Dynamical analysis,circuit realization,and application in pseudorandom number generators of a fractional-order laser chaotic system

A new five-dimensional fractional-order laser chaotic system (FOLCS) is constructed by incorporating complex variables and fractional calculus into a Lorentz-Haken-type laser system. Dynamical behavior of the system, circuit realization and application in pseudorandom number generators are studied. Many types of multi-stable states are discovered in the system. Interestingly, there are two types of state transition phenomena in the system, one is the chaotic state degenerates to a periodical state, and the other is the intermittent chaotic oscillation. In addition, the complexity of the system when two parameters change simultaneously is measured by the spectral entropy algorithm. Moreover, a digital circuit is design and the chaotic oscillation behaviors of the system are verified on this circuit. Finally, a pseudo-random sequence generator is designed using the FOLCS, and the statistical characteristics of the generated pseudo-random sequence are tested with the NIST-800-22. This study enriches the research on the dynamics and applications of FOLCS.     

一类非线性抛物型变分不等方程的全局吸引子及弱近似惯性流形

对由一类非线性抛物型变分不等方程所描述的无穷维动力系统，给出了存在全局吸引子及弱近似惯性流形的充分条件．     

Attractors for a Three-Dimensional Thermo-Mechanical Model of Shape Memory Alloys

Abstract In this note, we consider a Frémond model of shape memory alloys. Let us imagine a piece of a shape memory alloy which is fixed on one part of its boundary, and assume that forcing terms, e.g., heat sources and external stress on the remaining part of its boundary, converge to some time-independent functions, in appropriate senses, as time goes to infinity. Under the above assumption, we shall discuss the asymptotic stability for the dynamical system from the viewpoint of the global attractor. More precisely, we generalize the paper [12] dealing with the one-dimensional case. First, we show the existence of the global attractor for the limiting autonomous dynamical system; then we characterize the asymptotic stability for the non-autonomous case by the limiting global attractor. * Project supported by the MIUR-COFIN 2004 research program on “Mathematical Modelling and Analysis of Free Boundary Problems”.     

Hankel convolution operators on entire functions and distributions

In this paper we study the Hankel convolution operators on the space of even and entire functions and on Schwartz distribution spaces. We characterize the Hankel convolution operators as those ones that commute with Hankel translations and with a Bessel operator. Also we prove that the Hankel convolution operators are hypercyclic and chaotic on the spaces under consideration.     

GAUSSIAN WHITE NOISE CALCULUS OF GENERALIZED EXPANSION

A new framework of Gaussian white noise calculus is established, in line with generalized expansion in [3, 4, 7]. A suitable frame of Fock expansion is presented on Gaussian generalized expansion functionals being introduced here, which provides the integral kernel operator decomposition of the second quantization of Koopman operators for chaotic dynamical systems, in terms of annihilation operators dt and its dual, creation operators t*.     

Time Series Prediction Based on Chaotic Attractor

A new prediction technique is proposed for chaotic time series. The usefulness of the technique is thatit can kick off some false neighbor points which are not suitable for the local estimation of the dynamics systems. Atime-delayed embedding is used to reconstruct the underlying attractor, and the prediction model is based on the timeevolution of the topological neighboring in the phase space. We use a feedforward neural network to approximate thelocal dominant Lyapunov exponent, and choose the spatial neighbors by the Lyapunov exponent. The model is testedfor the Mackey-Glass equation and the convection amplitude of lorenz systems. The results indicate that this predictiontechnique can improve the prediction of chaotic time series.     

The long‐time behaviour of the thermoconvective flow in a porous medium

For the Boussinesq approximation of the equations of coupled heat and fluid flow in a porous medium we show that the corresponding system of partial differential equations possesses a global attractor. We give lower and upper bounds of the Hausdorff dimension of the attractor depending on a physical parameter of the system, namely the Rayleigh number of the flow. Numerical experiments confirm the theoretical findings and raise new questions on the structure of the solutions of the system. Copyright © 2004 John Wiley & Sons, Ltd.     

GLOBAL ATTRACTOR OF A SPATIALLY DISCRETIZED REACTION-DIFFUSION SYSTEM WITH HAMILTONIAN STRUCTURE

１ＨａｍｉｌｔｏｎｉａｎＳｔｒｕｃｔｕｒｅｉｎＲｅａｃｔｉｏｎ－ＤｉｆｕｓｉｏｎｓｙｓｔｅｍＣｏｎｓｉｄｅｒａｓｙｓｔｅｍｏｆｒｅａｃｔｉｏｎ－ｄｉｆｕｓｉｏｎｅｑｕａｔｉｏｎｓｕｔ＝ｕｘｘ＋ｆ（ｕ，ｖ）ｖｔ＝ｖｘｘ＋ｇ（ｕ，ｖ）｛（１．１）ｗｈｅｒ...     

Estimation of dynamic properties of attractors observed in hollow copper electrode atmospheric pressure arc plasma system

Understanding of the basic nature of arc root fluctuation is still one of the unsolved problems in thermal arc plasma physics. It has direct impact on myriads of thermal plasma applications being implemented at present. Recently, chaotic nature of arc root behavior has been reported through the analysis of voltages, acoustic and optical signals which are generated from a hollow copper electrode arc plasma torch. In this paper we present details of computations involved in the estimation process of various dynamic properties and show how they reflect chaotic behavior of arc root in the system.     

Dynamics of a two-frequency parametrically driven duffing oscillator

Summary We investigate the transition from two-frequency quasiperiodicity to chaotic behavior in a model for a quasiperiodically driven magnetoelastic ribbon. The model system is a two-frequency parametrically driven Duffing oscillator. As a driving parameter is increased, the route to chaos takes place in four distinct stages. The first stage is a torus-doubling bifurcation. The second stage is a transition from the doubled torus to a strange nonchaotic attractor. The third stage is a transition from the strange nonchaotic attractor to a geometrically similar chaotic attractor. The final stage is a hard transition to a much larger chaotic attractor. This latter transition arises as the result of acrisis, the characterization of which is one of our primary concerns. Numerical evidence is given to indicate that the crisis arises from the collision of the chaotic attractor with the stable manifold of a saddle torus. Intermittent bursting behavior is present after the crisis with the mean time between bursts scaling as a power law in the distance from the critical control parameter; τ ∼ (A-A_c)^-α. The critical exponent is computed numerically, yielding the value α=1.03±0.01. Theoretical justification is given for the computed critical exponent. Finally, a Melnikov analysis is performed, yielding an expression for transverse crossings of the stable and unstable manifolds of the crisis-initiating saddle torus.