Hiroya Nakao

TURING PATTERNS IN NETWORK-ORGANIZED ACTIVATOR-INHIBITOR SYSTEMS

Turing patterns formed by activator-inhibitor systems on networks are considered. The linear stability analysis shows that the Turing instability generally occurs when the inhibitor diffuses sufficiently faster than the activator. Numerical simulations, using a prey-predator model on a scale-free random network, demonstrate that the final, asymptotically reached Turing patterns can be largely different from the critical modes at the onset of instability, and multistability and hysteresis are typically observed. An approximate mean-field theory of nonlinear Turing patterns on the networks is constructed.

Multi-scaling properties of truncated Lévy flights

Abstract Multi-scaling properties of one-dimensional truncated Levy flights are studied. Due to the broken self-similarity of the distribution of jumps, they are expected to possess multi-scaling properties in contrast to the ordinary Levy flights. We argue this fact based on a smoothly truncated Levy distribution, and derive the functional form of the scaling exponents. Specifically, they exhibit bi-fractal behavior, which is the simplest case of multi-scaling.

Synchrony of limit-cycle oscillators induced by random external impulses

The mechanism of phase synchronization between uncoupled limit-cycle oscillators induced by common external impulsive forcing is analyzed. By reducing the dynamics of the oscillator to a random phase map, it is shown that phase synchronization generally occurs when the oscillator is driven by weak external impulses in the limit of large inter-impulse intervals. The case where the inter-impulse intervals are finite is also analyzed perturbatively for small impulse intensity. For weak Poissonian impulses, it is shown that the phase synchronization persists up to the first order approximation.

Stochastic phase reduction for a general class of noisy limit cycle oscillators.

We formulate a phase-reduction method for a general class of noisy limit cycle oscillators and find that the phase equation is parametrized by the ratio between time scales of the noise correlation and amplitude relaxation of the limit cycle. The equation naturally includes previously proposed and mutually exclusive phase equations as special cases. The validity of the theory is numerically confirmed. Using the method, we reveal how noise and its correlation time affect limit cycle oscillations.

Dynamics of Limit-Cycle Oscillators Subject to General Noise

The phase description is a powerful tool for analyzing noisy limit-cycle oscillators. The method, however, has found only limited applications so far, because the present theory is applicable only to Gaussian noise while noise in the real world often has non-Gaussian statistics. Here, we provide the phase reduction method for limit-cycle oscillators subject to general, colored and non-Gaussian, noise including a heavy-tailed one. We derive quantifiers like mean frequency, diffusion constant, and the Lyapunov exponent to confirm consistency of the results. Applying our results, we additionally study a resonance between the phase and noise.

Asymptotic power law of moments in a random multiplicative process with weak additive noise

It is well known that a random multiplicative process with weak additive noise generates a power-law probability distribution. It has recently been recognized that this process exhibits another type of power law: the moment of the stochastic variable scales as a function of the additive noise strength. We clarify the mechanism for this power-law behavior of moments by treating a simple Langevin-type model both approximately and exactly, and argue this mechanism is universal. We also discuss the relevance of our findings to noisy on-off intermittency and to singular spatio-temporal chaos recently observed in systems of non-locally coupled elements.

Power-law spatial correlations and the onset of individual motions in self-oscillatory media with non-local coupling

Abstract It is demonstrated numerically that power-law spatial correlations appear generically in self-oscillatory media with non-local coupling. This occurs in length scales smaller than the range of coupling when the turbulent fluctuations generated through the Benjamin-Feir instability spread deep into this regime. The associated exponent varies continuously with the coupling strength. However, the latter has a lower critical value below which the pattern can no longer sustain its spatial continuity giving way to individual motions of the oscillators. Our numerical analyses are carried out on a one-dimensional oscillator lattice where two types of model oscillators are examined. They are the complex Ginzburg-Landau type oscillators and the Brusselator. A non-trivial effect of breaking the special symmetry of the former model is also discussed.

Phase reduction approach to synchronisation of nonlinear oscillators

Systems of dynamical elements exhibiting spontaneous rhythms are found in various fields of science and engineering, including physics, chemistry, biology, physiology, and mechanical and electrical engineering. Such dynamical elements are often modelled as nonlinear limit-cycle oscillators. In this article, we briefly review phase reduction theory, which is a simple and powerful method for analysing the synchronisation properties of limit-cycle oscillators exhibiting rhythmic dynamics. Through phase reduction theory, we can systematically simplify the nonlinear multi-dimensional differential equations describing a limit-cycle oscillator to a one-dimensional phase equation, which is much easier to analyse. Classical applications of this theory, i.e. the phase locking of an oscillator to a periodic external forcing and the mutual synchronisation of interacting oscillators, are explained. Further, more recent applications of this theory to the synchronisation of non-interacting oscillators induced by common noise and the dynamics of coupled oscillators on complex networks are discussed. We also comment on some recent advances in phase reduction theory for noise-driven oscillators and rhythmic spatiotemporal patterns.

Phase synchronization between collective rhythms of globally coupled oscillator groups: Noiseless nonidentical case.

We theoretically study the synchronization between collective oscillations exhibited by two weakly interacting groups of nonidentical phase oscillators with internal and external global sinusoidal couplings of the groups. Coupled amplitude equations describing the collective oscillations of the oscillator groups are obtained by using the Ott-Antonsen ansatz, and then coupled phase equations for the collective oscillations are derived by phase reduction of the amplitude equations. The collective phase coupling function, which determines the dynamics of macroscopic phase differences between the groups, is calculated analytically. We demonstrate that the groups can exhibit effective antiphase collective synchronization even if the microscopic external coupling between individual oscillator pairs belonging to different groups is in-phase, and similarly effective in-phase collective synchronization in spite of microscopic antiphase external coupling between the groups.

Optimal phase response curves for stochastic synchronization of limit-cycle oscillators by common Poisson noise.

We consider optimization of phase response curves for stochastic synchronization of noninteracting limit-cycle oscillators by common Poisson impulsive signals. The optimal functional shape for sufficiently weak signals is sinusoidal, but can differ for stronger signals. By solving the Euler-Lagrange equation associated with the minimization of the Lyapunov exponent characterizing synchronization efficiency, the optimal phase response curve is obtained. We show that the optimal shape mutates from a sinusoid to a sawtooth as the constraint on its squared amplitude is varied.