Effect of resonant-frequency mismatch on attractors

Resonant perturbations are effective for harnessing nonlinear oscillators for various applications such as controlling chaos and inducing chaos. Of physical interest is the effect of small frequency mismatch on the attractors of the underlying dynamical systems. By utilizing a prototype of nonlinear oscillators, the periodically forced Duffing oscillator and its variant, we find a phenomenon: resonant-frequency mismatch can result in attractors that are nonchaotic but are apparently strange in the sense that they possess a negative Lyapunov exponent but its information dimension measured using finite numerics assumes a fractional value. We call such attractors pseudo-strange. The transition to pesudo-strange attractors as a system parameter changes can be understood analytically by regarding the system as nonstationary and using the Melnikov function. Our results imply that pseudo-strange attractors are common in nonstationary dynamical systems.     

Solitons as attractors of a forced dissipative nonlinear Schrödinger equation

A single soliton and a bound state of two solitons are shown to be approximate simple or strange attractors of a forced dissipative nonlinear Schrödinger equation when forcing and dissipative terms are small.     

Wada basins of strange nonchaotic attractors in a quasiperiodically forced system

Whether Wada basins of strange nonchaotic attractors (SNAs) can exist has been an open problem. Here we verify the existence of Wada basin for SNAs in a quasiperiodically forced Duffing map. We show that the SNAs? basins are full Wada for a set of parameters of positive measure. We identify two types of SNAs? Wada basins by the basin cell method. It suggests that SNAs cannot be predicted reliably for the specific initial conditions.     

Strange attractors and their periodic repetition

In this paper, we present some important findings regarding a comprehensive characterization of dynamical behavior in the vicinity of two periodically perturbed homoclinic solutions. Using the Duffing system, we illustrate that the overall dynamical behavior of the system, including strange attractors, is organized in the form of an asymptotic invariant pattern as the magnitude of the applied periodic forcing approaches zero. Moreover, this invariant pattern repeats itself with a multiplicative period with respect to the magnitude of the forcing. This multiplicative period is an explicitly known function of the system parameters. The findings from the numerical experiments are shown to be in great agreement with the theoretical expectations.     

Coexisting chaotic attractors,their basin of attractions and synchronization of chaos in two coupled Duffing oscillators

Two coupled Duffing oscillators equations with several coexisting chaotic attractors are considered. Six coexisting chaotic attractors are observed for the range of initial conditions considered in our study. For two attractors the basin of attraction is a simple disconnected straight line, while for the rest of the attractors it appears to be very complex. Phase portraits of (two) subsystems are found to be distinct for two attractors while identical for the other four attractors. Among these four attractors, state variables of the subsystems are perfectly synchronized for two attractors and asynchronized for the other two attractors. Synchronization of subsystems is studied by employing a continuous feedback method.     

A new 5D hyperchaotic system with stable equilibrium point,transient chaotic behaviour and its fractional-order form

In this paper, we construct a novel 4D autonomous chaotic system with four cross-product nonlinear terms and five equilibria. The multiple coexisting attractors and the multiscroll attractor of the system are numerically investigated. Research results show that the system has various types of multiple attractors, including three strange attractors with a limit cycle, three limit cycles, two strange attractors with a pair of limit cycles, two coexisting strange attractors. By using the passive control theory, a controller is designed for controlling the chaos of the system. Both analytical and numerical studies verify that the designed controller can suppress chaotic motion and stabilise the system at the origin. Moreover, an electronic circuit is presented for implementing the chaotic system.     

Structure and evolution of strange attractors in non-elastic triangular billiards

We study non-elastic billiard dynamics in an equilateral triangular table. In such dynamics, collisions with the walls of the table are not elastic, as in standard billiards; rather, the outgoing angle of the trajectory with the normal vector to the boundary at the point of collision is a uniform factor λ < 1 smaller than the incoming angle. This leads to contraction in phase space for the discrete-time dynamics between consecutive collisions, and hence to attractors of zero Lebesgue measure, which are almost always fractal strange attractors with chaotic dynamics, due to the presence of an expansion mechanism. We study the structure of these strange attractors and their evolution as the contraction parameter λ is varied. For λ∈(0,1/3), we prove rigorously that the attractor has the structure of a Cantor set times an interval, whereas for larger values of λ gaps arise in the Cantor structure. For λ close to 1, the attractor splits into three transitive components, whose basins of attraction have fractal boundaries.     

Two-parameter families of strange attractors

Periodically driven two-dimensional nonlinear oscillators can generate strange attractors that are periodic. These attractors are mapped in a locally 1-1 way to entire families of strange attractors that are indexed by a pair of relatively prime integers (n(1),n(2)), with n(1)>/=1. The integers are introduced by imposing periodic boundary conditions on the entire strange attractor rather than individual trajectories in the attractor. The torsion and energy integrals for members of this two parameter family of locally identical strange attractors depend smoothly on these integers.     

A lot of strange attractors: chaotic or not?

Iterations on R given by quasiperiodic displacement are closely linked with the quasiperiodic forcing of an oscillator. We begin by recalling how these problems are related. It enables us to predict the possibility of appearance of strange nonchaotic attractors (SNAs) for simple increasing maps of the real line with quasiperiodic displacement. Chaos is not possible in this case (Lyapounov exponents cannot be positive). Studying this model of iterations on R for larger variations, beyond critical values where it is no longer invertible, we can get chaotic motions. In this situation we can get a lot of strange attractors because we are able to smoothly adjust the value of the Lyapounov exponent. The SNAs obtained can be viewed as the result of pasting pieces of trajectories, some of which having positive local Lyapounov exponents and others having negative ones. This leads us to think that the distinction between these SNAs and chaotic attractors is rather weak.     

Symbolic Dynamics of the Two-Well Duffing Equation

At weak force range the two-well Duffing equation has a closed bifurcation region in theparameter plane, inside which there are many different periodic windows. By means ofone-dimensional and two-dimensional symbolic dynamics, these periodic windows and thedynamical behavior of chaotic attractors are analyzed systematically.     

Strange attractors are classified by bounding Tori

There is at present a doubly discrete classification for strange attractors of low dimension, d(L)<3. A branched manifold describes the stretching and squeezing processes that generate the strange attractor, and a basis set of orbits describes the complete set of unstable periodic orbits in the attractor. To this we add a third discrete classification level. Strange attractors are organized by the boundary of an open set surrounding their branched manifold. The boundary is a torus with g holes that is dressed by a surface flow with 2(g-1) singular points. All known strange attractors in R3 are classified by genus, g, and flow type.     

旋转对称的广义Lorenz奇怪吸引子

阐述了计算微分方程组最大Lyapunov指数的技术,介绍了由一维可观察量计算系统关联维数的方法.利用Lyapunov指数作判据,通过坐标变换,构造了具有旋转对称性的广义Lorenz奇怪吸引子,分析了奇怪吸引子的运动特征并计算了奇怪吸引子的关联维数.     

二维logistic映射的动力学行为和奇怪吸引子的分形特征

研究了二维logistic映射的动力学行为和奇怪吸引子的分形特征.利用分岔图、相图和Lyapunov指数谱分析系统的分岔过程,研究系统通向混沌的道路并确定系统处于混沌运动的参数区间;采用G-P算法计算奇怪吸引子的关联维数和Kolmogorov熵,对奇怪吸引子的分形特征定量刻画;采用逃逸时间算法构造奇怪吸引子的彩色广义M-J集,对奇怪吸引子的分形特征定性表征.结果表明,这些分析方法的配合使用可以更全面、形象地描述奇怪吸引子的分形特征.     

Chaotic Motion in a Harmonically Excited Soliton System

The influence of a soliton system under an external harmonic excitation is considered. We take the compound KdV-Burgers equation as an example, and investigate numerically the chaotic behavior of the system with a periodic forcing. Different routes to chaos such as period doubling, quasi-periodic routes, and the shapes of strange attractors are observed by using bifurcation diagrams, the largest Lyapunov exponents, phase projections and Poincaré maps.     

Invariant distribution on strange attractors in highly dissipative systems

We obtain analytically by an iterative procedure the equilibrium invariant distribution for a class of strange attractors in highly dissipative systems. The zeroth order distribution is found by solving the phase-averaged Markov equation. Repeated iterations over the map yield the higher order distributions that reveal successively finer structures. The analytical results have been compared quantitatively to numerical results for the Zaslavskii map.     

From Limit Cycles to Strange Attractors

We define a quantitative notion of shear for limit cycles of flows. We prove that strange attractors and SRB measures emerge when systems exhibiting limit cycles with sufficient shear are subjected to periodic pulsatile drives. The strange attractors possess a number of precisely-defined dynamical properties that together imply chaos that is both sustained in time and physically observable.     

Structure-preserving algorithms for the Duffing equation

In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, based on the gradient- Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms （SPAs） for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge Kutta methods, this paper finds that there is an error term of order p＋l for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper. Finally, through numerical experiments, relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method （a second-order Runge-Kutta method） are compared. Computational results illustrate that the SPAs are evidently better than the Heun method when e is small or equal to zero.     

Intermittency at the onset of stochasticity in nonlinear resonant coupling processes

A simple model for the nonlinear saturation of a plasma instability is studied via numerical solution of the resonant three-wave coupling equations. When parameters are varied the attractors of motion undergo bifurcations of several types. Intermittency is shown to occur in a transition from a limit cycle to a strange attractor. Such a transition might be indicative of an intermittent onset of turbulence in certain plasma experiments.     

From Invariant Curves to Strange Attractors

We prove that simple mechanical systems, when subjected to external periodic forcing, can exhibit a surprisingly rich array of dynamical behaviors as parameters are varied. In particular, the existence of global strange attractors with fully stochastic properties is proved for a class of second order ODEs. Received: 10 January 2001 / Accepted: 10 July 2001     

Statistics of strange attractors by generalized cell mapping

It is proposed in this paper to use the generalized cell mapping to locate strange attractors of dynamical systems and to determine their statistical properties. The cell-to-cell mapping method is based upon the idea of replacing the state space continuum by a large collection of state space cells and of expressing the evolution of the dynamical system in terms of a cell-to-cell mapping. This leads to a Markov chain which in turn allows us to compute all the statistical properties as well as the invariant distribution. After a general discussion, the method is applied in this paper to strange attractors of a variety of systems governed either by point mappings or by differential equations. The results indicate that it is a viable, effective and attractive method. Some comments on this method in comparison with the method of direct iteration will also be made.