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.     

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.     

Stability analysis and design of amplitude death induced by a time-varying delay connection

The present Letter considers amplitude death in a pair of oscillators coupled by a time-varying delay connection. A linear stability analysis is used to derive the boundary curves for amplitude death in a connection parameters space. The delay time can be arbitrarily long for certain amplitude of delay variation and coupling strength. A simple systematic procedure for designing such variation and strength is provided. The theoretical results are verified by a numerical simulation.     

Self-Stable Chaos Control of dc-dc Converter

A new concept related to self-stable chaos control is first put forward, and its theoretical basis and realization are presented from the frequency-domain perspective. With a new analogous-circuit realization of this control its applications in the voltage-mode Buck converter is discussed. The harmonic-balance method is applied to determine the control range of the control parameter. The experiment results given in the last part confirm the validity of the proposed control method.     

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.     

Adaptive chaos control and synchronization in only locally Lipschitz systems

In the existing results on chaos control and synchronization based on the adaptive controlling technique (ACT), a uniform Lipschitz condition on a given dynamical system is always assumed in advance. However, without this uniform Lipschitz condition, the ACT might be failed in both theoretical analysis and in numerical experiment. This Letter shows how to utilize the ACT to get a rigorous control for the system which is not uniformly Lipschitz but only locally Lipschitz, and even for the system which has unbounded trajectories. In fact, the ACT is proved to possess some limitation, which is actually induced by the nonlinear degree of the original system. Consequently, a piecewise ACT is proposed so as to improve the performance of the existing techniques.     

Filter based non-invasive control of chaos in Buck converter

A non-invasive method for controlling chaos in the voltage-mode Buck converter is proposed by using a hybrid active filter based feedback controller in this Letter. The harmonic balance method is applied to obtaining the bifurcation-point equations of the controlled system. Hence, a stability-boundary diagram is constructed, through which the control parameters are chosen correctly. The results of simulation and experiment are given after all.     

Direct transition from a stable equilibrium to quasiperiodicity in non-smooth systems

The purpose of this Letter is to show how a border-collision bifurcation in a piecewise-smooth dynamical system can produce a direct transition from a stable equilibrium point to a two-dimensional invariant torus. Considering a system of nonautonomous differential equations describing the behavior of a power electronic DC/DC converter, we first determine the chart of dynamical modes and show that there is a region of parameter space in which the system has a single stable equilibrium point. Under variation of the parameters, this equilibrium may collide with a discontinuity boundary between two smooth regions in phase space. When this happens, one can observe a number of different bifurcation scenarios. One scenario is the continuous transformation of the stable equilibrium into a stable period-1 cycle. Another is the transformation of the stable equilibrium into an unstable period-1 cycle with complex conjugate multipliers, and the associated formation of a two-dimensional (ergodic or resonant) torus.     

Dynamic Feedback Controlling Chaos in Current-Mode Boost Converter

A method for the control of chaos in the current-mode boost converter is presented by using the first-order dynamic feedback control. The feedback part consists of a resistance and a capacitance in series. The system to be controlled is treated as a third-order model, and then the discrete mapping model is obtained by using the data-sampling method. By analysing the position of the maximum norm eigenvalue, the stable range of feedback gain is ascertained out and its optimization is also carried out. Finally, the results of simulation and experiment confirm the correctness of the theoretical analysis and the validity of the proposed means.     

Detecting phase synchronization between coupled non-phase-coherent oscillators

We compare two methods for detecting phase synchronization in coupled non-phase-coherent oscillators. One method is based on the locking of self-sustained oscillators with an irregular signal. The other uses trajectory recurrences in phase space. We identify the pros and cons of both methods and propose guidelines to detect phase synchronization in data series.     

Equilibrium-torus bifurcation in nonsmooth systems

Considering a set of two coupled nonautonomous differential equations with discontinuous right-hand sides describing the behavior of a DC/DC power converter, we discuss a border-collision bifurcation that can lead to the birth of a two-dimensional invariant torus from a stable node equilibrium point. We obtain the chart of dynamic modes and show that there is a region of parameter space in which the system has a single stable node equilibrium point. Under variation of the parameters, this equilibrium may disappear as it collides with a discontinuity boundary between two smooth regions in the phase space. The disappearance of the equilibrium point is accompanied by the soft appearance of an unstable focus period-1 orbit surrounded by a resonant or ergodic torus.Detailed numerical calculations are supported by a theoretical investigation of the normal form map that represents the piecewise linear approximation to our system in the neighbourhood of the border. We determine the functional relationships between the parameters of the normal form map and the actual system and illustrate how the normal form theory can predict the bifurcation behaviour along the border-collision equilibrium-torus bifurcation curve.     

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.     

Analysis of heartbeat asymmetry based on multi-scale time irreversibility test

We investigate the asymmetry of heart rate control system and suggest a simple index to quantify this asymmetry by performing high-dimensional time irreversibility tests to heartbeat interval time series over multiple scales. The results provide strong evidence to the concept that the asymmetry is an intrinsic property of heart rate control system. As a simple and visual method, it is proved to be effective in classifying physiologic and synthetic subjects while the maximum scale is selected within a proper range, and also provides a new way to analyze the time irreversibility for other high-dimensional systems.     

Preserving synchronization under matrix product modifications

In this article we present a methodology under which stability and synchronization of a dynamical master/slave system configuration are preserved under modification through matrix multiplication. The objective is to show that under a defined multiplicative group, hyperbolic critical points are preserved along the stable and unstable manifolds. The properties of this multiplicative group were determined through the use of simultaneous Jordan decomposition. It is also shown that a consequence of this approach is the preservation of the signature of the Jacobian matrix associated with the dynamical system. To illustrate the results we present several examples of different modified systems.     

Prediction based chaos control via a new neural network

In this Letter, a new chaos control scheme based on chaos prediction is proposed. To perform chaos prediction, a new neural network architecture for complex nonlinear approximation is proposed. And the difficulty in building and training the neural network is also reduced. Simulation results of Logistic map and Lorenz system show the effectiveness of the proposed chaos control scheme and the proposed neural network.     

Synchronization and control of chaos in coupled chaotic multimode Nd:YAG lasers

In this Letter we numerically investigate the dynamics of a system of two coupled chaotic multimode Nd:YAG lasers with two mode and three mode outputs. Unidirectional and bidirectional coupling schemes are adopted; intensity time series plots, phase space plots and synchronization plots are used for studying the dynamics. Quality of synchronization is measured using correlation index plots. It is found that for laser with two mode output bidirectional direct coupling scheme is found to be effective in achieving complete synchronization, control of chaos and amplification in output intensity. For laser with three mode output, bidirectional difference coupling scheme gives much better chaotic synchronization as compared to unidirectional difference coupling but at the cost of higher coupling strength. We also conclude that the coupling scheme and system properties play an important role in determining the type of synchronization exhibited by the system.     

Impact of Viscosity on DNA Dynamics

We study the influence of viscosity on DNA dynamics. By employing the nonlinear Peyrard-Bishop-Dauxois （PBD） model, it is shown that the DNA dynamics can be explained by a solution of a complex nonlinear Schrǒdinger equation （CNLSE）. This is the nonlinear Schrǒdinger equation （NLSE） with a nonlinear parameter being a complex number. We compare real and imaginary parts of this nonlinear parameter and show that the latter one should not be negligible, which means that the CNLSE should be solved numerically,     

Impulsive synchronization of complex networks with non-delayed and delayed coupling

This Letter investigates the impulsive synchronization between two complex networks with non-delayed and delayed coupling. Based on the stability analysis of impulsive differential equation, the criteria for the synchronization is derived, and a linear impulsive controller and the simple updated laws are designed. Particularly, the weight configuration matrix is not necessarily symmetric or irreducible, and the inner coupling matrix need not be symmetric. Numerical examples are presented to verify the effectiveness and correctness of the synchronization criteria.     

Experimental study on tracking the state of analog Chua's circuit with particle filter for chaos synchronization

The aim of this Letter is to study by experiment the feasibility of tracking the state of analog Chua's circuit with particle filter for chaos synchronization. The structure of this digital-filter-based synchronization approach is proposed and depicted. The transmitted signals, generated from PSpice simulations and a physically implemented Chua's circuit, get through an additive white Gaussian noise (AWGN) channel to a receiver consisting of an analog-to-digital converter (ADC) and a particle filter (PF). The ADC is used to sample the analog signals and the PF is used to estimate and track the state of the transmitting circuit to obtain synchronized oscillation signals. Experiment results indicate that it is possible to use this approach to synchronize analog Chua's circuit, even if the circuit parameters are not known exactly and time-varying. In addition, the effects of different signal-to-noise ratio (SNR), ADC sampling frequency and PF state sampling time on synchronization performance are also studied for further understanding of this synchronization approach.