Qualitative analysis of the Rössler equations: Bifurcations of limit cycles and chaotic attractors

In this paper we study different aspects of the paradigmatic Rössler model. We perform a detailed study of the local and global bifurcations of codimension one and two of limit cycles. This provides us a global idea of the three-parametric evolution of the system. We also study the regions of parameters where we may expect a chaotic behavior by the use of different Chaos Indicators. The combination of the different techniques gives an idea of the different routes to chaos and the different kinds of chaotic attractors we may found in this system.     

On exact N-loop soliton solution to nonlinear coupled dispersionless evolution equations

We investigate some nonlinear coupled dispersionless evolution equations (NLCDEE) modelling the dynamics of a current-fed string within an external magnetic field in 2D-space. Using a blend of transformations of independent variables, we derive from the previous equations a Schäfer-Wayne short pulse equation (SWSPE). By means of a transformation back to the original independent variables, we find the N-loop soliton solution to the coupled equations. We give some detail on the scattering behavior of two-loop solitons.     

Synchronization in coupled nonidentical incommensurate fractional-order systems

Synchronization of fractional-order nonlinear systems has received considerable attention for many research activities in recent years. In this Letter, we consider the synchronization between two nonidentical fractional-order systems. Based on the open-plus-closed-loop control method, a general coupling applied to the response system is proposed for synchronizing two nonidentical incommensurate fractional-order systems. We also derive a local stability criterion for such synchronization behavior by utilizing the stability theory of linear incommensurate fractional-order differential equations. Feasibility of the proposed coupling scheme is illustrated through numerical simulations of a limit cycle system, a chaotic system and a hyperchaotic system.     

Projective Synchronization in Time-Delayed Chaotic Systems

For the first time, we report on projective synchronization between two time delay chaotic systems with single time delays. It overcomes some limitations of the previous work, where projective synchronization has been investigated only in finite-dimensional chaotic systems, so we can achieve projective synchronization in infinitedimensional chaotic systems. We give a general method with which we can achieve projective synchronization in time-delayed chaotic systems. The method is illustrated using the famous delay-differential equations related to optical bistability. Numerical simulations fully support the analytical approach.     

Controlling global stochasticity of Hamiltonian systems by nonlinear perturbation

A method of controlling global stochasticity in Hamiltonian systems by applying nonlinear perturbation is proposed. With the well-known standard map we demonstrate that this control method can convert global stochasticity into regular motion in a wide chaotic region for arbitrary initial condition, in which the control signal remains very weak after a few kicks. The system in which chaos has been controlled approximates to the original Hamiltonian system, and this approach appears robust against small external noise. The mechanism underlying this high control efficiency is intuitively explained. Received 15 January 2002 Published online 6 June 2002     

Noise-enhanced trapping in chaotic scattering

We show that noise enhances the trapping of trajectories in scattering systems. In fully chaotic systems, the decay rate can decrease with increasing noise due to a generic mismatch between the noiseless escape rate and the value predicted by the Liouville measure of the exit set. In Hamiltonian systems with mixed phase space we show that noise leads to a slower algebraic decay due to trajectories performing a random walk inside Kolmogorov-Arnold-Moser islands. We argue that these noise-enhanced trapping mechanisms exist in most scattering systems and are likely to be dominant for small noise intensities, which is confirmed through a detailed investigation in the Hénon map. Our results can be tested in fluid experiments, affect the fractal Weyl's law of quantum systems, and modify the estimations of chemical reaction rates based on phase-space transition state theory.     

Entanglement, quantum phase transition and fixed-point bifurcation in the N-atom Jaynes-Cummings model with an additional symmetry breaking term

In the present work we analyze the quantum phase transition (QPT) in the N-atom Jaynes-Cummings model (NJCM) with an additional symmetry breaking interaction term in the Hamiltonian. We show that depending on the type of symmetry breaking term added the transition order can change or not and also the fixed point associated to the classical analogue of the Hamiltonian can bifurcate or not. We present two examples of symmetry broken Hamiltonians and discuss based on them, the interconnection between the transition order, appearance of bifurcation and the behavior of the entanglement.     

On bifurcations in a chaotically stirred excitable medium

We study a one-dimensional filamental model of a chaotically stirred excitable medium. In a numerical simulation we systematically explore its rich bifurcation scenarios involving saddle-nodes, Hopf bifurcations and hysteresis loops. The bifurcations are described in terms of two parameters signifying the excitability of the reacting medium and the strength of the chaotic stirring, respectively. The solution behaviour, in particular at the bifurcation points, is analytically described by means of a nonperturbative variational method. Using this method we reduce the partial differential equations to either algebraic equations for stationary solutions and bifurcations, or to ordinary differential equations in the case of non-stationary solutions and bifurcations. We present numerical simulations corroborating our analytical results.     

Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order

This Letter further investigates the full state hybrid projective synchronization (FSHPS) of chaotic and hyper-chaotic systems with fully unknown parameters. Based on the Lyapunov stability theory, a unified adaptive controller and parameters update law can be designed for achieving the FSHPS of chaotic and/or hyper-chaotic systems with the same and different order. Especially, for two chaotic systems with different order, reduced order MFSHPS (an acronym for modified full state hybrid projective synchronization) and increased order MFSHPS are first studied in this Letter. Five groups numerical simulations are provided to verify the effectiveness of the proposed scheme. In addition, the proposed FSHPS scheme is quite robust against the effect of noise.     

On the viability of local criteria for chaos

Recently, Zscze¸sny and Dobrowolski proposed a geometrical criterion for local instability based on the geodesic deviation equation. Although such a criterion can be useful in some cases, we show here that, in general, it is neither necessary nor sufficient for the occurrence of chaos. To this purpose, we introduce a class of chaotic two-dimensional systems with Gaussian curvature everywhere positive and, hence, locally stable. We show explicitly that chaotic behavior arises from some trajectories that reach certain non-convex parts of the boundary of the effective Riemannian manifold. Our result questions, once more, the viability of local, curvature-based criteria to predict chaotic behavior.     

Exponential recovery of low frequency fluctuations in a diode laser with optical feedback

We show that the recovery after each power drop on the chaotic low frequency fluctuations in a semiconductor laser with optical feedback follows an exponential envelope. The time constant for such exponential behavior was experimentally measured. This recovery time constant and the average time interval between consecutive drops have different dependences when measured as function of the pump current.     

Impulsive control of a class of vibro-impact systems

In this Letter, the impulsive control method is developed to stabilize the chaotic motions in a class of vibro-impact systems. The strategy of the control is to implement the pulses just when the impact occurs. As applications of this method, we present the numerical simulations of two impact oscillators. Our numerical results indicate that the method used here could suppress chaos into periodic orbits which embedded in the chaotic attractor effectively, and also show that the method is robust even for high levels of multiplicative noise or additive noise.     

Grazing-induced crises in hybrid dynamical systems

In hybrid dynamical systems including both continuous and discrete components, an interplay between a continuous trajectory and a discontinuity boundary can trigger a sudden qualitative change in the system dynamics. Grazing phenomena, which occur when a continuous trajectory hits a boundary tangentially, are well known as a representative of such phenomena. We demonstrate that a grazing phenomenon of a chaotic attractor can result in its sudden disappearance and initiate chaotic transients. The mechanism of this grazing-induced crisis is revealed in an illustrative example. Furthermore, we derive a formula to obtain the critical exponent of the power law on the mean duration of chaotic transients.     

Universality of level spacing distributions in classical chaos

We suggest that random matrix theory applied to a matrix of lengths of classical trajectories can be used in classical billiards to distinguish chaotic from non-chaotic behavior. We consider in 2D the integrable circular and rectangular billiard, the chaotic cardioid, Sinai and stadium billiard as well as mixed billiards from the Limaçon/Robnik family. From the spectrum of the length matrix we compute the level spacing distribution, the spectral auto-correlation and spectral rigidity. We observe non-generic (Dirac comb) behavior in the integrable case and Wignerian behavior in the chaotic case. For the Robnik billiard close to the circle the distribution approaches a Poissonian distribution. The length matrix elements of chaotic billiards display approximate GOE behavior. Our findings provide evidence for universality of level fluctuations—known from quantum chaos—to hold also in classical physics.     

Stochastic synchronization of coupled neural networks with intermittent control

In this Letter, we study the exponential stochastic synchronization problem for coupled neural networks with stochastic noise perturbations. Based on Lyapunov stability theory, inequality techniques, the properties of Weiner process, and adding different intermittent controllers, several sufficient conditions are obtained to ensure exponential stochastic synchronization of coupled neural networks with or without coupling delays under stochastic perturbations. These stochastic synchronization criteria are expressed in terms of several lower-dimensional linear matrix inequalities (LMIs) and can be easily verified. Moreover, the results of this Letter are applicable to both directed and undirected weighted networks. A numerical example and its simulations are offered to show the effectiveness of our new results.     

Integrable discrete static Hamiltonian with a parametric double-well potential

A one-dimensional discrete conservative Hamiltonian with a generalized form of the Schmidt potential, is constructed with the help of a non-integrable discrete Hamiltonian whose parametrized double-well potential can be reduced to the ?⁴ potential. The new conservative Hamiltonian is completely integrable in the discrete static regime, and the associate exact nonlinear solution is shown to coincide with the continuum nonlinear periodic solution of the non-integrable Hamiltonian. Numerical simulations and nonlinear stability analysis suggest that the discrete mapping derived from the completely integrable Hamiltonian undergoes a bifurcation which does not leads to the chaotic phase with randomly pinned states, but instead to a phase where real solutions become rare forming a cluster of periodic points around an elliptic fixed point.     

Empirical analysis of individual popularity and activity on an online music service system

Quantitative understanding of human behaviors supplies basic comprehension of the dynamics of many socio-economic systems. Based on the log data of an online music service system, we investigate the statistical characteristics of individual activity and popularity, and find that the distributions of both of them follow a stretched exponential form which interpolates between exponential and power law distribution. We also study the human dynamics on the online system and find that the distribution of interevent time between two consecutive listenings of music shows the fat tail feature. Besides, with the reduction of user activity the fat tail becomes more and more irregular, indicating different behavior patterns for users with diverse activities. The research results may shed some light on the in-depth understanding of collective behaviors in socio-economic systems.     

Image encryption with chaotically coupled chaotic maps

We present a novel secure cryptosystem for direct encryption of color images, based on chaotically coupled chaotic maps. The proposed cipher provides good confusion and diffusion properties that ensures extremely high security because of the chaotic mixing of pixels’ colors. Information is mixed and distributed over a complete image using a complex strategy that makes known plaintext attack unfeasible. The encryption algorithm guarantees the three main goals of cryptography: strong cryptographic security, short encryption/decryption time, and robustness against noise and other external disturbances. Due to the high speed, the proposed cryptosystem is suitable for application in real-time communication systems.     

Function projective synchronization of two-cell quantum-CNN chaotic oscillators by nonlinear adaptive controller

In this Letter, we investigate function projective synchronization of two-cell quantum-CNN chaotic oscillators using nonlinear adaptive controller. Based on Lyapunov stability theory, the nonlinear adaptive control law is derived to make the state of two chaotic systems function projective synchronized. Two numerical simulations are presented to illustrate the effectiveness of the proposed nonlinear adaptive control scheme, which is more effective than that in previous literature.     

Noise-induced bifurcations and chaos in the average motion of globally-coupled oscillators

A system of coupled master equations simplified from a model of noise-driven globally coupled bistable oscillators under periodic forcing is investigated. In the thermodynamic limit, the system is reduced to a set of two coupled differential equations. Rich bifurcations to subharmonics and chaotic motions are found. This behavior can be found only for certain intermediate noise intensities. Noise with intensities which are too small or too large will certainly spoil the bifurcations. In a system with large though finite size, the bifurcations to chaos induced by noise can still be detected to a certain degree. Received 6 April 1999 and Received in final form 1 November 1999