Intermittent Chaos Research Articles

Band-edge localization as intermittent chaos.

Detailed statistical properties of the wave functions corresponding to the near-band-edge states, of the diagonal disorder Anderson model on a one-dimensional lattice are derived. The calculations are based on an exact map of the Anderson model to the problem of intermittency near a tangent bifurcation. Specifically, it is shown that a variable that equals the ratio of the wave function at neighboring sites satisfies a recursion relation similar to those investigated in the context of tangent bifurcations. As in the latter case one obtains intermittent dynamics of the pertinent variable, which is characterized by long laminar regions, followed by intermittent events and reinjection into a laminar region. Each intermittent event is shown to correspond to a node of the wave function. The wavelength of the wave function is thus space dependent, being determined by the number of steps (sites) between consecutive intermittent bursts. The correlation of the dynamical variables pertaining to different laminar regions is shown to vanish in the noiseless limit. It is shown that the amplitudes of the wave functions satisfy log-normal statistics whose parameters are computed and the corresponding phases obey normal statistics whose parameters are calculated as well. These properties are shown to be of a universal nature, to leading order in the strength of the noise. Beyond the results mentioned above our formulation reproduces in a straightforward manner the known values of the corresponding Lyapunov exponents and density of states and the appropriate scaling emerges in a natural way. Some physical implications of the above results are discussed.

Strongly intermittent chaos and scaling in an earthquake model.

We discuss the relation between scaling laws and dynamical behavior for earthquakes in the framework of a Burridge-Knopoff model. Due to the nontrivial interaction among many degrees of freedom, a new type of strongly intermittent chaos is found. The dynamics is dominated by wild fluctuations, implying exponential tails in the probability distributions. This is caused by very slow relaxation of time correlations, which gives rise to an anomalous behavior for the effective Lyapunov exponent and for the time signal of the earthquake magnitude.

Sinai disorder: intermittency for random maps

The properties of the first passage probability for a random walk with Sinai disorder are shown to be similar to those observed for deterministic intermittent chaos.

From simple to complex oscillatory behaviour via intermittent chaos in the Rose-Hindmarsh model for neuronal activity

The Rose-Hindmarsh equations are a system of three nonlinear ordinary differential equations that provide a phenomenological model for repetitive, patterned and irregular activity in molluscan neurones. We obtain bifurcation diagrams for this system, and obtain interval maps that reproduce the behaviour of the differential system. These maps are used to explore the bifurcations from simple to complex oscillatory behaviour.

Nonlinear behavior of a parametrically forced temperature-dependent model for a mite predator-prey interaction

The nonlinear behavior of a particular predator-prey differential equation system, with model parameters chosen appropriately for a temperature-dependent mite interaction on fruit trees and with forcing incorporated through environmental temperature, is examined numerically by means of the birfurcation code AUTO-86 as well as by calculation of Lyapunov exponents. A variety of dynamical behavior is demonstrated, including periodic, quasiperiodic, and chaotic attractors, multiple attractors, intermittency, and transient chaos. Resiliency is discussed with respect to basins of attraction and population perturbations in the parameter regions with multiple attractors. The ecological implications of the demonstrated nonlinear behavior are discussed.

Study on Dynamical Properties of Intermittent Phenomena in Turbulent Boundary Layer

Following the idea proposed previously that describes the phenomena observed in the outer region of the turbulent boundary layer in terms of a one-dimensional map generating the intermittent chaos, we attempt to analyze the problem with the use of statistical mechanical methodology applied frequently to dynamical systems. From the binary sequences constructed by processing measured data of instantaneous streamwise velocities, we obtain the singularity spectrum or the fluctuation of the dynamical scaling indices. In particular, the Kolmogorov entropy is given quantitatively as a function of the distance from the wall. The difference from random motions is emphasized.

Intermittently chaotic oscillations for a differential-delay equation with Gaussian nonlinearity.

For a differential-delay equation the time dependence of the variable is a function of the variable at a previous time. We consider a differential-delay equation with Gaussian nonlinearity that displays intermittent chaos. Although not the first example of a differential-delay equation that displays such behavior, for this example the intermittency is classified as type III, and the origin of the intermittent chaos may be qualitatively understood from the limiting forms of the equation for large and small variable magnitudes.

Proposal of chaotic steepest descent method for neural networks and analysis of their dynamics

AbstractMost of the neural network models based on the Lyapunov stability contain various problems such as the trap of the local minimum, and the limit of their dynamic performances has been pointed out.This paper attempts to provide a new dynamic performance to such a neural network model by introducing chaos dynamics. The features of the chaotic dynamic model proposed here is that the dynamical equation describing the trajectory on the energy curve has a periodically varying nonlinear resistance in the dissipation term. By iterating the stable and unstable phases, the chaotic transitions of the state can be realized.The proposed dynamic model is applied to the error backpropagation learning and the memory recall in the Hopfield‐type network, and the chaotic minimum transitions in the dynamic process are verified, it is verified further that the intermittent chaos generated by an appropriate parameter manipulation can introduce useful dynamic behaviors into the network, e.g., the flexible learning and the memory recall with a structure containing both stability and plasticity.

1/f noise generator using logarithmic and antilogarithmic amplifiers

A simple switched-capacitor circuit that generates 1/f/sup delta / noise is presented. The circuit realizes a discrete nonlinear one-dimensional map, that generates intermittent chaos with the 1/f/sup delta / power spectrum. An integrated circuit with a couple of logarithmic and antilogarithmic amplifiers is used to realize a nonlinear term x/sup Z/ in the map, which plays a key role in generating 1/f/sup delta / noise.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Intermittent Chaos Generated by Logarithmic Map

Intermittent Chaos Generated by Logarithmic Map

Intermittent Chaos Generated by Logarithmic Map

One-dimensional map described by x n+1 =1n (a|x n |) produces the type I and III intermittencies near the marginal points a=e and e −1 , respectively. Chaos appears between these parameter values. The power spectrum in the low-frequency region exhibits the equally-spaced Lorentzian lines near a=e, and f −ν -spectrum with exponent ν=0.95±0.01 near a=e −1

Reconsideration of the renormalization-group theory on intermittent chaos

The renormalization-group theory has made us believe that the average laminar length for type III intermittency is inversely proportional to the control parameter. However this does not agree with the result of numerical computations. This contradiction is removed through studying the scaling relation for the length of each laminar.

Renormalization-Group Theory on Intermittent Chaos in Relation to Its Universality

In order to compensate an imperfection in a previous theory, an extended renormalisation-group theory is constructed without using a local approximation. We show that the time evolution map of an ordinary differential equation displays the universal behavior of iterations

Renormalization-Group Theory on Intermittent Chaos in Relation to Its Universality

Renormalization-Group Theory on Intermittent Chaos in Relation to Its Universality

Is intermittent motion of outer flow in the turbulent boundary layer deterministic chaos?

Intermittent phenomena observed in the outer regions of the turbulent boundary layer are studied. A new idea is proposed in which the dynamics generating the intermittent phenomena can be described by a one-dimensional map; Xi+1=(1+ε)Xi+uXzi, for 0≤Xi≤Xc&amp;lt;1 (mode 1, z&amp;gt;1, u&amp;gt;0). The binary sequence {si} is constructed from carefully processed data of instantaneous streamwise velocities, and also from the map. The encoding is made in such a way that the turbulent state is referred to as si=1 and the nonturbulent state as si=0 at a discretized time i. Both sequences are found to have the common essential properties of intermittent chaos; the probability P(l) of finding nonturbulent states with length l exhibits the power law with an exponent r for l&amp;lt;lc, lc being a cutoff. From the exponent r, one can assign z almost equal to 4, while the control parameter ε is closely related to the distance from the wall. The scaling function of P(l) agrees very well with the one predicted from the map. The cutoff length lc is found to be associated with the average period of bursting.

Spatiotemporal intermittency on the sandpile.

The self-organized critical state exhibited by a sandpile model is shown to correspond to motion on an attractor characterized by an invariant distribution of the conserved variable. The largest Lyapunov exponent is equal to zero. Yet over time scales of the order of the linear size of the system, the model displays intermittent chaos. The divergence of local histories is found to exhibit intermittency in both time and space.

A Relationship between Local Error Growth and Quasi-stationary States: Case Study in the Lorenz System

Abstract Properties of the local predictability in the Lorenz system of three variables are investigated as a first step to develop a dynamical method for the skill prediction in the numerical weather forecasts instead of conventional statistical and empirical methods. As a measure of the local predictability, we adopt Lorenz's index which gives the amplification rate of the root-mean-square error during a prescribed time interval. In particular, we exert ourselves to understand a role of the quasi-stationary state in determining the variation of the Lorenz index. In an intermittent chaos regime, the Lorenz index determined for a time interval of the one return in the Poincare section has a minimum value at the onset of the laminar phase, gradually increases during the laminar phase, and abruptly attains a large value at the break of the laminar phase. If we consider the laminar phase as a quasi-stationary state generated by a local minimum point in the one-dimensional Poincare map, this characteristic ev...

Analytical approach to critical phenomena for dynamical properties of type-I intermittent chaos

A phase transition associated with dynamical properties of a type-I intermittency is analytically studied on the basis of the statistical-mechanical formalism previously proposed. At the saddle-node bifurcation point the continuous phase transition is found to occur at ${\mathit{q}}_{\mathit{c}}$ less than unity, due to the coexistence of different types of strange attractors with different measures. For nonvanishing shift parameter, crossover phenomena are discussed to obtain scaling exponents for the generalized entropies and the singularity spectrum. The exponents including a logarithmic correction are shown to depend on ${\mathit{q}}_{\mathit{c}}$, which means an absence of universality, in contrast to the usual critical phenomena.

Anomalous Convergence of the Time Average in Semi-Markovian Chaotic Dynamics

Two state semi-Markovian process and basic properties General theory of the semiMarkovian chaotic dynamics was developed in the previous paper,I) where the origin of long time tails was clearly explained_ In this article we are going to elucidate the large deviation for a dichotomous semi-Markovian process, which is the simplest model of intermittent chaos. We consider a random process a(t) which can take two sates A and B. The residence time r in each state is the statistically independent random variable. Denote the residence time distributions by a( r) and b( r) for A and B states respectively, and let a(t) be the indicator defined by a(t)=O in state A and a(t)=l in state B. We define a random variable AN = (l/N)JoNa(t)dt, which means the time average of aU) during the interval [0, N] and O::::::AN::::::L Denote the probability of the start at A (or B) state by PA (or PB) (h + PB=l), and the first pausing time distribution in each state by al(r) and bl(r) respectively, then the distribution density defined by p(N; il) =prob {il<AN::::::il+dil}/dil becomes

Statistical mechanics and crossover scaling for Pomeau-Manneville type intermittent chaos

On the basis of the statistical mechanics formalism, we study analytically intermittent chaos, concerning ourselves with the q-phase transition near the bifurcation point. The crossover exponents for the Renyi entropies, the singularity spectrum and the q-weighted rate of bursts are rigorously obtained.