Noisy one-dimensional maps near a crisis. II. General uncorrelated weak noise

The escape rate for one-dimensional noisy maps near a crisis is investigated. A previously introduced perturbation theory is extended to very general kinds of weak uncorrelated noise, including multiplicative white noise as a special case. For single-humped maps near the boundary crisis at fully developed chaos an asymptotically exact scaling law for the rate is derived. It predicts that transient chaos is stabilized by basically any noise of appropriate strength provided the maximum of the map is of sufficiently large order. A simple heuristic explanation of this effect is given. The escape rate is discussed in detail for noise distributions of Lévy, dichotomous, and exponential type. In the latter case, the rate is dominated by an exponentially leading Arrhenius factor in the deep precritical regime. However, the preexponential factor may still depend more strongly than any power law on the noise strength.     

Fluctuations and simple chaotic dynamics

We describe the effects of fluctuations on the period-doubling bifurcation to chaos. We study the dynamics of maps of the interval in the absence of noise and numerically verify the scaling behavior of the Lyapunov characteristic exponent near the transition to chaos. As previously shown, fluctuations produce a gap in the period-doubling bifurcation sequence. We show that this implies a scaling behavior for the chaotic threshold and determine the associated critical exponent. By considering fluctuations as a disordering field on the deterministic dynamics, we obtain scaling relations between various critical exponents relating the effect of noise on the Lyapunov characteristic exponent. A rule is developed to explain the effects of additive noise at fixed parameter value from the deterministic dynamics at nearby parameter values.     

Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity

Dissipative one-dimensional maps may exhibit special points (e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently, the sensitivity to the initial conditions has a power-law time dependence, instead of the usual exponential one. The associated exponent can be identified with 1/(1-q), where q characterizes the nonextensivity of a generalized entropic form currently used to extend standard, Boltzmann-Gibbs statistical mechanics in order to cover a variety of anomalous situations. It has been recently proposed (Lyra and Tsallis, Phys. Rev. Lett. 80, 53 (1998)) for such maps the scaling law , where and are the extreme values appearing in the multifractal function. We generalize herein the usual circular map by considering inflexions of arbitrary power z, and verify that the scaling law holds for a large range of z. Since, for this family of maps, the Hausdorff dimension d_f equals unity for all z in contrast with q which does depend on z, it becomes clear that d_f plays no major role in the sensitivity to the initial conditions. Received 5 February 1999     

Nonextensivity at the edge of chaos of a new universality class of one-dimensional unimodal dissipative maps

We introduce a new universality class of one-dimensional unimodal dissipative maps. The new family, from now on referred to as the (z₁, z₂)-logarithmic map, corresponds to a generalization of the z-logistic map. The Feigenbaum-like constants of these maps are determined. It has been recently shown that the probability density of sums of iterates at the edge of chaos of the z-logistic map is numerically consistent with a q-Gaussian, the distribution which, under appropriate constraints, optimizes the nonadditive entropy S_q. We focus here on the presently generalized maps to check whether they constitute a new universality class with regard to q-Gaussian attractor distributions. We also study the generalized q-entropy production per unit time on the new unimodal dissipative maps, both for strong and weak chaotic cases. The q-sensitivity indices are obtained as well. Our results are, like those for the z-logistic maps, numerically compatible with the q-generalization of a Pesin-like identity for ensemble averages.     

Lyapunov number for a noisy 2 n cycle

A stochastic one-dimensional map which produces a sequence of period doubling bifurcations is theoretically studied. We obtain analytic expressions, to a second-order approximation, of the local distribution function of fluctuating orbital points and the Lyapunov number for a noisy 2 ⁿ cycle. The expressions satisfy scaling laws and well agree with the results of numerical experiments when the external noise is weak. A scaling factor for the noise level is formulated in terms of the derivatives of a deterministic map. From it, the scaling factor is refined to be 6.6190 .... The Lyapunov number shows that, when the external noise is weaker than some extent, the noisy orbit is more stable rather than the deterministic one.     

Experimental characterization of transition to chaos in the presence of noise

Transition to chaos in the presence of noise is an important problem in nonlinear and statistical physics. Recently, a scaling law has been theoretically predicted which relates the Lyapunov exponent to the noise variation near the transition. Here we present experimental observation of noise-induced chaos in an electronic circuit and obtain the fundamental scaling law characterizing the transition. The experimentally obtained scaling exponent agrees very well with that predicted by theory.     

Controlling Strong Chaos by an Aperiodic Perturbation in Area Preserving Maps

We demonstrate a method for controlling strong chaos by an aperiodic perturbation in two-dimensional Hamiltonian systems.The method has the advantages that the controlled system remains conservative property and the selection of the perturbation has a considerable diversity.We illustrate this method with two area preserving maps:the non-monotonic twist map which is a mixed system and the perturbed cat map which exhibits hard chaos.Numerical results show that the strong chaos can be effectively controlled into regular motions,and the final states are always quasiperiodic ones.The method is robust against the presence of weak external noise.     

Ensemble Dynamics of Intermittency and Power-Law Decay

A spectral decomposition of the Frobenius–Perron operator is constructed for one-dimensional maps with intermittent chaos, using the method of coherent states. A technique using the spectral density function is applied to the the well-known cusp map, which generates weak type-II intermittency. Higher-order corrections are obtained to the leading 1/t long-time behavior of the x–x autocorrelation.     

Markov chains and discrete chaos

The connection between the statistics of a one-dimensional system exhibiting discrete chaos and its underlying deterministic law is formulated in terms of Markov chains. The correlation functions of piecewise linear Markovian maps are computed exactly using linear recursion formulae. An example of a non-Markovian map is given which can be approximated by Markovian maps.     

Stochastic dynamics of the cubic map: A study of noise-induced transition phenomena

The effects of finite-amplitude, additive noise on the dynamics generated by a one-dimensional, two-parameter cubic map are considered. The underlying deterministic system exhibits bistability and hysteresis, and noise-induced processes associated with these phenomena are studied. If a bounded noise source is applied to this system, trajectories may be confined to a finite region. Mechanisms are given for the merging transitions between different parts of this region and the eventual escape from it as the noise level is increased. The noisy dynamics is also represented by an integral evolution operator, with an equilibrium density function with finite support. The operator's spectrum is determined as a function of map parameters and noise amplitude. Such noisy one-dimensional maps can provide models for the study of noise-induced phenomena described by stochastic differential equations.     

Effect of boundary conditions on the invariant density of noisy maps at fully-developed chaos

The invariant density of one-dimensional maps in the regime of fully-developed chaos with uncorrelated additive noise is considered. Boundary conditions are shown to play a significant role in determining the precise form of the invariant density, via the manner in which they handle the spill-over, caused by the noise, of orbits beyond the interval. The known case of periodic boundary conditions is briefly recapitulated. Analytic solutions for the invariant density that are possible under certain conditions are presented with applications to specific well-known maps. The case of ‘sticky’ boundaries is generalized to ‘re-injection at the nearest boundary’, and the exact functional equations determining the invariant density are derived. Interesting boundary layer effects are shown to occur, that lead to significant modifications of the invariant density corresponding to the unperturbed (noise-free) case, even when the latter is a constant — as illustrated by an application of the formalism to the noisy tent map. All our results are non-perturbative, and hold good for any noise amplitude in the interval.     

Exponential decay and scaling laws in noisy chaotic scattering

In this Letter we present a numerical study of the effect of noise on a chaotic scattering problem in open Hamiltonian systems. We use the second order Heun method for stochastic differential equations in order to integrate the equations of motion of a two-dimensional flow with additive white Gaussian noise. We use as a prototype model the paradigmatic Hénon-Heiles Hamiltonian with weak dissipation which is a well-known example of a system with escapes. We study the behavior of the scattering particles in the scattering region, finding an abrupt change of the decay law from algebraic to exponential due to the effects of noise. Moreover, we find a linear scaling law between the coefficient of the exponential law and the intensity of noise. These results are of a general nature in the sense that the same behavior appears when we choose as a model a two-dimensional discrete map with uniform noise (bounded in a particular interval and zero otherwise), showing the validity of the algorithm used. We believe the results of this work be useful for a better understanding of chaotic scattering in more realistic situations, where noise is presented.     

A Class of Quantum System with Random Perturbation

In the presence of colored Gaussian noise, the quantum correspondence of Power system and its chaos anti-control are investigated. Some properties about the quantum chaos and classical chaos of the system are analyzed by adding the colored Gaussian noise to the phase of the system. Firstly, the quantization method is used to analyze some properties about the quantum correspondence of the classical chaos Power system. Then, the macroscopic dynamic behavior of the perturbed Power system is investigated. In addition, by using the computer simulation, we plot the Poincaré map and phase portraits to detect whether the system is chaos or not.     

Dynamics of simple one-dimensional maps under perturbation

It is known that the one-dimensional discrete maps having single-humped nonlinear functions with the same order of maximum belong to a single class that shows the universal behaviour of a cascade of period-doubling bifurcations from stability to chaos with the change of parameters. This paper concerns studies of the dynamics exhibited by some of these simple one-dimensional maps under constant perturbations. We show that the “universality” in their dynamics breaks down under constant perturbations with the logistic map showing different dynamics compared to the other maps. Thus these maps can be classified into two types with respect to their response to constant perturbations. Unidimensional discrete maps are interchangeably used as models for specific processes in many disciplines due to the similarity in their dynamics. These results prove that the differences in their behaviour under perturbations need to be taken into consideration before using them for modelling any real process.     

Effects of noise on periodic orbits of the logistic map

Noise can induce an inverse period-doubling transition and chaos. The effects of noise on each periodic orbit of three different period sequences are investigated for the logistic map. It is found that the dynamical behavior of each orbit, induced by an uncorrelated Gaussian white noise, is different in the mergence transition. For an orbit of the period-six sequence, the maximum of the probability density in the presence of noise is greater than that in the absence of noise. It is also found that, under the same intensity of noise, the effects of uncorrelated Gaussian white noise and exponentially correlated colored (Gaussian) noise on the period-four sequence are different.      

Stochastic resonance and chaotic resonance in bimodal maps: A case study

We present the results of an extensive numerical study on the phenomenon of stochastic resonance in a bimodal cubic map. Both Gaussian random noise as well as deterministic chaos are used as input to drive the system between the basins. Our main result is that when two identical systems capable of stochastic resonance are coupled, the SNR of either system is enhanced at an optimum coupling strength. Our results may be relevant for the study of stochastic resonance in biological systems.     

Routes to chaos,universality and glass formation

We review recent results obtained for the dynamics of incipient chaos. These results suggest a common picture underlying the three universal routes to chaos displayed by the prototypical logistic and circle maps. Namely, the period doubling, intermittency, and quasiperiodicity routes. In these situations the dynamical behavior is exactly describable through infinite families of Tsallis’ q-exponential functions. Furthermore, the addition of a noise perturbation to the dynamics at the onset of chaos of the logistic map allows to establish parallels with the behavior of supercooled liquids close to glass formation. Specifically, the occurrence of two-step relaxation, aging with its characteristic scaling property, and subdiffusion and arrest is corroborated for such a system.     

Front propagation in chaotic and noisy reaction-diffusion systems: a discrete-time map approach

We study the front propagation in reaction-diffusion systems whose reaction dynamics exhibits an unstable fixed point and chaotic or noisy behaviour. We have examined the influence of chaos and noise on the front propagation speed and on the wandering of the front around its average position. Assuming that the reaction term acts periodically in an impulsive way, the dynamical evolution of the system can be written as the convolution between a spatial propagator and a discrete-time map acting locally. This approach allows us to perform accurate numerical analysis. They reveal that in the pulled regime the front speed is basically determined by the shape of the map around the unstable fixed point, while its chaotic or noisy features play a marginal role. In contrast, in the pushed regime the presence of chaos or noise is more relevant. In particular the front speed decreases when the degree of chaoticity is increased, but it is not straightforward to derive a direct connection between the chaotic properties (e.g. the Lyapunov exponent) and the behaviour of the front. As for the fluctuations of the front position, we observe for the noisy maps that the associated mean square displacement grows in time as t ^1/2 in the pushed case and as t ^1/4 in the pulled one, in agreement with recent findings obtained for continuous models with multiplicative noise. Moreover we show that the same quantity saturates when a chaotic deterministic dynamics is considered for both pushed and pulled regimes. Received 17 July 2001     

Sensitivity to initial conditions, entropy production, and escape rate at the onset of chaos

We analytically link three properties of nonlinear dynamical systems, namely sensitivity to initial conditions, entropy production, and escape rate, in z-logistic maps for both positive and zero Lyapunov exponents. We unify these relations at chaos, where the Lyapunov exponent is positive, and at its onset, where it vanishes. Our result unifies, in particular, two already known cases, namely (i) the standard entropy rate in the presence of escape, valid for exponential functionality rates with strong chaos, and (ii) the Pesin-like identity with no escape, valid for the power-law behavior present at points such as the Feigenbaum one.