Chaotic dynamics of a parametrically modulated Josephson junction with quadratic damping

We study the chaotic dynamics of a parametrically modulated Josephson junction with quadratic damping. The Melnikov chaotic criteria are presented. When the perturbation conditions cannot be satisfied, numerical simulations demonstrate that the system can step into chaos via a quasi-periodic route with the increasing of the dc component of the modulation. However, it is numerically demonstrated that adding a feedback to the system can effectively suppress the chaos.     

The width of a chaotic layer

A model of nonlinear resonance as a periodically perturbed pendulum is considered, and a new method of analytical estimating the width of a chaotic layer near the separatrices of the resonance is derived for the case of slow perturbation (the case of adiabatic chaos). The method turns out to be successful not only in the case of adiabatic chaos, but in the case of intermediate perturbation frequencies as well.     

Controlling Chaos Probability of a Bose-Einstein Condensate in a Weak Optical Superlattice

The spatial chaos probability of a Bose-Einstein condensate perturbed by a weak optical superlattice is studied. It is demonstrated that the spatial. chaotic solution appears with a certain probability in a given parameter region under a random boundary condition. The effects of the lattice depths and wave vectors on the chaos probability are illustrated, and different regions associated with different chaos probabilities are found. This suggests a feasible scheme for suppressing and strengthening chaos by adjusting the optical superlattice experimentaJly.     

Chaotic phase oscillation of a proton beam in a synchrotron

We investigate the chaotic phase oscillation of a proton beam in a cooler synchrotron. By using direct perturbation method, we construct the general solution of the 1st-order equation. It is demonstrated that the general solution is bounded under some initial and parameter conditions. From these conditions, we get a Melnikov function which predicts the existence of Smale-horseshoe chaos iff it has simple zeros. Our result under the 1st-order approximation is in good agreement with that in [H. Huang et al., Phys. Rev. E 48 (1993) 4678]. When the perturbation method is not suitable for the system, numerical simulation shows the system may present transient chaos before it goes into periodical oscillation; changing the damping parameter can result in or suppress stationary chaos.     

Chaotic Dynamics of a Josephson Junction with a Ratchet Potential and Current-Modulating Damping

The chaotic dynamics of a Josephson junction with a ratchet potential and current-modulating damping are studied. Under the first-order approximation, we construct the general solution of the first-order equation whose boundedness condition contains the famous Melnikov chaotic criterion. Based on the general solution, the incomputability and unpredictability of the system’s chaotic behavior are discussed. For the case beyond perturbation conditions, the evolution of stroboscopic Poincaré sections shows that the system undergoes a quasi-periodic transition to chaos with an increasing intensity of the rf-current. Through a suitable feedback controlling strategy, the chaos can be effectively suppressed and the intensity of the controller can vary in a large range. It is also found that the current between the two separated superconductors increases monotonously in some specific parameter spaces.     

Adaptive lag synchronization in unknown stochastic chaotic neural networks with discrete and distributed time-varying delays

In this Letter, we have dealt with the problem of lag synchronization and parameter identification for a class of chaotic neural networks with stochastic perturbation, which involve both the discrete and distributed time-varying delays. By the adaptive feedback technique, several sufficient conditions have been derived to ensure the synchronization of stochastic chaotic neural networks. Moreover, all the connection weight matrices can be estimated while the lag synchronization is achieved in mean square at the same time. The corresponding simulation results are given to show the effectiveness of the proposed method.     

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.     

Chaos synchronization of uncertain Genesio-Tesi chaotic systems with deadzone nonlinearity

In this Letter, the concept of practical synchronization is introduced and the chaos synchronization of uncertain Genesio-Tesi chaotic systems with deadzone nonlinearity is investigated. Based on the time-domain approach, a tracking control is proposed to realize chaos synchronization for the uncertain Genesio-Tesi chaotic systems with deadzone nonlinearity. Moreover, the guaranteed exponential convergence rate and convergence radius can be pre-specified. Finally, a numerical example is provided to illustrate the feasibility and effectiveness of the obtained result.     

Partial state feedback control of chaotic neural network and its application

The chaos control in the chaotic neural network is studied using the partial state feedback with a control signal from a few control neurons. The controlled CNN converges to one of the stored patterns with a period which depends on the initial conditions, i.e., the set of control neurons and other control parameters. We show that the controlled CNN can distinguish between two initial patterns even if they have a small difference. This implies that such a controlled CNN can be feasibly applied to information processing such as pattern recognition.     

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     

Analytical and Numerical Studies for Chaotic Dynamics of a Duffing Oscillator with a Parametric Force

The chaotic dynamics of a Duffing oscillator with a parametric force is investigated. By using the direct perturbation technique, we analytically obtain the general solution of the 1st-order equation. Through the boundedness condition of the general solution we get the famous Melnikov function predicting the onset of chaos. When the parametric and external forces are strong, numerical simulations show that increasing the amplitude of the parametric or external force can lead the system into chaos via period doubling.     

Analytical and Numerical Studies for Chaotic Dynamics of a Duffing Oscillator with a Parametric Force

The chaotic dynamics of a Duffing oscillator with a parametric force is investigated. By using the direct perturbation technique, we analytically obtain the general solution of the lst-order equation. Through the boundedness condition of the general solution we get the famous Melnikov function predicting the onset of chaos. When the parametric and external forces are strong, numerical simulations show that increasing the amplitude of the parametric or external force can lead the system into chaos via period doubling.     

Experimental Investigation of Chaos Synchronization in DFB Diode Lasers with Unsymmetrical Scheme

We experimentally generate high dimension chaotic waveforms with smooth spectrum using a distributed feedback （DFB） semiconductor laser with unidirectional fibre ring long-cavity feedback, and implement the stable chaos synchronization when the chaotic light is injected into a solitary DFB laser diode. The synchronization quality is investigated by time-domain and frequency-domain analysis separately, The frequency-domain analysis indicates that the synchronization has higher quality in the high frequency band. The influences of the injection strength and the frequency detuning on the synchronization are measured. Our experimental results show that the robust synchronization can be maintained with the optical frequency detuning from -11GHz to 40 GHz.     

A search for the simplest chaotic partial differential equation

A numerical search for the simplest chaotic partial differential equation (PDE) suggests that the Kuramoto-Sivashinsky equation is the simplest chaotic PDE with a quadratic or cubic nonlinearity and periodic boundary conditions. We define the simplicity of an equation, enumerate all autonomous equations with a single quadratic or cubic nonlinearity that are simpler than the Kuramoto-Sivashinsky equation, and then test those equations for chaos, but none appear to be chaotic. However, the search finds several chaotic, ill-posed PDEs; the simplest of these, in the discrete approximation of finitely many, coupled ordinary differential equations (ODEs), is a strikingly simple, chaotic, circulant ODE system.     

Theory and Applications of Discontinuous State Feedback Generating Chaos for Linear Systems

We investigate a kind of chaos generating technique on a type of n-dimensional linear differential systems by adding feedback control items under a discontinuous state. This method is checked with some examples of numeric simulation. A constructive theorem is proposed for generalized synchronization related to the above chaotic system.     

Soliton and chaotic structures of dust ion-acoustic waves in quantum dusty plasmas

The quantum hydrodynamic model is employed to study the soliton and chaotic structures of dust ion-acoustic waves in quantum dusty plasmas consisting of electrons, ions, and negatively/positively charged dust particles. By means of the reductive perturbation technique, two-dimensional Davey-Stewartson (DS) system is derived. By improving the extended projective method and the extended tanh-function method, a separation of variables solution with arbitrary functions for the Davey-Stewartson system is obtained. Many soliton and chaotic structures such as localized nonlinear coherent structure, line-soliton structure, periodic wave pattern structure, Rössler and Lorenz chaotic structures are given. It is found that these structures are effected by the quantum effects.     

LMC-complexity and various chaotic regimes

A variant of the statistical complexity originally advanced by López-Ruiz et al. [R. López-Ruiz, H.L. Mancini, X. Calbet, Phys. Lett. A 209 (1995) 321], is here used in conjunction with Fisher's information measure so as to explore fine details of chaotic dynamics. As a main result we can easily distinguish between (i) periodicity and chaos or (ii) between distinct chaotic dynamics belonging to different attractors.     

Torus destruction via global bifurcations in a piecewise-smooth, continuous map with square-root nonlinearity

It has been shown recently that torus formation in piecewise-smooth maps can occur through a special type of border collision bifurcation in which a pair of complex conjugate Floquet multipliers “jump” from the inside to the outside of the unit circle. It has also been shown that a large class of impacting mechanical systems yield piecewise-smooth maps with square-root singularity. In this Letter we investigate the dynamics of a two-dimensional piecewise-smooth map with square-root type nonlinearity, and describe two new routes to chaos through the destruction of two-frequency torus. In the first scenario, we identify the transition to chaos through the destruction of a loop torus via homoclinic bifurcation. In the other scenario, a change of structure in the torus occurs via heteroclinic saddle connections. Further parameter changes lead to a homoclinic bifurcation resulting in the creation of a chaotic attractor. However, this scenario is much more complex, with the appearance of a sequence of heteroclinic and homoclinic bifurcations.     

Message filtering characteristics of semiconductor laser as receiver in optical chaos communication

The message filtering characteristics of the receiver in closed-loop chaotic optical communication system are numerically studied based on laser rate equations. A pair of external cavity semiconductor lasers was employed as the chaotic carrier transmitter and the synchronized chaos receiver. We examined the filtering properties of the semiconductor laser receiver for message encoded with chaos masking. Our results demonstrate that, the lower the message frequency, the more easily the receiver filters out the message from chaotic carrier. We also analyzed the effects of each parameter mismatches between the transmitter and the receiver on the quality of the recovered message. Comparing the synchronization quality with the signal-to-noise ratio affected by parameter mismatches, we find that the quality of the recovered message depends not only on the synchronization quality but also on the filtering characteristics of the receiver. The filtering characteristics of receiver will be playing an important role on the quality of the recovered message in the case of large mismatches.     

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.