Chaotic attitude oscillations of a satellite filled with a rotating ellipsoidal mass of liquid subject to gravity-gradient torques

The Hamiltonian equations of a liquid-filled satellite subject to gravity-gradient torques in terms of the generalized Deprit variables are established for applying the Melnikov theory formally. The heteroclinic orbits of the torque-free symmetric liquid-filled satellite are found. A criterion for the heteroclinic transversal intersections at the onset of chaotic attitude is formulated using Melnikov's integral. The results from the Melnikov theory are crosschecked with simulation.     

A study of a liquid-filled lens module with a flexible sensor for aberration correction

This study introduces the concept of a liquid-filled lens module; the module is composed of a flexible lens membrane, a flexible sensor membrane that is constructed by attaching sensor chips onto a flexible substrate and transparent liquid that fills the space between the flexible lens membrane and the sensor membrane. The image quality of the liquid-filled lens module is comparable to that of a solid lens module with a flat-type image sensor. However, the liquid-filled lens module has the advantages of requiring fewer parts, having better optical aberration compensation, a wider image range. This study identifies the optimized conditions for the sensor membrane to fulfill the Petzval surface of the optical module and the compensation mechanism of the flexible lens and the sensor membranes. The simulation results show that the distortion aberration, the field of curvature and the Modulation Transfer Function are good.     

Chaotic atomic tunneling between two periodically driven Bose-Einstein condensates

The chaotic coherent atomic tunneling between two periodically driven and weakly coupled Bose-Einstein condensates has been investigated. The perturbed correction to the homoclinic orbit is constructed and its boundedness conditions are established that contain the Melnikov criterion for the onset of chaos. We analytically reveal that the chaotic coherent atomic tunneling is deterministic but not predictable. Our numerical calculation shows good agreement with the analytical result and exhibits nonphysically numerical instability. By adjusting the initial conditions, we propose a method to control the unboundedness, which leads the quantum coherent atomic tunneling to predictable periodical oscillation.     

Exponential synchronization of stochastic impulsive perturbed chaotic Lur''e systems with time-varying delay and parametric uncertainty

This paper is devoted to investigating the scheme of exponential synchronization for uncertain stochastic impulsive perturbed chaotic Lur＇e systems. The parametric uncertainty is assumed to be norm bounded. Based on the Lyapunov function method, time-varying delay feedback control technique and a modified Halanay inequality for stochastic differential equations, several sufficient conditions are presented to guarantee the exponential synchronization in mean square between two identical uncertain chaotic Lur＇e systems with stochastic and impulsive perturbations. These conditions are expressed in terms of linear matrix inequalities （LMIs）, which can easily be checked by utilizing the numerically efficient Matlab LMI toolbox. It is worth pointing out that the approach developed in this paper can provide a more general framework for the synchronization of multi-perturbation chaotic Lur＇e systems, which reflects a more realistic dynamics. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.     

Transitions between chaos and order in rf-driven Josephson junction

We investigate transitions between the chaotic and regular states through a perturbed chaotic solution of a rf-driven Josephson system. It is shown that the transition from order to chaos may occur when the system initially near the heteroclinic points. The chaotic solution tends to an unstable periodic one for the initial values sufficiently nearing the heteroclinic orbit but going beyond the heteroclinic points. Thus the Josephson chaos can be analytically and numerically controlled, by adjusting the initial conditions.     

Homoclinic Bifurcations in Simple Parametrically Driven Systems

In this paper we consider the onset of chaotic particle motion in a perturbed Morse potential and the homoclinic bifurcations in a parametrically driven Lorenz system. The Melnikov-Keener method is used to derive bifurcation conditions for the parameters of the dynamical systems. For some selected parameter values the theoretical predictions are checked by numerical experiments.     

Complex dynamical behaviors of compact solitary waves in the perturbed mKdV equation

In this paper, we give a detailed discussion about the dynamical behaviors of compact solitary waves subjected to the periodic perturbation. By using the phase portrait theory, we find one of the nonsmooth solitary waves of the mKdV equation, namely, a compact solitary wave, to be a weak solution, which can be proved. It is shown that the compact solitary wave easily turns chaotic from the Melnikov theory. We focus on the sufficient conditions by keeping the system stable through selecting a suitable controller. Furthermore, we discuss the chaotic threshold for a perturbed system. Numerical simulations including chaotic thresholds, bifurcation diagrams, the maximum Lyapunov exponents, and phase portraits demonstrate that there exists a special frequency which has a great influence on our system; with the increase of the controller strength, chaos disappears in the perturbed system. But if the controller strength is sufficiently large, the solitary wave vibrates violently.     

Chaotic Dynamics of a Periodically Modulated Josephson Junction

We study the chaotic dynamics of a periodically modulated Josephson junction with damping. The general solution of the first-order perturbed equation is constructed by using the direct perturbation technique. It is theoretically found that the boundedness conditions of the general solution contain the Melnikov chaotic criterion. When the perturbation conditions cannot be satisfied, numerical simulations demonstrate that the system can step into chaos through a period doubling route with the increase of the amplitude of the modulating term. Regulating specific parameters can effectively suppress the chaos.     

Ionisation of metastable 3P-state hydrogen atom by electron with exchange effects

Propagation of nonlinear shock waves for the generalised Oskolkov equation and dynamic motions of the perturbed Oskolkov equation are investigated. Employing the unified method, a collection of exact shock wave solutions for the generalised Oskolkov equations is presented. Collocation finite element method is applied to the generalised Oskolkov equation for checking the accuracy of the proposed method by two test problems including the motion of shock wave and evolution of waves with Gaussian and undular bore initial conditions. Considering an external periodic perturbation, the dynamic motions of the perturbed generalised Oskolkov equation are studied depending on the system parameters with the help of phase portrait and time series plot. The perturbed generalised Oskolkov equation exhibits period-3, quasiperiodic and chaotic motions for some special values of the system parameters, whereas the generalised Oskolkov equation presents shock waves in the absence of external periodic perturbation.     

一类带有未知参数的受扰混沌系统的观测器同步

在系统参数未知的情形下,观测器方法和自适应方法相结合,实现了一类受扰混沌系统的同步.借助于Lyapunov稳定性理论和Barbalat引理,给出了观测器的设计方法.此方法约束条件较少,适应于大部分常见的混沌系统.最后,通过对典型混沌系统的同步数值仿真,证实了方法的有效性和正确性. 关键词： 混沌系统 外界干扰 同步 观测器法     

Chaotic Behavior and Instability of Perturbed sine-Gordon Solitons

Solitons of the sine-Gordon system interacting with a disturbed external field are handled by using a direct method. Theoretical analysis reveals that the single soliton perturbed by a periodic field leads to chaotic behavior of the system, and the perturbed double soliton is unstable for most of physically interesting spacetime disturbances.     

Quasi-Hamiltonian principle of liquid-filled elastic body dynamics

Liquid-filled elastic body dynamics is an important branch of fluid-solid coupling mechanics. It deals with the study of motion of a body and the liquid contained in the body under the interaction between the two,for example,a liquid-filled satellite,a fuel tank of an airplane,etc. The research on liquid-filled elastic body dynamics is usually done by the variational method since the method has a feature of treating things as a whole. Considering the elastic motion of the liquid-filled body and the surface tension effect on the liquid-gas interface,liquid-solid interface and gas-solid interface,the present paper establishes a quasi-Hamiltonian variational principle for the above-mentioned system. After finding the stationary-value conditions of its functional obtained,a complete system of governing equations consisting of the stationary value conditions,preconditions and constraint conditions is established,and then the equations are reduced into some known ones in a special case.     

一种基于washout滤波器技术的参数受扰混沌系统的控制

基于washout滤波器技术的混沌控制方法不仅能用于参数不变的混沌系统的控制，还可用于参数受扰动时的混沌系统的控制.给出了参数受扰动情况下基于washout滤波器技术的控制器设计的一个充分条件.数值仿真表明，该控制方法易于实现、控制代价小、能使原混沌系统的不稳定不动点稳定，而且具有一定的鲁棒性，很适合于工程应用. 关键词： 混沌控制 参数受扰动 washout滤波器     

Spatial chaos of trapped Bose-Einstein condensate in one-dimensional weak optical lattice potential

The spatially chaotic attractor in an elongated cloud of Bose-Einstein condensed atoms perturbed by a weak optical lattice potential is studied. The analytical insolvability and numerical incomputability of the atomic number density are revealed by a perturbed solution that illustrates the unpredictability of the deterministic chaos. Although this could lead the nonphysical explosion and unboundedness to the numerical solution, the theoretical analysis offers a criterion to avoid them. Moreover, the velocity field is investigated that exhibits the superfluid property of the chaotic system.     

Nodal domain statistics for quantum maps, percolation, and stochastic Loewner evolution

We develop a percolation model for nodal domains in the eigenvectors of quantum chaotic torus maps. Our model follows directly from the assumption that the quantum maps are described by random matrix theory. Its accuracy in predicting statistical properties of the nodal domains is demonstrated for perturbed cat maps and supports the use of percolation theory to describe the wave functions of general Hamiltonian systems. We also demonstrate that the nodal domains of the perturbed cat maps obey the Cardy crossing formula and find evidence that the boundaries of the nodal domains are described by stochastic Loewner evolution with diffusion constant close to the expected value of 6, suggesting that quantum chaotic wave functions may exhibit conformal invariance in the semiclassical limit.     

Wannier–Stark chaos of a Bose–Einstein condensate in 1D optical lattices

Using the idea of the macroscopic quantum wave function and the definition of the Melnikov chaos, we investigate the spatially chaotic features of a Bose–Einstein condensate (BEC) in a Wannier–Stark potential for the trivial phase and the non-trivial phase cases. The perturbed chaotic solutions are constructed, and the chaotic and unstable regions on the parameter space are illustrated. Numerical calculations to the spatial evolutions of the atomic number density and the energy density demonstrate the analytical results and exhibit the chaotic spatial distribution and energy distribution of the BEC atoms.     

Stochastic Perturbation of Integrable Systems: A Window to Weakly Chaotic Systems

Integrable non-linear Hamiltonian systems perturbed by additive noise develop a Lyapunov instability, and are hence chaotic, for any amplitude of the perturbation. This phenomenon is related, but distinct, from Taylor’s diffusion in hydrodynamics. We develop expressions for the Lyapunov exponents for the cases of white and colored noise. The situation described here being ‘multi-resonance’—by nature well beyond the Kolmogorov–Arnold–Moser regime, it offers an analytic glimpse on the regime in which many near-integrable systems, such as some planetary systems, find themselves in practice. We show with the aid of a simple example, how one may model in some cases weakly chaotic deterministic systems by a stochastically perturbed one, with good qualitative results.     

Chaotic solitons in Sine-Gordon system

We extend the constant-variation method to the case of partial differential equations. Applying the method to periodically perturbed Sine-Gordon system, we find some novel solitons, which are embedded in a chaotic attractor and possess controllable velocity of motion. Taking periodically driven long Josephson junction as an example the corresponding chaotic region in parameter space and chaotic orbit are obtained analytically and numerically. Received 25 December 2000     

Stochastic resonance in chaotic systems

The phenomenon of stochastic resonance (SR) is investigated for chaotic systems perturbed by white noise and a harmonic force. The bistable discrete map and the Lorenz system are considered as models. It is shown that SR in chaotic systems can be realized via both parameter variation (in the absence of noise) and by variation of the noise intensity with fixed values of the other parameters.     

Periodic and chaotic behaviour in a reduced form of the perturbed generalized Korteweg–de Vries and Kadomtsev–Petviashvili equations

The dynamical behaviour of a reduced form of the perturbed generalized Korteweg–de Vries and Kadomtsev–Petviashvili equations (extension of the Korteweg–de Vries equation to two space variables) are studied in this paper. Harmonic solutions of non-resonance and primary resonance are obtained using the perturbation method. Chaotic motion under harmonic excitations is studied using the Melnikov method.A wide range of solutions for the reduced perturbed generalized Korteweg–de Vries equations, in which non-linear phenomena appearing within transition from regular harmonic response (periodic solutions) to chaotic motion, are obtained using the time integration Runge–Kutta method. When chaos is found, it is detected by examining the phase plane, the Poincaré map, the sensitivity solution of the solution to initial conditions, and by calculating the largest Lyapunov exponent.