A new procedure for exploring chaotic attractors in nonlinear dynamical systems under random excitations

Due to uncertain push-pull action across boundaries between different attractive domains by random excitations, attractors of a dynamical system will drift in the phase space, which readily leads to colliding and mixing with each other, so it is very difficult to identify irregular signals evolving from arbitrary initial states. Here, periodic attractors from the simple cell mapping method are further iterated by a specific Poincare map in order to observe more elaborate structures and drifts as well as possible dynamical bifurcations. The panorama of a chaotic attractor can also be displayed to a great extent by this newly developed procedure. From the positions and the variations of attractors in the phase space, the action mechanism of bounded noise excitation is studied in detail. Several numerical examples are employed to illustrate the present procedure. It is seen that the dynamical identification and the bifurcation analysis can be effectively performed by this procedure.

Generation of countless embedded trumpet-shaped chaotic attractors in two opposite directions from a new three-dimensional system with no equilibrium point

A new three-dimensional （3D） continuous autonomous system with one parameter and three quadratic terms is presented firstly in this paper. Countless embedded trumpet-shaped chaotic attractors in two opposite directions are generated from the system as time goes on. The basic dynamical behaviors of the strange chaotic system are investigated. Another more complex 3D system with the same capability of generating countless embedded trumpet-shaped chaotic attractors is also put forward.

IMPULSIVE CONTROL OF CHAOTIC ATTRACTORS IN NONLINEAR CHAOTIC SYSTEMS

Based on the study from both domestic and abroad, an impulsive control scheme on chaotic attractors in one kind of chaotic system is presented.By applying impulsive control theory of the universal equation, the asymptotically stable condition of impulsive control on chaotic attractors in such kind of nonlinear chaotic system has been deduced, and with it, the upper bond of the impulse interval for asymptotically stable control was given. Numerical results are presented, which are considered with important reference value for control of chaotic attractors.

Novel two-directional grid multi-scroll chaotic attractors based on the Jerk system

A new method is presented to generate two-directional (2D) grid multi-scroll chaotic attractors via a specific form of the sine function and sign function series, which are applied to increase saddle points of index 2. The scroll number in the x-direction is modified easily through changing the thresholds of the specific form of the sine function, while the scroll number in the y-direction is controlled by the sign function series. Some basic dynamical properties, such as equilibrium points, bifurcation diagram, phase portraits, and Lyapunov exponents spectrum are studied. Furthermore, the electronic circuit of the system is designed and its simulation results are given by Multisim 10.

Novel four-wing and eight-wing attractors using coupled chaotic Lorenz systems

This paper presents the problem of generating four-wing （eight-wing） chaotic attractors. The adopted method consists in suitably coupling two （three） identical Lorenz systems. In analogy with the original Lorenz system, where the two wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four wings （eight wings） of these novel attractors axe located around the four （eight） equilibria with two （three） pairs of unstable complex-conjugate eigenvalues.

Novel four-dimensional autonomous chaotic system generating one-,two-,three- and four-wing attractors

In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the Poincare map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations axe in good agreement with the numerical simulation results.

New results of study on generating multiple-scroll chaotic attractors

We present a new method for generating multiple-scroll attractors. By constructing a piece-wise-linear odd function as the driving-point characteristic of Chua's diode with multiple breakpoints, and finding a recurrence formula for determining equilibrium point voltages and breakpoint voltages, we can calculate the voltages of all the equilibrium points and breakpoints, and the multiple-scroll chaotic attractors can be obtained. This method is based on Shil'nikov theorem and the cell model of chaotic attractors. The theoretical calculations are in good agreement with computer simulations and hardware experimental results. Compared with the results reported by Suykens, our method has some advantages that much more scrolls can be obtained, the scrolls are uniform in size, and a general recurrence formula for generating attractors with any number of scrolls is given. Furthermore, the observation of attractors with a maximum of 11-21 uniform scrolls obtained by hardware experiment and computer simulation, respectively, is reported for the first time.

Generating one-,two-,three-and four-scroll attractors from a novel four-dimensional smooth autonomous chaotic system

A new four-dimensional quadratic smooth autonomous chaotic system is presented in this paper, which can exhibit periodic orbit and chaos under the conditions on the system parameters. Importantly, the system can generate one-, two-, three- and four-scroll chaotic attractors with appropriate choices of parameters. Interestingly, all the attractors are generated only by changing a single parameter. The dynamic analysis approach in the paper involves time series, phase portraits, Poincare maps, a bifurcation diagram, and Lyapunov exponents, to investigate some basic dynamical behaviours of the proposed four-dimensional system.

A novel class of two-dimensional chaotic maps with infinitely many coexisting attractors

We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable.In particular,a computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex dynamics.Interestingly,this map contains infinitely many coexisting attractors which has been rarely reported in the literature.Further studies on these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan–Yorke)dimension.Bifurcation analysis reveals that the map has periodic and chaotic solutions,and more importantly,exhibits extreme multi-stability.

Design and implementation of a novel multi-scroll chaotic system

This paper proposes a novel approach for generating a multi-scroll chaotic system. Together with the theoretical design and numerical simulations, three different types of attractor are available, governed by constructing triangular wave, sawtooth wave and hysteresis sequence. The presented new multi-scroll chaotic system is different from the classical multi-scroll chaotic Chua system in dimensionless state equations, nonlinear functions and maximum Lyapunov exponents. In addition, the basic dynamical behaviours, including equilibrium points, eigenvalues, eigenvectors, eigenplanes, bifurcation diagrams and Lyapunov exponents, are further investigated. The success of the design is illustrated by both numerical simulations and circuit experiments.

COMPOUND STRUCTURE OF NEW FOUR-SCROLLS CHAOTIC SYSTEM

The finding of the compound structure of a new four-scrolls chaotic system is reported, which is obtained by merging together two symmetrical attractors. And the two symmetrical attractors are generated only by adding a constant gain to the original system. Also, the forming procedure of the new four-scrolls chaotic attractor is explored and the relation between the constant gain and the properties of the system is given.

Construction and Verification of a Simple Smooth Chaotic System

This article introduces a new chaotic system of three-dimensional quadratic autonomous ordinary differential equations, which can display different attractors with two unstable equilibrium points and four unstable equilibrium points respectively. Dynamical properties of this system are then studied. Furthermore, by applying the undetermined coefficient method, heteroclinic orbit of Shil＇nikov＇s type in this system is found and the convergence of the series expansions of this heteroclinic orbit are proved in this article. The Shil＇nikov＇s theorem guarantees that this system has Smale horseshoes and the horseshoe chaos.

Multiple attractors and generalized synchronization in delayed Mackey-Glass systems

Nonlinear dynamics of the time-delayed Mackey-Glass systems is explored. Coexistent multiple chaotic attractors are found. Attractors with double-scroll structures can be well classified in terms of different return times within one period of the delay time by constructing the Poincare section. Synchronizations of the drive-response Mackey-Glass oscillators are investigated. The critical coupling strength for the emergence of generalized synchronization against the delay time exhibits the interesting resonant behaviour. We reveal that stronger resonance effect may be observed when different attractors are applied to the drivers, i.e., more resonance peaks can be found.

A novel four-wing chaotic attractor generated from a three-dimensional quadratic autonomous system

This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms,and the system can generate a single four-wing chaotic attractor with wide parameter ranges. Through theoretical analysis,the Hopf bifurcation processes are proved to arise at certain equilibrium points.Numerical bifurcation analysis shows that the system has many interesting complex dynamical behaviours;the system trajectory can evolve to a chaotic attractor from a periodic orbit or a fixed point as the proper parameter varies. Finally,an analog electronic circuit is designed to physically realize the chaotic system;the existence of four-wing chaotic attractor is verified by the analog circuit realization.

A new multi-scroll chaotic generator

In this paper a new simple multi-scroll chaotic generator is studied. The characteristic of this new multi-scroll chaotic generator is that it is easy to generate different number of scroll chaotic attractors through modifying the nature number n after fixing the suitable system parameters and it does not need complex mathematical derivation. Various number of scroll chaotic attractors are illustrated not only by computer simulation but also by the realization of an electronic circuit experiment on Electronic Workbench （EWB）.

A class of two-dimensional rational maps with self-excited and hidden attractors

This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stability of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov(Kaplan–Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.

Generation of a novel spherical chaotic attractor from a new three-dimensional system

A new three-dimensional （3D） system is constructed and a novel spherical chaotic attractor is generated from the system. Basic dynamical behaviors of the chaotic system are investigated respectively. Novel spherical chaotic attractors can be generated from the system within a wide range of parameter values. The shapes of spherical chaotic attractors can be impacted by the variation of parameters. Finally, a simpler 3D system and a more complex 3D system with the same capability of generating spherical chaotic attractors are put forward respectively.

Chaotic attractor transforming control of hybrid Lorenz-Chen system

Based on passive theory, this paper studies a hybrid chaotic dynamical system from the mathematics perspective to implement the control of system stabilization. According to the Jacobian matrix of the nonlinear system, the stabilization control region is gotten. The controller is designed to stabilize fast the minimum phase Lorenz-Chen chaotic system after equivalently transforming from chaotic system to passive system. The simulation results show that the system not only can be controlled at the different equilibria, but also can be transformed between the different chaotic attractors.

Local Lyapunov Exponents and characteristics of fixed/periodic points embedded within a chaotic attractor

A chaotic dynamical system is characterized by a positive averaged exponential separation of two neighboring tra- jectories over a chaotic attractor. Knowledge of the Largest Lyapunov Exponent λ1 of a dynamical system over a bounded attractor is necessary and sufficient for determining whether it is chaotic (λ1>0) or not (λ1≤0). We intended in this work to elaborate the connection between Local Lyapunov Exponents and the Largest Lyapunov Exponent where an alternative method to calculate λ1 has emerged. Finally, we investigated some characteristics of the fixed points and periodic orbits embedded within a chaotic attractor which led to the conclusion of the existence of chaotic attractors that may not embed in any fixed point or periodic orbit within it.

SPECIAL DYNAMIC BEHAVIORS OF A TEMPORAL CHAOTIC SYSTEM

When dynamic behaviors of temporal chaotic system are analyzed, we find that a temporal chaotic system has not only generic dynamic behaviors of chaotic reflection, but also has phenomena influencing two chaotic attractors by original values. Along with the system parameters changing to certain value, the system will appear a break in chaotic region, and jump to another orbit of attractors. When it is opposite that the system parameters change direction, the temporal chaotic system appears complicated chaotic behaviors.