Nonlinear dynamics and circuit realization of a new chaotic flow: A variant of Lorenz,Chen and Lü

In this paper, a new three-dimensional autonomous chaotic system is presented, and the range of the parameters which can induce the system to be unstable is analyzed. The dynamical behavior of this system is further investigated in some detail, including equilibria and stability, various attractors, together with the maximally complex attractor, Poincaré maps, bifurcations, and Lyapunov-exponent spectrum. The oscillator circuit of the new chaotic system is afterwards designed by using EWB software and a typical chaotic attractor is experimentally demonstrated.     

Analysis and circuitry realization of a novel three-dimensional chaotic system

In this paper a new three-dimensional chaotic system is introduced. Some basic dynamical properties are analyzed to show chaotic behavior of the presented system. These properties are covered by dissipation of system, instability of equilibria, strange attractor, Lyapunov exponents, fractal dimension and sensitivity to initial conditions. Through altering one of the system parameters, various dynamical behaviors are observed which included chaos, periodic and convergence to an equilibrium point. Eventually, an analog circuit is designed and implemented experimentally to realize the chaotic system.     

A novel three-dimensional autonomous chaos system

This letter reports a novel three-dimensional autonomous chaotic system. Some basic dynamical properties, such as Lyapunov exponents, Poincare mapping, fractal dimension, continuous spectrum and chaotic behaviors of this new transverse butterfly attractor are studied. Furthermore, the forming mechanism of its compound structure obtained by merging together two simple attractors after performing one mirror operation has been investigated by detailed numerical as well as theoretical analysis.     

Coexisting chaotic attractors in a single neuron model with adapting feedback synapse

In this paper, we consider the nonlinear dynamical behavior of a single neuron model with adapting feedback synapse, and show that chaotic behaviors exist in this model. In some parameter domain, we observe two coexisting chaotic attractors, switching from the coexisting chaotic attractors to a connected chaotic attractor, and then switching back to the two coexisting chaotic attractors. We confirm the chaoticity by simulations with phase plots, waveform plots, and power spectra.     

A new 4-D non-equilibrium fractional-order chaotic system and its circuit implementation

A new 4-D fractional-order chaotic system without equilibrium point is proposed in this paper. There is no chaotic behavior for its corresponding integer-order system. By computer simulations, we find complex dynamical behaviors in this system, and obtain that the lowest order for exhibiting a chaotic attractor is 3.2. We also design an electronic circuit to realize this 4-D fractional-order chaotic system and present some experiment results.     

A new butterfly-shaped attractor of Lorenz-like system

In this letter a new butterfly-shaped chaotic attractor is reported. Some basic dynamical properties, such as Poincare mapping, Lyapunov exponents, fractal dimension, continuous spectrum and chaotic dynamical behaviors of the new chaotic system are studied. Furthermore, we clarify that the chaotic attractors of the system is a compound structure obtained by merging together two simple attractors through a mirror operation.     

Optimizing the dynamics of a two-cell DC–DC buck converter by time delayed feedback control

A study of the dynamical behavior of a two-cell DC–DC buck converter under a digital time delayed feedback control (TDFC) is presented. Various numerical simulations and dynamical aspects of this system are illustrated in the time domain and in the parameter space. Without TDFC, the system may present many undesirable behaviors such as sub-harmonics and chaotic oscillations. TDFC is able to widen the stability range of the system. Optimum values of parameters giving rise to fast response while maintaining stable periodic behavior are given in closed form. However, it is detected that in a certain region of the parameter space, the stabilized periodic orbit may coexist with a chaotic attractor. Boundary between basins of attraction are obtained by means of numerical simulations.     

Bifurcation structure of chaotic attractor in switched dynamical systems with spike noise

High-frequency ripple (spike noise) effects in the qualitative properties of DC/DC converter circuits. This study investigates the bifurcation structure of a chaotic attractor in a switched dynamical system with spike noise. First, we introduce the system dynamics and derive the associated Poincaré map. Next, we show the bifurcation structure of the chaotic attractor in a system with spike noise. Finally, we investigate the dynamical effect of spike noise in the existence region of the chaotic attractor compare with that of a chaotic attractor in a system with ideal switching. The results suggest that spike noise enlarges an invariant set and generates a new bifurcation structure of the chaotic attractor.     

A four-wing chaotic attractor generated from a new 3-D quadratic autonomous system

This paper introduces a new 3-D quadratic autonomous system, which can generate two coexisting single-wing chaotic attractors and a pair of diagonal double-wing chaotic attractors. More importantly, the system can generate a four-wing chaotic attractor with very complicated topological structures over a large range of parameters. Some basic dynamical behaviors and the compound structure of the new 3-D system are investigated. Detailed bifurcation analysis illustrates the evolution processes of the system among two coexisting sinks, two coexisting periodic orbits, two coexisting single-wing chaotic attractors, major and minor diagonal double-wing chaotic attractors, and a four-wing chaotic attractor. Poincaré-map analysis shows that the system has extremely rich dynamics. The physical existence of the four-wing chaotic attractor is verified by an electronic circuit. Finally, spectral analysis shows that the system has an extremely broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.     

Effect of the phase on the dynamics of a perturbed bouncing ball system

The phase control method is a non-feedback control technique which has been used for different purposes in continuous periodically driven dynamical systems. One of the main goals of this paper is to apply this control technique to the bouncing ball system, which can be seen as a paradigmatic periodically driven discrete dynamical system, and has a rather simple physical interpretation. The main idea is to apply a periodic control signal including a phase difference with respect to the periodic forcing of the initial system and to analyze its effect on the dynamics of the bouncing ball system. The numerical simulations we have carried out clearly show the strong effect of the phase of the control signal in suppressing or generating chaotic behavior and in changing the period of a periodic orbit. We have also analyzed the effect of the phase in the phenomenon of the crisis-induced intermittency, showing how the phase enhances the size of the attractor near a crisis and can induce the intermittent behavior. Finally we have analyzed the scaling behavior of the crisis by varying the phase difference between the perturbation and the external forcing.     

A new hyperchaotic system and its synchronization

In this paper, a four-dimensional (4D) continuous autonomous hyperchaotic system is introduced and analyzed. This hyperchaotic system is constructed by adding a linear controller to the 3D autonomous chaotic system with a reverse butterfly-shape attractor. Some of its basic dynamical properties, such as Lyapunov exponents, Poincare section, bifurcation diagram and the periodic orbits evolving into chaotic, hyperchaotic dynamical behavior by varying parameter d are studied. Furthermore, the full state hybrid projective synchronization (FSHPS) of new hyperchaotic system with unknown parameters including the unknown coefficients of nonlinear terms is studied by using adaptive control. Numerical simulations are presented to show the effective of the proposed chaos synchronization scheme.     

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

The nonlinear reaction‐diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor ?? in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ?‐entropy. Upper and lower bounds of this entropy are obtained. Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ?ⁿ. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space ??. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy. In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz‐continuous) embedding of this system into the spatial shifts on the attractor. Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.     

Bifurcations of forced oscillators with fuzzy uncertainties by the generalized cell mapping method

In this paper, bifurcations in dynamical systems with fuzzy uncertainties are studied by means of the fuzzy generalized cell mapping (FGCM) method. A bifurcation parameter is modeled as a fuzzy set with a triangular membership function. We first study a boundary crisis resulting from a collision of a fuzzy chaotic attractor with a fuzzy saddle on the basin boundary. The fuzzy chaotic attractor together with its basin of attraction is eradicated as the fuzzy control parameter reaches a critical point. We also show that a saddle-node bifurcation is caused by the collision of a fuzzy period-one attractor with a fuzzy saddle on the basin boundary. The fuzzy attractor together with its basin of attraction suddenly disappears as the fuzzy parameter passes through a critical value.     

Crisis transitions in excitable cell models

It is believed that sudden changes both in the size of chaotic attractor and in the number of unstable periodic orbits on chaotic attractor are sufficient for interior crisis. In this paper, some interior crisis phenomena were discovered in a class of physically realizable dissipative dynamical systems. These systems represent the oscillatory activity of membrane potentials observed in excitable cells such as neuronal cells, pancreatic β-cells, and cardiac cells. We examined the occurrence of interior crises in these systems by two means: (i) constructing bifurcation diagrams and (ii) calculating the number of unstable periodic orbits on chaotic attractor. Bifurcation diagrams were obtained by numerically integrating the simultaneous differential equations which simulate the activity of excitable membranes. These bifurcation diagrams have shown an apparent crisis activity. We also demonstrate in terms of the associated Poincaré maps that the number of unstable periodic orbits embedded in a chaotic attractor suddenly increases or decreases at the crisis.     

Multiple attractors and crisis route to chaos in a model food-chain

An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out. The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors. Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics. The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain. We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a₁=1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation. Multiple attractors with riddled basins and fractal boundaries are also observed. If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation. Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable.     

A periodically forced,piecewise linear system,Part II: The fragmentation mechanism of strange attractors and grazing

In the first part of this work, the local singularity of non-smooth dynamical systems was discussed and the criteria for the grazing bifurcation were presented mathematically. In this part, the fragmentation mechanism of strange attractors in non-smooth dynamical systems is investigated. The periodic motion transition is completed through grazing. The concepts for the initial and final grazing, switching manifolds are introduced for six basic mappings. The fragmentation of strange attractors in non-smooth dynamical systems is described mathematically. The fragmentation mechanism of the strange attractor for such a non-smooth dynamical system is qualitatively discussed. Such a fragmentation of the strange attractor is illustrated numerically. The criteria and topological structures for the fragmentation of the strange attractor need to be further developed as in hyperbolic strange attractors. The fragmentation of the strange attractors extensively exists in non-smooth dynamical systems, which will help us better understand chaotic motions in non-smooth dynamical systems.     

Localization of hidden attractors of the generalized Chua system based on the method of harmonic balance

Chua’s circuits, which were introduced by Leon Ong Chua in 1983, are simplest electric circuits operating in the mode of chaotic oscillations. Systems of differential equations describing the behavior of Chua’s circuits are three-dimensional autonomous dynamical systems with scalar nonlinearity. In the standard Chua system, chaotic oscillations are excited in the classical manner, namely, starting from the neighborhood of the unstable zero equilibrium, after the transient process, the system trajectory tends to a Chua attractor.     

Complex dynamics in Duffing system with two external forcings

Duffing's equation with two external forcing terms have been discussed. The threshold values of chaotic motion under the periodic and quasi-periodic perturbations are obtained by using second-order averaging method and Melnikov's method. Numerical simulations not only show the consistence with the theoretical analysis but also exhibit the interesting bifurcation diagrams and the more new complex dynamical behaviors, including period-n (n=2,3,6,8) orbits, cascades of period-doubling and reverse period doubling bifurcations, quasi-periodic orbit, period windows, bubble from period-one to period-two, onset of chaos, hopping behavior of chaos, transient chaos, chaotic attractors and strange non-chaotic attractor, crisis which depends on the frequencies, amplitudes and damping. In particular, the second frequency plays a very important role for dynamics of the system, and the system can leave chaotic region to periodic motions by adjusting some parameter which can be considered as an control strategy of chaos. The computation of Lyapunov exponents confirm the dynamical behaviors.     

Chaos in a generalized Lorenz system

A three-component dynamic system describing a quantum cavity electrodynamic device with a pumping and nonlinear dissipation is studied. Various dynamical regimes are investigated in terms of divergent trajectories approaches and fractal statistics. It has been shown that stable and unstable dissipative structures type of limit cycles can be formed in such system, with variation of pumping and nonlinear dissipation rates. Transitions to chaotic regime and the corresponding chaotic attractor are studied in detail.     

Hausdorff dimension of a random invariant set

The notion of random attractor for a dissipative stochastic dynamical system has recently been introduced. It generalizes the concept of global attractor in the deterministic theory. It has been shown that many stochastic dynamical systems associated to a dissipative partial differential equation perturbed by noise do possess a random attractor. In this paper, we prove that, as in the case of the deterministic attractor, the Hausdorff dimension of the random attractor can be estimated by using global Lyapunov exponents. The result is obtained under very natural assumptions. As an application, we consider a stochastic reaction-diffusion equation and show that its random attractor has finite Hausdorff dimension.