We study the dynamics of a pair of two uncoupled identical chaotic elements driven by common noise. When each element exhibits type-I intermittency, we observe that two uncoupled elements synchronize each other after a finite time of interval for a certain range of the noise intensity. In order to clarify the mechanism of this noise-induced synchronization phenomenon, we focus on the effect of external noise on the fluctuation of the local expansion rate of orbits to perturbations of type-I intermittency. It is found that the probability that the finite-time Liapunov exponent (FTLE) takes a negative value may increase due to the introduction of noise, whereas the Liapunov exponent itself remains to be positive. We show that this noise-induced enhancement of fluctuation may cause the synchronization and also discuss the relation between the statistical properties of relaxation process of the synchronization and the fluctuation properties of FTLE in terms of the thermodynamic formalism.
We show that when paced chaotic oscillators, which can be flows or maps, are coupled appropriately, phase slips produced by each oscillator are synchronized. If a periodically driven chaotic oscillator strays from a phase synchronization region, the phase difference between the oscillator and the pacer jumps intermittently by 2π, which is called a phase slip. When two sinusoidally forced Roessler oscillators are coupled appropriately, phase slips produced by the two oscillators occur simultaneously, that is the phase slips are synchronized. We also show that if the coupled oscillator deviates slightly from the slip synchronization region, a portion of the simultaneous phase slips are desynchronized, namely only one of the two oscillators produces a phase slip. Such phenomena as synchronized phase slips and partially synchronized phase slips can be reproduced by a coupled map system. We investigate some statistical properties and dynamical structures of the phenomena by investigating the coupled map system.
We show that deterministic stochastic resonance (DSR) can be enhanced by coupling of chaotic oscillators. We study periodic-forced chaotic oscillators coupled to each other. One oscillator phase and periodic force phase are synchronized with each other when the force strength is larger than a critical value. When we set the force strength below the critical value, the phase synchronization occasionally fails. We can observe a quick jump in phase difference between one oscillator and periodic force. In this study, we focus on this phase slip and consider one forced chaotic oscillator as a resonator using the phase slips. When we consider coupled resonators, the coupling strength becomes a bifurcation parameter that has a critical point between asynchronous and synchronous phase slip state. Increases in coupling strength leads to a higher degree of phase slip synchronization. The coupling helps to synchronize the slips with a cooperative effect. Therefore, it can enhance the coincident response to the signal. Optimal coupling strength maximizes the resonance response. This enhancement provides some advantages for signal detection applications using DSR. It is considered that intrinsic fluctuations are important for information processing in biological system. This coupled system may be useful for a model study of neural information processing.
We study the dynamics of assemblies of “uncoupled” identical chaotic elements under the influence of external noisy filed. It is numerically demonstrated that in the case where each chaotic element exhibits type-I intermittency, the degree of the temporal regularity of the mean-field dynamics of the system reaches a maximum at a certain optimal noise intensity. Moreover, we also report that inhomogeneous noise which drives each element partly independently enhances the coherence of the mean-field more than that of the case where all elements of the system receive a completely identical noisy input, and the degree of the coherence as a function against the degree of inhomogeneity of noise shows a convex curve. In noisy uncoupled systems, the common part of noise which drives each element can be regarded as the interaction among elements which corresponds to the coupling term in the case of coupled systems, so our finding that some degree of inhomogeneity enhances the coherence of the dynamics is not trivial.
We show deterministic stochastic resonance (DSR) in chaotic diffusion when the diffusion map is modulated by a sinusoid. In chaotic diffusion, the map parameter determines the state transition rate and the diffusion coefficient. The transition rate shows the diffusion intensity. Therefore, the parameter represents the intensity of the internal fluctuation. By this fact, increase of the parameter maximizes the response of DSR as in standard stochastic resonance (SR) where the external noise intensity optimizes the response. Sinusoidally modulated diffusion is regarded as a stochastic process whose transition rate is modulated by the sinusoid. Therefore, the transition dynamics can be approximated by a time-dependent random walk process. Using the mean transition rate function against the map parameter, we can derive the DSR response depending on the parameter. Our approach is based on the rate modulation theory for SR. Even when the diffusion map is modulated by the sinusoid and noise from an external environment, the increasing parameter can also maximize the DSR response. We can calculate the DSR response depending on the external noise intensity and the map parameter. DSR takes advantage of applications to signal detection because the system has the control parameter corresponding to the internal fluctuation intensity.