Attractors of convex maps with positive Schwarzian derivative in the presence of noise

One-dimensional maps have proved to be useful models for understanding the transition to turbulence. We investigate a smooth perturbation of the well-known logistic system in order to examine numerically the change in the bifurcation behavior which is observed, when the Schwarzian derivative is allowed to become positive. We find coexistence of a fixed point attractor and a periodic or chaotic two-band-attractor. The chaotic two-band attractor can disappear by yielding a preturbulent state which will asymptotically settle down to a fixed-point. The chaotic behavior of some systems can be destroyed by arbitrarily small amounts of external noise. The concept of (ε, δ)-diffusions is used to describe the sensitivity of attractors against external noise. We also observe a direct transition from a fixed-point to a chaotic one-band attractor. This can be interpreted as type-III-intermittency of Pomeau and Manneville but with an almost linear scaling behavior of the Lyapunov exponent.     

Noise enhanced phase synchronization and coherence resonance in sets of chaotic oscillators with weak global coupling

The effect of noise on phase synchronization in small sets and larger populations of weakly coupled chaotic oscillators is explored. Both independent and correlated noise are found to enhance phase synchronization of two coupled chaotic oscillators below the synchronization threshold; this is in contrast to the behavior of two coupled periodic oscillators. This constructive effect of noise results from the interplay between noise and the locking features of unstable periodic orbits. We show that in a population of nonidentical chaotic oscillators, correlated noise enhances synchronization in the weak coupling region. The interplay between noise and weak coupling induces a collective motion in which the coherence is maximal at an optimal noise intensity. Both the noise-enhanced phase synchronization and the coherence resonance numerically observed in coupled chaotic R?ssler oscillators are verified experimentally with an array of chaotic electrochemical oscillators.     

Chaotic solitons in the quadratic-cubic nonlinear Schrödinger equation under nonlinearity management

We analyze the response of rational and regular (hyperbolic-secant) soliton solutions of an extended nonlinear Schro?dinger equation (NLSE) which includes an additional self-defocusing quadratic term, to periodic modulations of the coefficient in front of this term. Using the variational approximation (VA) with rational and hyperbolic trial functions, we transform this NLSE into Hamiltonian dynamical systems which give rise to chaotic solutions. The presence of chaos in the variational solutions is corroborated by calculating their power spectra and the correlation dimension of the Poincare? maps. This chaotic behavior (predicted by the VA) is not observed in the direct numerical solutions of the NLSE when rational initial conditions are used. The solitary-wave solutions generated by these initial conditions gradually decay under the action of the nonlinearity management. On the contrary, the solutions of the NLSE with exponentially localized initial conditions are robust solitary-waves with oscillations consistent with a chaotic or a complex quasiperiodic behavior.     

Nonmonotonicity in the quantum-classical transition: chaos induced by quantum effects

The classical-quantum transition for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative variant Planck's over 2pi is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the double-well Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effective variant Planck's over 2pi is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantum-classical transition in this case is nonmonotonic in variant Planck's over 2pi.     

The origin of chaos

This paper deals with the origin of observed chaotic behavior in nonlinear systems and the meaning of a solution when chaos is present.     

Wave trains in a model of gypsy moth population dynamics

A recent model of gypsy moth [Lymantria dispar (Lepidoptera: Lymantriidae)] populations led to the observation of traveling waves in a one-dimensional spatial model. In this work, these waves are studied in more detail and their nature investigated. It was observed that when there are no spatial effects the model behaves chaotically under certain conditions. Under the same conditions, when diffusion is allowed, traveling waves develop. The biomass densities involved in the model, when examined at one point in the spatial domain, are found to correspond to a limit cycle lying on the surface of the chaotic attractor of the spatially homogeneous model. Also observed are wave trains that have modulating maxima, and which when examined at one point in the spatial domain show a quasiperiodic temporal behavior. This complex behavior is determined to be due to the interaction of the traveling wave and the chaotic background dynamics. (c) 1995 American Institute of Physics.     

Synchronization of chaotic oscillator time scales

We consider chaotic oscillator synchronization and propose a new approach for detecting the synchronized behavior of chaotic oscillators. This approach is based on analysis of different time scales in the time series generated by coupled chaotic oscillators. We show that complete synchronization, phase synchronization, lag synchronization, and generalized synchronization are particular cases of the synchronized behavior called time-scale synchronization. A quantitative measure of chaotic oscillator synchronous behavior is proposed. This approach is applied to coupled Rössler systems.     

Noise-induced order

A new noise effect on chaos in one-dimensional mappings is reported. The transition from chaotic behavior to ordered behavior induced by external noise is observed in a certain class of one-dimensional mappings. This transition is clearly shown in terms of the Lyapunov number, entropy, power spectrum, and the nature of orbits.     

Correlation between jerky flow and jerky dynamics in a nanoscratch on a metallic glass film

Chaos has been well understood in dynamic system, however, how the chaotic behavior occur in jerky flow in material, is still not clear, and is lack of specific chaotic attractor. Here the jerky evolution of lateral force and the stair-like fluctuation of lateral displacement are observed for Ni62 Nb38(at.%) metallic glass film during nanoscratch process. This jerky flow is investigated by using the largest Lyapunov exponent, Kolmogorov entropy and fractal dimension, and chaotic behavior of lateral force-time and normal displacement-lateral displacement sequences is verified. In addition to time series analysis, it is found that jerk equation can be used to describe the jerky flow of the metallic-glass film during nanoscratch. More importantly, unambiguous chaotic attractor is presented by jerky dynamics using "jerk"-singularities, namely the total change rate of lateral force relative to scratch time. These reveal an inner connection between jerky flow and jerky dynamics in nanoscratch of a metallic-glass film.     

Underlying dynamics of the pedestal components in a modulated laser diode with noise

We demonstrate that the pedestal components observed in the power spectra of a directly modulated laser diode, which were interpreted as a sign of instability of the periodic regime, are an indication of the coexistence of a chaotic regime with the periodic one. We present the underlying dynamics behind the rise of these pedestals, showing two different situations in which the pedestals appear. In both, a periodic regime coexists with another attractor, a saddle cycle in one case and a chaotic attractor in the other. The random fluctuations included in the laser diode model allow the coexisting attractors to merge in the observed behavior of the laser.     

一种双光子量子光学模型的混沌行为

研究了一种二能级原子与辐射场的双光子相互作用模型的混沌行为。结果表明,该模型在不加扰动项或调制项时,即存在着混沌演化行为。以正的李亚普诺夫指数和功率谱的宽峰以及强噪声背景为标志,对混沌行为的特性作了讨论,并证明了在旋转波近似下所得到的双光子Jaynes-Cummings模型并不存在混沌演化行为。 关键词：     

Theoretical derivation of 1/f noise in quantum chaos

It was recently conjectured that 1/f noise is a fundamental characteristic of spectral fluctuations in chaotic quantum systems. This conjecture is based on the power spectrum behavior of the excitation energy fluctuations, which is different for chaotic and integrable systems. Using random matrix theory, we derive theoretical expressions that explain without free parameters the universal behavior of the excitation energy fluctuations power spectrum. The theory gives excellent agreement with numerical calculations and reproduces to a good approximation the 1/f (1/f(2)) power law characteristic of chaotic (integrable) systems. Moreover, the theoretical results are valid for semiclassical systems as well.     

Chaotic rocking behavior of freestanding objects with sliding motion

This paper examines the influence of effects of sliding on the non-linear rocking response behavior of freestanding rigid objects (blocks) subjected to harmonic horizontal and vertical excitations. It is well known that the rocking responses depend strongly on the impact effect between object and the base, which takes place with abrupt reduction in kinetic energy. In this study, it is shown that the rocking behavior is significantly affected by the presence of the sliding motion. A parametric response analysis is carried out over a range of excitation amplitudes and frequencies. Chaotic responses are observed over a wide response region, particularly for the case of large vertical amplitude excitation with significant sliding motions. The chaotic characteristics are demonstrated using time histories, Poincaré sections, power spectral density and Lyapunov exponents of the rocking responses. The complex chaotic response behavior is illustrated by Poincaré section in the phase space. The distribution of various types of rocking responses and the effects of sliding motion are examined via bifurcation diagrams and examples of typical rocking responses.     

Dependence of chaotic actuation dynamics of Casimir oscillators on optical properties and electrostatic effects

With Casimir and electrostatic forces playing a crucial role for the performance and stability of microelectromechanical systems (MEMS), the presence of chaotic behavior, which is often unavoidable, leads to device malfunction due to stiction. Therefore, we investigate here how the optical properties of different materials influence the chaotic behavior of electrostatic torsional MEMS due to changes in magnitude of the Casimir forces and torques. We consider the materials Au, which is a good conductor, AIST, which is a phase change material being close to metal in the crystalline state, and finally doped SiC as a very poor conductor. For the conservative systems, there is no chaotic behavior and the analysis of phase portraits and bifurcation diagrams reveal the strong sensitivity of stable actuation dynamics on the material optical properties, while applied electrostatic potentials lead faster to instability and stiction for higher conductivity materials. For the driven systems, the Melnikov method is used to study the chaotic behavior. The results from this method are supported by the study of the contours of the transient time to stiction in the phase plane, which reveal a substantially increased chaotic behavior for higher conductivity materials, associated with stronger Casimir torques and applied electrostatic potentials.     

An approach to chaotic synchronization

This paper deals with the chaotic oscillator synchronization. An approach to the synchronization of chaotic oscillators has been proposed. This approach is based on the analysis of different time scales in the time series generated by the coupled chaotic oscillators. It has been shown that complete synchronization, phase synchronization, lag synchronization, and generalized synchronization are the particular cases of the synchronized behavior called "time-scale synchronization." The quantitative measure of chaotic oscillator synchronous behavior has been proposed. This approach has been applied for the coupled R?ssler systems and two coupled Chua's circuits.     

Chaotic capture of vortices by a moving body. II. Bound pair model

Previously, we have presented a simple model for the interaction of a fluid vortex structure with a moving bluff body, and demonstrated the existence of a trapping mechanism related to chaotic scattering. This single point vortex model required explicit perturbation to generate chaos and the subsequent complex dynamics. Here, we present a model which attempts to introduce internal degrees-of-freedom in the vortex structure in the simplest manner, by replacing the single vortex with a like-signed pair. We show that this model exhibits chaotic trapping without the need of explicit perturbation, however, the region of parameter space for which trapping occurs is exceedingly small due to the spatially dependent form of the perturbation. We claim that this result explains some the behavior observed in Navier-Stokes simulations of the same vortex-body system, where we find close correspondence between the dynamics of an extended vorticity distribution and the single vortex model. Finally, we generalize the model to unequal strength vortex pairs, and find more complex behavior which includes "partial" capture of the weaker vortex by the body. (c) 1994 American Institute of Physics.     

Phase order in one-dimensional piecewise linear discontinuous map

The phase order in a one-dimensional(1 D) piecewise linear discontinuous map is investigated. The striking feature is that the phase order may be ordered or disordered in multi-band chaotic regimes, in contrast to the ordered phase in continuous systems. We carried out an analysis to illuminate the underlying mechanism for the emergence of the disordered phase in multi-band chaotic regimes, and proved that the phase order is sensitive to the density distribution of the trajectories of the attractors. The scaling behavior of the net direction phase at a transition point is observed. The analytical proof of this scaling relation is obtained. Both the numerical and analytical results show that the exponent is 1, which is controlled by the feature of the map independent on whether the system is continuous or discontinuous. It extends the universality of the scaling behavior to systems with discontinuity. The result in this work is important to understanding the property of chaotic motion in discontinuous systems.     

n-scroll chaotic attractors from a first-order time-delay differential equation

In this paper, a novel first-order delay differential equation capable of generating n-scroll chaotic attractor is presented. Hopf bifurcation of the introduced n-scroll chaotic system is analytically and numerically determined. The bifurcation diagram and Lyapunov spectrum of the system are calculated and the results show that the system has a chaotic regime in a wider parameter range. Furthermore, period-3 behavior has been observed on the system. Circuit realizations of two-, three-, four-, and five-scroll chaotic attractors are also presented.     

共轭Chen混沌系统的分岔分析及基于该系统的超混沌生成研究

分析了一个三维自治混沌系统的Hopf分岔现象,该系统的混沌吸引子属于共轭Chen混沌系统.通过引入一个控制器,基于该混沌系统构建了一个四维自治超混沌系统.该超混沌系统含有一个单参数,在一定的参数范围内呈现超混沌现象.通过Lyapunov指数和分岔分析,随着参数的变化该系统轨道呈现周期轨道、准周期轨道、混沌和超混沌的演化过程. 关键词： 混沌 超混沌生成 Hopf分岔 分岔分析