Andronov-Hopf bifurcations in planar, piecewise-smooth, continuous flows

An equilibrium of a planar, piecewise-C¹, continuous system of differential equations that crosses a curve of discontinuity of the Jacobian of its vector field can undergo a number of discontinuous or border-crossing bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to λL±iωL on one side of the discontinuity and −λR±iωR on the other, with λL,λR>0, and the quantity Λ=λL/ωL−λR/ωR is nonzero, then a periodic orbit is created or destroyed as the equilibrium crosses the discontinuity. This bifurcation is analogous to the classical Andronov-Hopf bifurcation, and is supercritical if Λ<0 and subcritical if Λ>0.     

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.     

Analysis and circuit realization of intermittency with multiple laminar states

In this Letter, an electrical circuit is built for realizing the multi-state intermittency generated by a simple force-driven chaotic system. The intermittency phenomenon and its underlaying mechanism are analyzed, and the experimental results are discussed. It is shown that, with two classes of invariant subspaces, the number of the laminar states and the distance between the adjacent laminar states of the created multi-state on-off intermittency can be arbitrarily changed by manipulating the control parameters.     

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.     

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.     

Adaptive global synchronization of a general complex dynamical network with non-delayed and delayed coupling

This Letter investigates the global synchronization of a general complex dynamical network with non-delayed and delayed coupling. Based on Lasalle's invariance principle, adaptive global synchronization criteria is obtained. Analytical result shows that under the designed adaptive controllers, a general complex dynamical network with non-delayed and delayed coupling can globally asymptotically synchronize to a given trajectory. What is more, the node dynamic need not satisfy the very strong and conservative uniformly Lipschitz condition and the coupling matrix is not assumed to be symmetric or irreducible. Finally, numerical simulations are presented to verify the effectiveness of the proposed synchronization criteria.     

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.     

Existence of chaos in a piecewise smooth two-dimensional contractive map

Piecewise smooth maps occur in a variety of physical systems. We show that in a two-dimensional continuous map a chaotic orbit can exist even when the map is contractive (eigenvalues less than unity in magnitude) at every point in the phase space. In this Letter we explain this peculiar feature of piecewise smooth continuous maps.     

Coupled cantilever array with tunable on-site nonlinearity and observation of localized oscillations

A macro-mechanical cantilever array is newly proposed for experimental investigations of intrinsic localized mode (ILM). There has never reported such a macro-system which show ILMs mechanically. The array consists of cantilevers, electromagnets faced on the cantilevers, elastic rods for coupling between cantilevers, and a voice coil motor for external excitation. Nonlinearity appears in the magnetic interaction, that is, the restoring force of cantilever. Therefore it is tunable. In the array, ILMs are experimentally generated by the sinusoidal forced excitation. Observed ILMs are also identified in the model equation for the array.     

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.     

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.     

Stabilizability of discrete chaotic systems via unified impulsive control

This Letter is concerned with the asymptotical stabilization problem of discrete chaotic systems by using a novel unified impulsive control scheme. Sufficient conditions for asymptotical stability of the impulsive controlled discrete systems are obtained by means of the Lyapunov stability theory and algebraic inequality techniques. Finally, numerical simulations on the Hénon and Ushio discrete chaotic systems are presented to illustrate the effectiveness and usefulness of the unified impulsive control scheme.     

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.     

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.     

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.     

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.     

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.     

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.     

On global asymptotic stability of 2-D discrete systems with state saturation

A criterion for the global asymptotic stability of 2-D discrete systems described by the Roesser model employing state saturation nonlinearities is presented. The criterion is a less restrictive version of an earlier criterion due to Liu and Michel.     

On global attractors for a class of nonhyperbolic piecewise affine maps

Dynamical behavior of a class of nonhyperbolic discrete systems are considered. These systems are generated by iterating planar maps that are piecewise isometries, and they arise as mathematical models for signal processing, digital filters and modulator dynamics. Planar piecewise isometries may be discontinuous and/or non-invertible. First, the authors consider attraction caused by discontinuity in planar piecewise isometries. Namely, they have shown that the maximal invariant set can induce an invariant measure, and all the Lyapunov exponents are zero under this invariant measure. Second, they discuss various definitions of global attractors and their existence and uniqueness for discontinuous maps, and introduce a few examples in which the attractors are created due to discontinuity. Third, they study the relation between invariance and invertibility for various nonhyperbolic maps, and finally they investigate decomposability of global attractors for certain nonhyperbolic systems.