Perturbed on-off intermittency

Detecting and characterizing high-frequency oscillations in epilepsy: a case study of big data analysis

We develop a framework to uncover and analyse dynamical anomalies from massive, nonlinear and non-stationary time series data. The framework consists of three steps: preprocessing of massive datasets to eliminate erroneous data segments, application of the empirical mode decomposition and Hilbert transform paradigm to obtain the fundamental components embedded in the time series at distinct time scales, and statistical/scaling analysis of the components. As a case study, we apply our framework to detecting and characterizing high-frequency oscillations (HFOs) from a big database of rat electroencephalogram recordings. We find a striking phenomenon: HFOs exhibit on-off intermittency that can be quantified by algebraic scaling laws. Our framework can be generalized to big data-related problems in other fields such as large-scale sensor data and seismic data analysis.

Motor learning characterized by changing Lévy distributions

The probability distributions for changes in transverse plane fingertip speed are Lévy distributed in human pole balancing. Six subjects learned to balance a pole on their index finger over three sessions while sitting and standing. The Lévy or decay exponent decreased as a function of learning, showing reduced decay in the probability for large speed steps and was significantly smaller in the sitting condition. However, the probability distribution for changes in fingertip speed was truncated so that the probability for large steps was reduced in this condition. These results show a learning-induced tolerance for large speed step sizes and demonstrate that motor learning in continuous tasks may be characterized by changing distributions that reflect sensorimotor skill acquisition.

Beneficial role of noise in promoting species diversity through stochastic resonance

There is an increasing recognition that patterns in species diversity cannot be understood without reference to nonequilibrial or unstable dynamics. Recently, through a realistic ecological model that involves dispersal, we have addressed the positive role of noise in promoting species coexistence [Phys. Rev. Lett. 94, 038102 (2005)]. Here we present a physical theory to account for the main scaling law responsible for this phenomenon.

Mechanism for the partial synchronization in three coupled chaotic systems

We investigate the dynamical mechanism for the partial synchronization in three coupled one-dimensional maps. A completely synchronized attractor on the diagonal becomes transversely unstable via a blowout bifurcation, and then a two-cluster state, exhibiting on-off intermittency, appears on an invariant plane. If the newly created two-cluster state is transversely stable, then partial synchronization occurs on the invariant plane; otherwise, complete desynchronization takes place. It is found that the transverse stability of the intermittent two-cluster state may be determined through the competition between its laminar and bursting components. When the laminar (bursting) component is dominant, partial synchronization (complete desynchronization) occurs through the blowout bifurcation. This mechanism for the occurrence of partial synchronization is also confirmed in three coupled multidimensional invertible systems, such as coupled He non maps and coupled pendula.

Noise promotes species diversity in nature

Species diversity in nature is accomplished by coexistence. In a spatial environment, inferior but rapidly moving species can coexist with superior but relatively stationary species. Recent work showed that chaotic dynamics can provide the spatiotemporal variation in the fitness required for coexistence, via the dynamical mechanism of synchronization and intermittency. Utilizing a realistic model that consists of two interacting species in a two-patch environment, we discovered a stochastic-resonance phenomenon where noise can significantly enhance the coexistence and thereby promote species diversity.

Statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability

Severe obstruction to shadowing of computer-generated trajectories can occur in nonhyperbolic chaotic systems with unstable dimension variability. That is, when the dimension of the unstable eigenspace changes along a trajectory in the invariant set, no true trajectory of reasonable length can be found to exist near any numerically generated trajectory. An important quantity characterizing the shadowability of numerical trajectories is the shadowing time, which measures for how long a trajectory remains valid. This time depends sensitively on initial condition. Here we show that the probability distribution of the shadowing time contains two distinct scaling behaviors: an algebraic scaling for short times and an exponential scaling for long times. The exponential behavior depends on system details but the small-time algebraic behavior appears to be universal. We describe the computational procedure for computing the shadowing time and give a physical analysis for the observed scaling behaviors.

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.

Universal and nonuniversal features in shadowing dynamics of nonhyperbolic chaotic systems with unstable-dimension variability

An important quantity characterizing the shadowability of computer-generated trajectories in nonhyperbolic chaotic system is the shadowing time, which measures for how long a numerical trajectory remains valid. This time depends sensitively on an initial condition. Here, we show that for nonhyperbolic systems with unstable-dimension variability, the probability distribution of the shadowing time contains two distinct scaling behaviors: an algebraic scaling for short times and an exponential scaling for long times. The exponential behavior depends on the system details but the small-time algebraic behavior appears to be universal.

Noise-induced unstable dimension variability and transition to chaos in random dynamical systems

Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapunov exponent becomes positive from zero in a continuous manner, and its scaling with the variation of the noise amplitude is algebraic. Formulas for the scaling exponent are derived in all dimensions. Numerical support using both low- and high-dimensional systems is provided.

Can noise make nonbursting chaotic systems more regular?

It has been known that noise can enhance the temporal regularity of dynamical systems that exhibit a bursting behavior--the phenomenon of coherence resonance. But can the phenomenon be expected for nonbursting chaotic systems? We present a theoretical argument based on the idea of time-scale matching and provide experimental evidence with a chaotic electronic circuit for coherence resonance in nonbursting chaotic systems.

Dynamical mechanism for coexistence of dispersing species

Dispersal of organisms may play an essential role in the coexistence of species. Recent studies of the evolution of dispersal in temporally varying environments suggest that clones differing in dispersal rates can coexist indefinitely. In this work, we explore the mechanism permitting such coexistence for a model of dispersal in a patchy environment, where temporal heterogeneity arises from endogenous chaotic dynamics. We show that coexistence arises from an extreme type of intermittent behavior, namely the phenomenon known as on-off intermittency. In effect, coexistence arises because of an alternation between synchronized and de-synchronized dynamical behaviors. Our analysis of the dynamical mechanism for on-off intermittency lends strong credence to the proposition that chaotic synchronism may be a general feature of species coexistence, where competing species differ only in dispersal rate.