Kuniyasu Shimizu

Pulse Wave Propagation in a Large Number of Coupled Bistable Oscillators

A simple model of inductor-coupled bistable oscillators is shown to exhibit pulse wave propagation. We demonstrate numerically that there exists a pulse wave which propagates with a constant speed in comparatively wide parameter region. In particular, the propagating pulse wave can be observed in non-uniform lattice with noise. The propagating pulse wave can be observed for comparatively strong coupling case, and for weak coupling case no propagating pulse wave can be observed (propagation failure). We also demonstrate various interaction phenomena between two pulses.

Averaging method analysis of a new mutual synchronization method from living organism

In this paper, we propose averaging method in analyzing two mutually-coupled van der Pol oscillators with a new coupling scheme from living organism. This scheme was invented by S. Nagano, and its characteristic existed in the special nonlinear coupling which was completely different from the conventional linear coupling. Unfortunately, only computer simulation results are given so far. We clarify that this system can be reduced to the simpler averaged equation, and that synchronization range can be obtained theoretically by investigating the stability of the equilibrium point of the averaged equation. As a result, we confirm theoretically that synchronization range of this scheme is much larger than conventional scheme, and hence superior to it

Bifurcation and transitional dynamics in asymmetrical two-coupled oscillators with hard type nonlinearity

The paper investigates various global bifurcations and related changes of dynamics, including the transitional phenomenon, in asymmetrical two-coupled oscillators with hard type nonlinearity. There exist periodic attractors for comparatively large /spl epsiv/ (a parameter showing the degree of nonlinearity), and they disappear for small /spl epsiv/ via saddle-node bifurcation. We draw nodes and saddles with their unstable manifold near the bifurcation point on the Poincare section for various values of k/sup 2/ (deviation between two oscillators intrinsic frequencies). It is recognized that the unstable manifold changes a lot with parameter variation around the bifurcation point, and hence the associated dynamics change drastically.

Bifurcation and transitional dynamics in three-coupled oscillators with hard type nonlinearity

In this paper, we investigate various bifurcation and related dynamics of certain periodic attractors in a three-coupled oscillator system with hard type nonlinearity. The periodic attractors exist for comparatively large /spl epsiv/(=parameter showing the degree of nonlinearity), and they disappear via saddle-node (S-N) bifurcation when /spl epsiv/ becomes small. Sometimes, there exist a heteroclinic and a homoclinic cycle near the bifurcation parameter value. In such cases, a quasi-periodic attractor appears generally after the S-N bifurcation. In particular, it presents intermittent phenomenon just after the S-N bifurcation. We clarify the existence of the heteroclinic and homoclinic cycles by drawing an unstable manifold of saddles on the Poincare section, and demonstrate the intermittent phenomenon by simulation.

Pulse Wave Propagation in Bistable Oscillator Array

A study of propagating wave phenomenon in coupled systems is one of the familiar topics of research. There exist propagating wave phenomena in various systems such as reaction-diffusion systems (Nishiura et al.; 2005; Comte et al.; 2001), coupled map lattice (Kaneko; 1993) and coupled oscillator systems (Hikihara et al.; 2001; Yamauchi et al.; 1999), etc.. They are important not only in a pure nonlinear science viewpoint but also from the viewpoint of various applications such as engineering purpose (Sato et al.; 2003) and biological information processing (Keener; 1987). A basic question concerning these systems is the condition under which propagating wave can emerge. It is known as propagation failure phenomenon that propagating wave fails to propagate below a certain critical coupling strength (Comte et al.; 2001). In this chapter, a simple model of one-dimensinally coupled bistable oscillators is shown to exhibit wave propagation phenomena. The propagating wave consists of several adjacent oscillators oscillating with large amplitude, and the part of large amplitude oscillation in the array propagates with a constant speed. In particular, we pay attention to the formation mechanism of the propagating wave related to disappearance of a certain kind of standing wave. When a coupling strength is weak, a standing pulse wave exists. This solution disappears when the coupling strength exceeds a certain critical value, and around this point the propagating wave begins to exist. On the basis of the observation, one of the onset mechanisms of the propagating waves is investigated paying attention to global phase-space structure around the bifurcation point. At first, we will introduce the ring of coupled bistable oscillator system, and derive the fundamental equation. Then, a typical propagating wave phenomenon and its characteristic features are stated briefly. Next, we make a study on the onset mechanism of the propagating wave for 6 coupled oscillator case. As a result, we have found that a global bifurcation of maps based on the heteroclinic tangle converts the fixed point (= standing wave) into the invariant circle (= propagating pulse wave).

Pulse Wave Propagation in Bistable Oscillator Arrays under Both Fluctuation and Noise

In this paper, we investigate pulse wave propagation phenomenon in a ring of a large number of coupled bistable oscillators. In particular, we focus our attention to such a practical condition that each oscillator has a slightly different intrinsic oscillation frequency in addition to noise. We elucidate numerically that the propagating pulse can be observed by increasing the the coupling strength even for such a practical condition. Moreover, we present the interaction phenomena between two pulses and confirm pulse unification, vanishing, repulsion and passing through phenomena.

Phase Synchronization in Injection-'Un'locking Oscillator Arrays

A new class of phase synchronization patterns is discovered and analyzed systematically. This pattern is generated in certain `unlocking coupled oscillator arrays and it turns out to be quite robust. Applications of this phenomena is proposed for beam-scanning control using coupled oscillator arrays.

Averaging Method Analysis of Synchronization Characteristics of a Large Number of Nonlinearly Coupled van der Pol Oscillators

The averaged equation for an arbitrary number of oscillators coupled by nonlinear coupling scheme invented by S. Nagano, is derived. This system is invented as a model of uni-cellular slime amoeba. By using the averaged equation, we investigate the synchronization characteristics of five coupled oscillators and a large number of coupled oscillators. In particular, we present the statistical property of coupled oscillators in terms of coupling factor γ. We also investigate the effect of linear and nonlinear coupling terms for achieving synchronization, and confirm that the nonlinear coupling term plays an important role for strong synchronization than linear coupling term does.

Stability of the quasi-periodic and periodic attractors constrained to the hyper-plane in a ring of three-coupled hard-type oscillators

In this paper, we investigate quasi-periodic and periodic attractors constrained on the invariant subspace H in a ring of three coupled oscillators. In H two of three oscillators are synchronized. We elucidate the stability of the solution in H by calculating average largest transversal Lyapunov exponent, and find the stable region of periodic and quasi-periodic attractors. We also demonstrate intermittency for parameters in which average largest transversal Lyapunov exponent takes a small positive value.