators, we discover the existence of a stable three-phase oscillation. By the numerical bifurcation analysis of generated periodic oscillations we find out successive period-doubling bifurcations as the route to chaos and show some of them are symmetry-breaking bifurcations. As a result of the symmetry-breaking period-doubling bifurcations, a periodic solution with complete synchronization becomes a chaotic solution with phase synchronization.

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.

A data-parallel processing approach is promising for real-time volume rendering because of the massive parallelism in volume rendering. In data-parallel volume rendering, local results processing elements(PEs) generate from allocated subvolumes are integrated to form a final image. Generally, the integration causes an overhead unavoidable in data-parallel volume rendering due to communications among PEs. This paper proposes a data-parallel shear-warp volume rendering algorithm combined with an adaptive volume subdivision method to reduce the communication overhead and improve processing efficiency. We implement the parallel algorithm on a message-passing multiprocessor system for performance evaluation. The experimental results show that the adaptive volume subdivision method can reduce the overhead and achieve higher efficiency compared with a conventional slab subdivision method.

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.

In this paper, we study bifurcations of equilibrium points and periodic solutions observed in a resistively coupled oscillator with voltage ports. We classify equilibrium points and periodic solutions into four and eight different types, respectively, according to their symmetrical properties. By calculating D-type of branching sets (symmetry-breaking bifurcations) of equilibrium points and periodic solutions, we show that all types of equilibrium points and periodic solutions are systematically found. Possible oscillations in two coupled oscillators are presented by calculating Hopf bifurcation sets of equilibrium points. A parameter region in which chaotic oscillations exist is also shown by obtaining a cascade of period-doubling bifurcation sets.

We consider a system of coupled two oscillators with external force. At first we introduce the symmetrical property of the system. When the external force is not applied, the two oscillators are synchronized at the opposite phase. We obtain a bifurcation diagram of periodic solutions in the coupled system when the single oscillator has a stable anti-phase solution. We find that the synchronized oscillations eventually become in-phase when the amplitude of the external force is increased.

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.