Convergence of attractors

The system of coupled oscillators and its time-discretization (with constant stepsize h) are considered in this paper. Under some conditions, it is showed that the discrete systems have one-dimensional global attractors l(h) converging to l which is the global attractor of continuous system.

Symmetry of chaotic attractors and kink-antikink structures in CMLs

In this paper the symmetries of coupled map lattices (CMLs) and their attractors are investigated by group and dynamical system theory, as well as numerical simulation, by means of which the kink-antikink patterns of CMLs in space-amplitude plots are discussed.

Chaotic dynamic analysis of viscoelastic plates

The dynamic buckling of viscoelastic plates with large deflection is investigated in this paper by using chaotic and fractal theory. The material behavior is given in terms of the Boltzmann superposition principle. in order to obtain accurate computation results, the nonlinear integro-differential dynamic equation is changed into an autonomic four-dimensional dynamical system. The numerical time integrations of equations are performed by using the fourth-order Runge-Kutta method. And the Lyapunov exponent spectrum, the fractal dimension of strange attractors and the time evolution of deflection are obtained. The influence of geometry nonlinearity and viscoelastic parameter on the dynamic buckling of viscoelastic plates is discussed.

Rotation Numbers Of Invariant Manifolds Around Unstable Periodic Orbits For The Diamagnetic Kepler Problem

In this paper, a method to construct topological template in terms of symbolic dynamics for the diamagnetic Kepler problem is proposed. To confirm the topological template, rotation numbers of invariant manifolds around unstable periodic orbits in a phase space are taken as an object of comparison. The rotation numbers are determined from the definition and connected with symbolic sequences encoding the periodic orbits in a reduced Poincare section. Only symbolic codes with inverse ordering in the forward mapping can contribute to the rotation of invariant manifolds around the periodic orbits. By using symbolic ordering, the reduced Poincare section is constricted along stable manifolds and a topological template, which preserves the ordering of forward sequences and can be used to extract the rotation numbers, is established. The rotation numbers computed from the topological template are the same as those computed from their original definition.

On properties of hyperchaos: Case study

Some properties of hyperchaos are exploited by studying both uncoupled and coupled CML. In addition to usual properties of chaotic strange attractors, there are other interesting properties, such as: the number of unstable periodic points embedded in the strange attractor increases dramatically increasing and a large number of low-dimensional chaotic invariant sets are contained in the strange attractor. These properties may be useful for regarding the edge of chaos as. the origin of complexity of dynamical systems.

Streamline topology and dilute particle dynamics in a Karman vortex street flow

Three types of streamline topology in a Karman vortex street flow are shown under the variation of spatial parameters. For the motion of dilute particles in the Karman vortex street flow, there exist a route of bifurcation to a chaotic orbit and more attractors in a bifurcation diagram for the proportion of particle density to fluid density. Along with the increase of spatial parameters in the flow field, the bifurcation process is suspended, as well as more and more attractors emerge. In the motion of dilute particles, a drag term and gravity term dominate and result in the bifurcation phenomenon.

Predicting chaotic time-series with wavelet networks

A new technique, wavelet network, is introduced to predict chaotic time series. By using this technique, firstly, we make accurate short-term predictions of the time series from chaotic attractors. Secondly, we make accurate predictions of the values and bifurcation structures of the time series from dynamical systems whose parameter values are changing with time. Finally we predict chaotic attractors by making long-term predictions based on remarkably few data points, where the correlation dimensions of predicted attractors are calculated and are found to be almost identical to those of actual attractors.

Direct dynamical test for deterministic chaos and optimal embedding of a chaotic time-series

We propose here a local exponential divergence plot which is capable of providing an alternative means of characterizing a complex time series. The suggested plot defines a time-dependent exponent and a ''plus'' exponent. Based on their changes with the embedding dimension and delay time, a criterion for estimating simultaneously the minimal acceptable embedding dimension, the proper delay time, and the largest Lyapunov exponent has been obtained. When redefining the time-dependent exponent LAMBDA(k) curves on a series of shells, we have found that whether a linear envelope to the LAMBDA(k) curves exists can serve as a direct dynamical method of distinguishing chaos from noise.

Direct dynamical test for deterministic chaos

We present a direct and dynamical method to distinguish low-dimensional deterministic chaos from noise. We define a series of time-dependent curves which are closely related to the largest Lyapunov exponent. For a chaotic time series, there exists an envelope to the time-dependent curves, while for a white noise or a noise with the same power spectrum as that of a chaotic time series, the envelope cannot be defined. When a noise is added to a chaotic time series, the envelope is eventually destroyed with the increasing of the amplitude of the noise.

Local exponential divergence plot and optimal embedding of a chaotic time-series

we propose here a local exponential divergence plot which is capable of providing a new means of characterizing chaotic time series. The suggested plot defines a time dependent exponent LAMBDA and a ''plus'' exponent LAMBDA+ which serves as a criterion for estimating simultaneously the minimal acceptable embedding dimension, the proper delay time and the largest Lyapunov exponent.

Information Preservation Modeling of Rayleigh-Benard Transition from Thermal Conduction to Convection

Onset and evolution of the Rayleigh-Benard (R-B) convection are investigated using the Information Preservation (IP) method. The information velocity and temperature are updated using the Octant Flux Splitting (OFS) model developed by Masters & Ye based on the Maxwell transport equation suggested by Sun & Boyd. Statistical noise inherent in particle approaches such as the direct simulation Monte Carlo (DSMC) method is effectively reduced by the IP method, and therefore the evolutions from an initial quiescent fluid to a final steady state are shown clearly. An interesting phenomenon is observed: when the Rayleigh number (Ra) exceeds its critical value, there exists an obvious incubation stage. During the incubation stage, the vortex structure clearly appears and evolves, whereas the Nusselt number (Nu) of the lower plate is close to unity. After the incubation stage, the vortex velocity and Nu rapidly increase, and the flow field quickly reaches a steady, convective state. A relation of Nu to Ra given by IP agrees with those given by DSMC, the classical theory and experimental data.

Monte Carlo Simulation Of Thermal Fluctuations Below The Onset Of Rayleigh-Benard Convection

The density fluctuations below the onset of convection in the Rayleigh-Benard problem are studied with the direct simulation Monte Carlo method. The particle simulation results clearly show the connection between the static correlation functions of fluctuations below the critical Rayleigh number and the flow patterns above the onset of convection for small Knudsen number flows (Kn=0.01 and Kn=0.005). Furthermore, the physical nature for no convection in the Rayleigh-Benard problem under large Knudsen number conditions (Kn>0.028) is explained based on the dynamics of fluctuations.