Narrow Results

This paper aims at studying the basic properties and chaotic synchronization of complex Lorenz system:

Numerically we show that this system is a chaotic system and exhibits chaotic attractors. The necessary conditions for system (⋆) to generate chaos are obtained. Analytical and numerical calculations are presented to achieve synchronization. Active control technique is used to synchronize chaotic attractors of equations (⋆).

In this paper, the chaotic phenomenon is reported in a bearing system. The chaotic dynamics of the new system is first derived from an available bearing system. The proposed system is then analyzed using various tools to observe its dynamical behaviors. Theoretical and simulation results confirm the presence of chaotic and periodic behaviors in the new bearing system.

Traffic flow shows various complex behaviors. Numerous empirical data and numerical simulation results demonstrate the existence of distinct dynamics states. In order to obtain insight into the nature of the dynamics, we apply the nonlinear time series analysis approach to study the characteristic behavior of traffic flow at low and intermediate density values. A procedure called "embedding scheme" has been used to reconstruct the representation of the time series which is obtained from the deterministic NaSch traffic model. In the reconstructed phase space, we investigate the regular attractor and the chaotic attractor of traffic time series. Our results indicate that nonlinear technique can be successfully used for understanding of dynamics feature of traffic flow.

We study the effect of linear dissipative forces on the chaotic behavior of coupled quartic oscillators with two degrees of freedom. The effect of quadratic Rayleigh dissipation functions, one with diagonal coefficients only and the other with nondiagonal coefficients as well are studied. It is found that the effect of Rayleigh Dissipation function with diagonal coefficients is to suppress chaos in the system and to lead the system to its equilibrium state. However, with a dissipation function with nondiagonal elements, other types of behaviors — including fixed point attractor, periodic attractors and even chaotic attractors — are possible even when there is no external driving. In such a system the route to chaos is through period-doubling bifurcations. This result contradicts the view that linear dissipation always causes decay of oscillations in oscillator models.

This letter reports the finding of a chaotic attractor in a very simple three-dimensional autonomous hybrid system, which is obtained by applying a discontinuous feedback to a stable linear system. This attractor is different from the known chaotic attractors investigated in the literature and is globally attractive.

A new chemical kinetic system for Chua's equation is presented. In contrast to the double-scroll Chua's attractor, the chaotic attractor obtained from this system has only one scroll.

This Letter reports a simple smooth quadratic autonomous chaotic system with four unstable equilibrium points and three homoclinic orbits, and can display a 3-layer attractor among some others. It is the first example of a simple smooth quadratic system that we know, which can exhibit Šhilnikov chaos without numerical evidence of a route to chaos through period-doubling bifurcations.

In this Letter, we propose a class of nonlinear dynamic translinear circuits as a viable tool for the rapid and systematic implementation of chaotic oscillators in analog integrated circuits. In this regard, our primary focus is on multiple-input translinear element (MITE) networks and their particular merits. We also describe, as an illustrative example, our monolithic implementation of the Lorenz equations, and report on results from a test chip fabricated in a 0.5 μm CMOS process through MOSIS. We also show that Chua's circuit can be implemented easily in our framework.

This paper presents a novel result on the effect of coupling through both analytical and numerical investigations on linearly coupled systems including chaotic and nonchaotic systems. It is found that when a single system has potential of oscillation but is currently in a "marginal" state to produce a limit cycle via Hopf bifurcation due to the change of a parameter, an appropriate coupling strength can excite the potential limit cycle such that the coupled system oscillates synchronously. Similarly, when a stable limit cycle is at the "margin" of a chaotic attractor in a single system, a certain coupling strength can induce the potential chaotic attractor such that the coupled system has a synchronous chaotic behavior. This excitation mechanism is different from the traditional function of coupling in that the latter mainly drives the coupled system to synchronize with the ongoing dynamics of a single system but does not recover its disappearing dynamics. This newly observed synchronization is called coherent synchronization to distinguish it from various common types of synchronization. Several numerical examples are presented for quantitative description of this interesting phenomenon.

Based on the generalized Lorenz system, a conjugate Lorenz-type system is introduced, and a new unified Lorenz-type system containing these two classes of systems is naturally constructed in the paper. Such a unified system is state-equivalent to a simple special form, which is parameterized by two parameters useful for chaos turning and system classification. More importantly, based on the parameterized form, three new chaotic attractors, called conjugate attractors, are found for the first time, which are conjugate to the Lorenz attractor, the Chen attractor, and the Lü attractor, respectively.

This letter reports a study on some topological properties of chaos using a generalized competitive mode (GCM). The Lorenz system and the Chen system are used as examples for comparison. It is shown that for typical parameter values used in the two systems, the Lorenz attractor has one pair of GCMs in competition, while the Chen attractor has two pairs of GCMs in competition. This explains why the two attractors are topologically different, and furthermore indicates that the Chen attractor is more complex than the Lorenz attractor from the dynamics point of view.

In this paper, a chaotic attractor gallery obtained from a new higher dimensional chaotic circuit combining LMT (Lindberg–Murali and Tamasevicious) circuit and Chua's circuit is presented. This circuit is capable of exhibiting a rich variety of chaotic attractors including temporary multiscroll attractors.

An electronic oscillator producing digital signals, formed by a second-order, nonlinear, nonautonomous, closed loop topology that is based on two integrators and nonlinear elements (a comparator and a XOR gate), was experimentally investigated. For a certain frequency and amplitude range of the periodic driving signal, the intermittency route to chaos was observed and studied.

This paper presents a novel chaotic four-wing attractor generated by coupling two identical Lorenz systems. An analysis of the proposed system shows that its equilibria have certain symmetries with respect to specific coordinate planes and the eigenvalues of the associated Jacobian matrices exhibit the property of similarity. 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 of this new attractor are located around four equilibria with four unstable complex-conjugate eigenvalues. A generalization of the proposed system to realize an eight-wing attractor is also described.

In this paper we study the local codimension one, two and three Hopf bifurcations which occur in the classical Chua's differential equations with cubic nonlinearity. A detailed analytical description of the regions in the parameter space for which multiple small periodic solutions bifurcate from the equilibria of the system is obtained. As a consequence, a complete answer for the challenge proposed in [Moiola & Chua, 1999] is provided.

The classical Šil'nikov homoclinic theorem provides an analytic criterion for proving the existence of chaos in three-dimensional autonomous systems, but it can only be applied to systems with fixed points of the saddle-focus type. This paper extends this powerful theorem to a degenerate case where one of the eigenvalues of the Jacobian evaluated at an equilibrium point is zero and the other two are a pair of conjugate complex numbers, and consequently establishes a set of criteria for proving the existence of chaos in the sense of having Smale horseshoes. Based on this new extended Šil'nikov homoclinic theorem, a new chaotic system is constructed, whose corresponding bounded chaotic attractor is first verified numerically through phase trajectories, Lyapunov exponents, bifurcation routes and Poincaré mappings, followed by theoretical analysis on the existence of one homoclinic orbit, the key component of the extended Šil'nikov homoclinic theorem.

In this paper, a modified generalized Lorenz-type system is introduced, which is state-equivalent to a simple and special form, and is parameterized by two parameters useful for chaos turning and system classification. More importantly, based on the parameterized form, two classes of new chaotic attractors are found for the first time in the literature, which are similar but nonequivalent in topological structure. To further understand the complex dynamics of the new system, some basic properties such as Lyapunov exponents, Hopf bifurcations and compound structure of the attractors are analyzed and demonstrated with careful numerical simulations.

We address a nonautonomous differential equation with a pulse function, whose moments of discontinuity depend on the initial moment. Existence of a quasi-minimal set is proved. An appropriate simulation of a chaotic attractor is presented.

This paper introduces a modified Chua's system which is a smooth four-dimensional continuous-time autonomous chaotic system with a cubic nonlinearity. Some dynamical behaviors of this 4-D Chua's system are further investigated by means of Poincaré mapping, parameter phase portraits, equilibrium points, bifurcations and calculated Lyapunov exponents. Moreover, using RC-opamp and analog multiplier we describe a simple electronic circuit for hardware implementation of the 4-D Chua's system which differ from previously reported Chua's circuits. Various attractors of experimental results from this chaotic oscillator are in good agreement with theoretical analysis. In particular, based on the approximation theory of fractional-order operator, a relevant analog circuit diagram of this fractional-order modified Chua's system is designed with α = 0.9. Observation results demonstrate that chaos exists indeed in this fractional-order modified Chua's system with an order as low as 3.6. This fractional-order oscillation circuit, for the first time in the literature, realizes high-dimensional Chua's chaotic system.

In this study, we develop a new dynamical model of cancer growth, which includes the interactions between tumour cells, healthy tissue cells, and activated immune system cells, clearly leading to chaotic behavior. We explain the biological relevance of our model and the ways in which it differs from the existing ones. We perform equilibria analysis, indicate the conditions where chaotic dynamics can be observed, and show rigorously the existence of chaos by calculating the Lyapunov exponents and the Lyapunov dimension of the system. Moreover, we demonstrate that Shilnikov's theorem is valid in the parameter range of interest.