On-off Intermittency Research Articles

Control of on-off intermittency by slow parametric modulation.

We study on-off intermittent behavior in two coupled double-well Duffing oscillators with stochastic driving and demonstrate that, by using slow harmonic modulation applied to an accessible system parameter, the intermittent attractors can be completely eliminated. The influence of noise is also investigated. Power-law scaling of the average laminar time with a critical exponent of -1 as a function of both the amplitude and frequency of the control modulation is found near the onset of intermittency, which is a signature of on-off intermittency.

Life on the Edge of Chaos

In this article we consider singularly perturbed systems of ordinary differential equations having one swift and one n (n ≥ 3) slow variable. Conditions for the existence of attractors of hard turbulence type and of on-off intermittency are formulated. It is shown that any finite-dimensional system with chaos can be complemented so that it will have one dimension more and hard turbulence will arise. In other words, we propose one possible way of taking into account rare catastrophic events in systems with complicated behavior.

Validity of numerical trajectories in the synchronization transition of complex systems.

We investigate the relationship between the loss of synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems. In the neighborhood of the critical transition to strongly nonhyperbolic behavior, the system undergoes on-off intermittency with respect to the synchronization state. There are potentially severe consequences of these facts on the validity of the computer-generated trajectories obtained from dynamical systems whose synchronization manifolds share the same nonhyperbolic properties.

Dynamical origin for the occurrence of asynchronous hyperchaos and chaos via blowout bifurcations.

We investigate the dynamical origin for the occurrence of asynchronous hyperchaos and chaos via blowout bifurcations in coupled chaotic systems. An asynchronous hyperchaotic or chaotic attractor with a positive or negative second Lyapunov exponent appears through a blowout bifurcation. It is found that the sign of the second Lyapunov exponent of the newly born asynchronous attractor, exhibiting on-off intermittency, is determined through competition between its laminar and bursting components. When the "strength" (i.e., a weighted second Lyapunov exponent) of the bursting component is larger (smaller) than that of the laminar component, an asynchronous hyperchaotic (chaotic) attractor appears.

Chaotic cluster itinerancy and hierarchical cluster trees in electrochemical experiments.

Experiments on an array of 64 globally coupled chaotic electrochemical oscillators were carried out. The array is heterogeneous due to small variations in the properties of the electrodes and there is also a small amount of noise. Over some ranges of the coupling parameter, dynamical clustering was observed. The precision-dependent cluster configuration is analyzed using hierarchical cluster trees. The cluster configurations varied with time: spontaneous changes of number of clusters and their configurations were detected. Simple transitions occurred with the switch of a single element or groups of elements. During more complicated transitions subclusters were exchanged among clusters but original cluster configurations were revisited. At weaker coupling the system itinerated among lower-dimensional quasistationary chaotic two-cluster states and higher-dimensional states with many clusters. In this region the transitions showed characteristics of on-off intermittency.

Bubbling and on-off intermittency in bailout embeddings.

We establish and investigate the conceptual connection between the dynamics of the bailout embedding of a Hamiltonian system and the dynamical regimes associated with the occurrence of bubbling and blowout bifurcations. The roles of the invariant manifold and the dynamics restricted to it, required in bubbling and blowout bifurcating systems, are played in the bailout embedding by the embedded Hamiltonian dynamical system. The Hamiltonian nature of the dynamics is precisely the distinctive feature of this instance of a bubbling or blowout bifurcation. The detachment of the embedding trajectories from the original ones can thus be thought of as transient on-off intermittency, and noise-induced avoidance of some regions of the embedded phase space can be recognized as Hamiltonian bubbling.

Chaotic Synchronization in a Five-Star-Coupled Phase-Locked Loop System

This paper describes a chaotic synchronization phenomenon observed experimentally in a five-star-coupled phase-locked loop (PLL) oscillator system. This system consists of five PLLs connected in the shape of a star. The state in this system itinerates between the completely synchronized state, in which all PLL oscillators synchronize, and the disordered state, in which all PLLs oscillate individually, via the partially synchronized state. The time-series of the voltage difference between arbitrary two PLL oscillators shows the on-off intermittency. When such a chaotic itinerancy occurs, the distribution function of the duration time of the completely synchronized state is expressed as a \(-\frac{3}{2}\) power law which is quite similar to the distribution function of the laminar duration time in the on-off intermittency.

On-off intermittency in chaotic rotation induced in liquid crystals by competition between spin and orbital angular momentum of light.

We observed on-off intermittency in the chaotic rotation induced by a cw laser beam in a thin liquid crystal film where the spin and the orbital angular momentum of light compete in reorienting the sample. We found that the azimuthal angle phi(t) of the molecular director increased linearly in time on large time scales but, occasionally, it exhibited large fluctuations about its average value omega(0)t, so that its angular velocity phi;(t) undergoes an on-off intermittent motion. The intermittent signal omega(t)=phi;(t)-omega(0) obeyed the scaling laws of on-off intermittency, including the symmetry between laminar and burst phases. The chaotic rotations were observed only when the spin and the angular momentum of light were transferred simultaneously to the sample.

Pattern dynamics associated with on-off convection in a one-dimensional system.

A numerical and theoretical analysis of the phenomenologically constructed nonlinear stochastic model of on-off intermittency experimentally observed by John et al. in the electrohydrodynamic convection in nematic liquid crystal under applied dichotomous electric field is carried out. The model has the structure of the one-dimensional Swift-Hohenberg equation with a fluctuating threshold which represents an applied electric field and either with or without additive noise which corresponds to thermal noise. It is found that the fundamental statistics of pattern dynamics without additive noise agree with those experimentally observed, and also with those reported previously in two-dimensional system. In contrast to that the presence of multiplicative noise generates an intermittent evolution of pattern intensity, whose statistics are in agreement with those of on-off intermittency so far known, the additive noise gives rise to the change of position of the convective pattern. It is found that the temporal evolution of the phase suitably introduced to describe the global convective pattern also shows an intermittent evolution. Its statistics are studied in a detailed way with numerical simulation and stochastic analysis. The comparison of these results turn out to be in good agreement with each other.

On–off intermittency and attractor bubbling in the Ising‐like cellular automaton with temperature random in time

AbstractThe Ising‐like cellular automaton with global ferromagnetic coupling and parallel thermal bath dynamics is considered, with uniform in space temperature varying randomly in time in the neighbourhood of the critical point. The system is studied both by numerical simulations with a large number of spins, and analytically in the mean field approximation. The time series of magnetisation show a sequence of laminar phases during which the magnetisation is close to zero, corresponding to paramagnetic phase, and large bursts with non‐zero magnetisation, corresponding to abrupt occurrence of ordered phase. It is shown that this behaviour is an example of on‐off intermittency and attractor bubbling, which appear as collective phenomena due to interaction of a large number of spins subject to random temperature.

Synchronization and intermittency in three-coupled chaotic oscillators.

Synchronization of three-coupled chaotic oscillators was studied with the use of a coupled map system derived for interacting kicked relaxators. Partial synchronization (PS), in which two of the three were synchronized, was observed in addition to complete synchronization. An intermittency associated with the breakdown of the PS, seemingly different from the conventional on-off intermittency, was found. We elucidated the statistics, observing the burst-size distribution, the laminar duration distribution, etc. It was found that the breakdown of the PS generated an anomalous diffusion different from that associated with on-off intermittency.

On-Off Intermittency in Relaxation Systems

On-Off Intermittency in Relaxation Systems

Bailout Embedding as a Blowout Bifurcation

We show that the bailout embedding of a Hamiltonian dynamical system provides an example of blowout bifurcation with conservative dynamics on the invariant manifold. The detachment of the embedding trajectories from the original ones can thus be thought of as transient on-off intermittency.

Slow Switching near a Blowout Bifurcation: Yet Another Mechanism

An example of slow switching behavior that occurs near a blowout bifurcation is presented. Long lasting chaotic states near an invariant manifold are observed, and their lifetime distribution is obtained for various parameter values. These long lasting states result from a mechanism closely related to a periodic window, in addition to the typical on-off intermittency mechanism.

On-off intermittency and intermingledlike basins in a granular medium.

Molecular dynamic simulations of a medium consisting of disks in a periodically tilted box yield two dynamic modes differing considerably in the total potential and kinetic energies of the disks. Depending on parameters, these modes display the following features: (i) hysteresis (coexistence of the two modes in phase space); (ii) intermingledlike basins of attraction (uncertainty exponent indistinguishable from zero); (iii) two-state on-off intermittency; and (iv) bimodal velocity distributions. Bifurcations are defined by a cross-shaped phase diagram.

Signature of on-off intermittency in measured signals.

On-off intermittency is a phase-space mechanism that allows dynamical systems to undergo bursting. As its name suggests, bursting is a phenomenon in which episodes of high activity are alternated with periods of inactivity. Here we attempt to see whether we can tell from the output of a signal when an observed bursting behavior is caused by the presence of on-off intermittency, using the example of a forced logistic map. The results of our study indicate that on-off intermittency can be readily distinguished from other mechanisms for bursting we know of, except for one. Many statistical properties of finite-length signals generated by on-off intermittency can in fact be mimicked by the output of a nonlinearly filtered, linear autoregressive random process.

Spatial orders appearing at instabilities of synchronous chaos of spatiotemporal systems

Various spatial orders introduced by the instabilities of synchronous chaotic state of spatiotemporal systems are investigated by considering coupled map lattice and chaotic partial differential equation. In particular, the motions of on-off intermittent states at the onset of the instabilities are studied in detail. The chaotic desynchronized patterns can be described by a simple universal form, including three parts: the synchronous chaos; a spatially ordered pattern, determined by the unstable mode of the reference synchronous chaos; and on-off intermittency of the scale of this given pattern.

Extended Self-Similarity in Modulational Intermittency Based on the Continuous-Time Random Walk Approach

Anomalous diffusion caused by modulational intermittency, which is also known as on-off intermittency, is studied on the basis of the continuous-time random walk (CTRW) approach. There exists a characteristic time scale τ . For the time region t � τ , anomalous subdiffusion is observed, which is followed by normal diffusion for t � τ . Higher-order moments are analytically obtained by use of the saddle-point method, and it is found that they obey scaling relations that are reminiscent of extended self-similarity (ESS) and generalized extended selfsimilarity (GESS) found in turbulent systems. The results are compared with those obtained using the numerical inverse Laplace transform and from model simulations employing a coupled chaotic map. Good agreement between these results is obtained even for lowerorder moments.

On-off intermittency in a human balancing task.

Motion analysis in three dimensions demonstrate that the fluctuations in the vertical displacement angle of a stick balanced at the fingertip obey a scaling law characteristic of on-off intermittency and that >98% of the corrective movements occur fast compared to the measured time delay. These experimental observations are reproduced by a model for an inverted pendulum with time-delayed feedback in which parametric noise forces a control parameter across a particular stability boundary. Our observations suggest that parametric noise is an essential, but up until now underemphasized, component of the neural control of balance.

Volatility clustering and scaling for financial time series due to attractor bubbling.

A microscopic model of financial markets is considered, consisting of many interacting agents (spins) with global coupling and discrete-time heat bath dynamics, similar to random Ising systems. The interactions between agents change randomly in time. In the thermodynamic limit, the obtained time series of price returns show chaotic bursts resulting from the emergence of attractor bubbling or on-off intermittency, resembling the empirical financial time series with volatility clustering. For a proper choice of the model parameters, the probability distributions of returns exhibit power-law tails with scaling exponents close to the empirical ones.