International Journal of Bifurcation and Chaos

In the first half of this paper, some general ideas will be developed on how to approach mathematically large systems of coupled limit-cycle oscillators. Two representative reduction techniques, namely, the phase reduction and the center-manifold reduction will be presented for a prototypal system of biological cell assembly with periodic activity. The evolution equation derived through each reduction method is further classified into three groups according to the range of the oscillator coupling (i.e. local, global and intermediate). As a consequence, six classes of model equations are obtained. In the second half of the paper, some new results from our recent study on non-locally coupled oscillators will be reported, and the generation of anomalous turbulent fluctuations obeying a power law will be discussed in some detail.

Macroscopic phase-locking in large populations of coupled nonlinear oscillators is an interesting phenomenon analogous to collective behaviors of conventional cooperative systems, being paid increasing attention in diverse fields of science. The behavior of such populations is known to be well described by what is called phase models, provided that both mutual interactions among oscillators and the dispersion of their intrinsic frequencies are comparatively weak. Here reviewed is one of recent developments in the study of macroscopic phase-locking based on phase models, which is the order function theory enabling us to understand and analyze that phenomenon in a general way for such populations that every element is coupled to all the other. With this theory, generic features of macroscopic phase-locking under such coupling will be clarified, in particular, as to scaling behaviors at the onset of macroscopic phase locking.

We obtained new characteristic relations in Type-II and III intermittencies according to the reinjection probability distribution. When the reinjection probability distribution is fixed at the lower bound of reinjection, the critical exponents are -1, as is well known. However when the reinjection probability distribution is uniform, the critical exponent is -1/2, and when it is of form , -3/4. On the other hand, if the square root of Δ, which represents the lower bound of reinjection, is much smaller than the control parameter ∊, i.e. ∊ ≫ Δ^1/2, critical exponent is always -1, independent of the reinjection probability distribution. Those critical exponents are confirmed by numerical simulation study.

An analytical solution for the well-known quadratic recursion, also known as the logistic map, is presented. Our derivation makes use of the analogy between this equation and Galton–Watson processes. The latter can be treated in a linear way using the transfer matrix technique. The solution is represented as a power of an appropriate transfer matrix.

Second order phase transition is numerically investigated in a Hamiltonian system with many degrees of freedom. Slow relaxations of power type are observed for some initial conditions at critical energy of phase transition. This is consisent with a result of a phenomenological theory of statistical mechanics. On the other hand, the slow relaxations show that the system stays in non-equilibrium states for a while, and that phenomenon does not agree with a result of the theory. To understand the slow relaxation, theories for perturbed systems cannot be applied since near the critical energy the system is highly chaotic rather than nearly integrable. The thresholds of the highly chaotic systems is different from the critical energy of phase transition.

We have observed an interesting bifurcation sequence at very low driving voltages applied to a diode resonator. As the driving voltage increases, an ordinary period-doubling bifurcation appears. Further increase of the driving voltage results in a reverse period-doubling bifurcation and a sudden jump of an amplitude of the current oscillation of the diode. The experimental results were compared with the results of numerical simulations of the equivalent circuit of the diode resonator and they were in good agreement. This suggests that these low-voltage bifurcations are due to nonlinearity of the capacitance and current of the diode to the applied voltage. A simple one-dimensional map explaining these low-voltage bifurcations of the diode resonator is introduced and its properties are discussed.

To study the coherent nature of chaos, two models are proposed. Model 1 is a simple nonlinear system and Model 2 is a linear harmonic oscillator , which are driven by a chaotic force f(t). The chaotic force f(t) is defined by for nτ < t ≤ (n + 1)τ(n = 0, 1, 2, …), where y_n+1 is a chaotic sequence of a map F(y; r) with the bifurcation parameter r: y_n+1 = F(y_n; r) (-0.5 ≤ y_n ≤ 0.5) and ŷ_n = y_n - < y₀>. In Model 1 the relaxation process of this system and the τ- and r-dependence of the stationary distribution of x are discussed. It is shown that the small change of the bifurcation parameter r causes the drastic change of the stationary distribution. In Model 2, resonance phenomena are investigated near the period 3 window of the logistic map, in particular, in the intermittent chaos region and the period doubling region. The theoretical results are shown to be in a good agreement with numerical ones, which have been done for the logistic map as F(y; r).

It is usually believed that strong diffusive coupling in one of the dynamical variables is well-suited for imposing synchronization of oscillators. But it was recently shown that weak diffusive coupling, counter-intuitively, can lead to dephasing of coupled neural oscillators.

In this paper, we investigate how diffusively coupled oscillators become dephasing. For this we study a system of coupled neural oscillators on a limit cycle generated through a homoclinic bifurcation. We examine the asymptotic behavior of diffusive coupling as the control parameter approaches the critical value for which the homoclinic bifurcation occurs. In this study, we show that the gradient of phase velocity near the limit cycle is essential in generating dephasing through diffusive interaction.

Recently it was shown that dephasing of diffusively coupled neural oscillators leads to a new class of bursting phenomena, where neural oscillators switch between high and low oscillation amplitudes.

To analyze this behavior we study a system of three-coupled neurons, which is the most simple one that shows chaotic bursting behavior. For the intermediate values of coupling constant k_c, the chaotic bursting behavior occurs. For a quantitative analysis of chaotic bursting, we introduce three mean activities of oscillators. From the Poincaré sections we find a period-doubling route to chaos. We illustrate the busting behavior in terms of competition of the single oscillator behavior with the collective one arising from the diffusive coupling of oscillators.

Synchrony and clustering in phase dynamics of two and three Hodgkin–Huxley neurons coupled globally by excitatory synapses with a time delay have been studied by numerical simulations and bifurcation analysis. In particular, we have obtained phase diagrams for the synchronous state and various cluster states in the parameter space of the synaptic coupling strength, G_syn, and the synaptic time delay, τ_d, by computing the phase shifts of action potential spikes. In the weak coupling limit the computed phase diagrams are found to be consistent with the results of phase model analysis. The bifurcation analysis shows the phase model predictions break down at G_syn~ 0.3 and new complex phase dynamics appear. For all numbers of neurons explored, the critical time delay at which the nature of synaptic coupling changes completely is found to be typically about 2–4 msec, which may have some implications in modeling cortical dynamics.

We have designed coupled diode resonators to study the effect of small perturbations due to weak symmetric coupling on chaotic dynamics. Our experiment clearly demonstrated that chaos of the diode resonators was suppressed so that chaotic motions were converted into periodic ones with small modifications to the attractor when an appropriate coupling signal perturbed the diode resonators. Many unstable periodic orbits were stabilized and they were very stable depending on the dynamical properties of the coupling signals. Our results suggest that coupling of signals belonging to the same class is effective in controlling chaos.

We demonstrate that return map control and adaptive tracking can be used together to locate, stabilize, and track unstable periodic orbits (UPOs). Through bifurcation studies as a function of some control parameters of return map control, we observe the control bifurcation (CB) phenomenon which exhibits another route to chaos. Nearby an UPO there are a lot of driven periodic orbits (DPOs) along the CB route. DPOs are not embedded in the original chaotic attractor, but they are generated artificially by driving the system slightly in a direction with feedback control. Based on the CB phenomenon, our adaptive tracking algorithm searches for the location and the exact control condition of the UPO by minimizing feedback perturbations. We discuss the universality of the CB phenomenon and the possibility of immediate control which does not require much prior analysis of the system.

We describe some universal features of dynamical systems that exhibit stochastic resonance. Examples of such systems show that stochastic resonance appears as the result of the interaction between an externally imposed rate and stochastic characteristics of the dynamic systems which are not restricted to be nonlinear.

The Nonequilibrium phenomena in a class of globally coupled phase oscillators systems with multiplicative noise are studied. It is shown that at the critical value of the noise intensity the systems undergo a phase transition and converge to clustered states. We also show that the time delay in the interaction between oscillators gives rise to the switching phenomena of clusters. These phenomena are noise-induced effects which cannot be seen in the deterministic systems or in the simple additive noise models.

Underdamped oscillators that possess a maximum or minimum in their dependence of eigenfrequency on energy have recently been shown to exhibit a range of unusual phenomena. Because they are associated with the presence of an extremum in whose vicinity the eigenfrequency is almost energy-independent, they have been named zero-dispersion phenomena. They manifest themselves both in the deterministic dynamics and in the presence of noise. When the oscillator is driven by a weak periodic force at a frequency close to that of the extremum, a novel type of nonlinear resonance, zero-dispersion nonlinear resonance (ZDNR) can occur. A giant response then arises even in the absence of resonance between the drive frequency and any eigenoscillation of the system. The properties of ZDNR, the nature of the transition from ZDNR to conventional nonlinear resonance as relevant parameters are varied, the occurrence of dynamical chaos associated with ZDNR, and the influence of noise, are analysed and discussed for both Hamiltonian and dissipative systems.

We introduce a theory on phase breaking time, which predicts the temperature dependence of phase breaking time as 1/τ_ϕ∝ k_BT and geometrical effects characteristic of ballistic quantum dots. We also compare the result with the experimental data.

The classical dynamics of a damped quartic oscillator driven by a sinusoidal force is investigated, with particular attention to the effects that arise when the motion of the oscillator becomes relativistic. Bifurcation diagrams constructed numerically indicate that, as relativistic effects become strong, chaotic behavior exhibited by the oscillator in the nonrelativistic limit at large force amplitude is replaced by a period-1 regular motion. At relativistic energies, therefore, a transition from chaos to a regular motion occurs as the force amplitude is increased beyond a critical value. The transition is seen to occur abruptly with a slight increase of the force amplitude from below to above the critical value. The sudden destruction of the chaotic attractor is probably triggered by the mechanism of crisis.

We apply controlling chaos techniques to select the desired sequence of impacts in a map that captures universal properties of impact oscillators near grazing. For instance, we can choose the period and then stabilize an unstable periodic orbit with, say, one impact per period involved in the grazing bifurcations that take place in the system.

An analog circuit which models an infinite-dimensional dynamical system is described. It provides an easy and rapid way to simulate periodic and chaotic behaviors in a delay differential equation, the Mackey-Glass mathematical model. Several methods of controlling unstable steady states are considered. Analog techniques for stabilizing non-zero stationary point in the Mackey-Glass system are developed.