Nearly-integrable dissipative systems and celestial mechanics

The influence of dissipative effects on classical dynamical models of Celestial Mechanics is of basic importance. We introduce the reader to the subject, giving classical examples found in the literature, like the standard map, the Hénon map, the logistic mapping. In the framework of the dissipative standard map, we investigate the existence of periodic orbits as a function of the parameters. We also provide some techniques to compute the breakdown threshold of quasi-periodic attractors. Next, we review a simple model of Celestial Mechanics, known as the spin-orbit problem which is closely linked to the dissipative standard map. In this context we present the conservative and dissipative KAM theorems to prove the existence of quasi-periodic tori and invariant attractors. We conclude by reviewing some dissipative models of Celestial Mechanics. Among the rotational dynamics we consider the Yarkovsky and YORP effects; within the three-body problem we introduce the so-called Stokes and Poynting–Robertson effects.     

Crisis-induced unstable dimension variability in a dynamical system

Unstable dimension variability is an extreme form of non-hyperbolic behavior in chaotic systems whose attractors have periodic orbits with a different number of unstable directions. We propose a new mechanism for the onset of unstable dimension variability based on an interior crisis, or a collision between a chaotic attractor and an unstable periodic orbit. We give a physical example by considering a high-dimensional dissipative physical system driven by impulsive periodic forcing.     

Unstable periodic orbits and templates of the Rossler system: Toward a systematic topological characterization

The Rossler system has been exhaustively studied for parameter values (a in [0.33,0.557],b=2,c=4). Periodic orbits have been systematically extracted from Poincare maps and the following problems have been addressed: (i) all low order periodic orbits are extracted, (ii) encoding of periodic orbits by symbolic dynamics (from 2 letters up to 11 letters) is achieved, (iii) some rules of growth and of pruning of the periodic orbits population are obtained, and (iv) the templates of the attractors are elaborated to characterize the attractors topology. (c) 1995 American Institute of Physics.     

Complex transient dynamics of hidden attractors in a simple 4D system

A simple four-dimensional system with only one control parameter is proposed in this paper. The novel system has a line or no equilibrium for the global control parameter and exhibits complex transient transition behaviors of hyperchaotic attractors, periodic orbits, and unstable sinks. Especially, for the nonzero-valued control parameter, there exists no equilibrium in the proposed system, leading to the formation of various hidden attractors with complex transient dynamics. The research results indicate that the dynamics of the system shows weak chaotic robustness and depends greatly on the initial states.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

共轭Chen混沌系统的分岔分析及基于该系统的超混沌生成研究

分析了一个三维自治混沌系统的Hopf分岔现象,该系统的混沌吸引子属于共轭Chen混沌系统.通过引入一个控制器,基于该混沌系统构建了一个四维自治超混沌系统.该超混沌系统含有一个单参数,在一定的参数范围内呈现超混沌现象.通过Lyapunov指数和分岔分析,随着参数的变化该系统轨道呈现周期轨道、准周期轨道、混沌和超混沌的演化过程. 关键词： 混沌 超混沌生成 Hopf分岔 分岔分析     

The Plateau and Symmetrical Structure of Lyapunov Exponents in 2-Dimensional Homogeneous. Dissipative Map

The universal crossover behavior of Lyapunov exponents in transition from conservative limit to dissipative limit of dynamical system is studied. We discover numerically and prove analytically that for homogeneous dissipative two-dimensional maps, along the equal dissipation line in parameter space, two Lyapunov exponents λ₁ and λ₂ of periodic orbits possess a plateau structure, and around this exponent plateau value, there is a strict symmetrical relation between λ₁ and λ₂ no matter whether the orbit is periodic, quasiperiodic, or chaotic.The method calculating stable window and Lyapunov exponent plateau widths is given. For Hénon map and 2-dimensional circle map, the analytical and numerical results of plateau structure of Lyapunov exponents for period-1,2 and 3 orbits are presented.     

Microcanonical variational transition-state theory for reaction rates in dissipative systems

Upper bounds for the classical escape rate of a particle trapped in a metastable well and interacting with a dissipative medium are derived based on the periodic orbits of a reduced two-degree-of-freedom Hamiltonian involving the unstable normal mode and a collective bath mode. It is shown that even in what is usually thought of as the spatial diffusion limit the reactive flux can involve an energy diffusion term due to energy transfer from the dissipative media, in addition to the standard spatial diffusion term.     

Bifurcations of the trajectories at the saddle level in a Hamiltonian system generated by two coupled Schrodinger equations

Bifurcations of the complex homoclinic loops of an equilibrium saddle point in a Hamiltonian dynamical system with two degrees of freedom are studied. It arises to pick out the stationary solutions in a system of two coupled nonlinear Schrodinger equations. Their relation to bifurcations of hyperbolic and elliptic periodic orbits at the saddle level is studied for varying structural parameters of the system. Series of complex loops are described whose existence is related to periodic orbits.     

Interaction of nonlinear energy sink with a two degrees of freedom linear system: Internal resonance

We investigate the dynamics of 2DOF linear subsystem with close frequencies with attached nonlinear energy sink (NES). In this system, simultaneous targeted energy transfer from both linear oscillators to the NES is possible. It was demonstrated that the process of the TET can be analytically described as transient beats of relaxation—like motion arising due to the internal resonance. Contrary to previously studied models, the approach based on Hamiltonian structure of the system (study of the periodic orbits in the absence of the damping) fails to provide insight into the TET process. The reason of that is large number of secondary resonances activated through interaction between two primary 1:1 resonances. In the damped system these resonances are eliminated and then averaging—based approach is applicable. It was shown by the Hilbert Vibration Decomposition (HVD) that in the damped case there is a single significant component of the response regarded to the 1:1:1 resonance. Analytical model was verified numerically and a fairly good correspondence was observed.     

Boundary crisis and transient in a dissipative relativistic standard map

Some dynamical properties for a problem concerning the acceleration of particles in a wave packet are studied. The model is described in terms of a two-dimensional nonlinear map obtained from a Hamiltonian which describes the motion of a relativistic standard map. The phase space is mixed in the sense that there are regular and chaotic regions coexisting. When dissipation is introduced, the property of area preservation is broken and attractors emerge. We have shown that a tiny increase of the dissipation causes a change in the phase space. A chaotic attractor as well as its basin of attraction are destroyed thereby leading the system to experience a boundary crisis. We have characterized such a boundary crisis via a collision of the chaotic attractor with the stable manifold of a saddle fixed point. Once the chaotic attractor is destroyed, a chaotic transient described by a power law with exponent −1 is observed.     

Logarithmic correction to the probability of capture for dissipatively perturbed Hamiltonian systems

Hamiltonian systems are analyzed with a double homoclinic orbit connecting a saddle to itself. Competing centers exist. A small dissipative perturbation causes the stable and unstable manifolds of the saddle point to break apart. The stable manifolds of the saddle point are the boundaries of the basin of attraction for the competing attractors. With small dissipation, the boundaries of the basins of attraction are known to be tightly wound and spiral-like. Small changes in the initial condition can alter the equilibrium to which the solution is attracted. Near the unperturbed homoclinic orbit, the boundary of the basin of attraction consists of a large sequence of nearly homoclinic orbits surrounded by close approaches to the saddle point. The slow passage through an unperturbed homoclinic orbit (separatrix) is determined by the change in the value of the Hamiltonian from one saddle approach to the next. The probability of capture can be asymptotically approximated using this change in the Hamiltonian. The well-known leading-order change of the Hamiltonian from one saddle approach to the next is due to the effect of the perturbation on the homoclinic orbit. A logarithmic correction to this change of the Hamiltonian is shown to be due to the effect of the perturbation on the saddle point itself. It is shown that the probability of capture can be significantly altered from the well-known leading-order probability for Hamiltonian systems with double homoclinic orbits of the twisted type, an example of which is the Hamiltonian system corresponding to primary resonance. Numerical integration of the perturbed Hamiltonian system is used to verify the accuracy of the analytic formulas for the change in the Hamiltonian from one saddle approach to the next. (c) 1995 American Institute of Physics.     

On a simple recursive control algorithm automated and applied to an electrochemical experiment

We review a simple recursive proportional feedback (RPF) control strategy for stabilizing unstable periodic orbits found in chaotic attractors. The method is generally applicable to high-dimensional systems and stabilizes periodic orbits even if they are completely unstable, i.e., have no stable manifolds. The goal of the control scheme is the fixed point itself rather than a stable manifold and the controlled system reaches the fixed point in d+1 steps, where d is the dimension of the state space of the Poincare map. We provide a geometrical interpretation of the control method based on an extended phase space. Controllability conditions or special symmetries that limit the possibility of using a single control parameter to control multiply unstable periodic orbits are discussed. An automated adaptive learning algorithm is described for the application of the control method to an experimental system with no previous knowledge about its dynamics. The automated control system is used to stabilize a period-one orbit in an experimental system involving electrodissolution of copper. (c) 1997 American Institute of Physics.     

周期参数扰动的T混沌系统同宿轨道分析

针对一类周期参数扰动的T混沌系统, 通过变换将系统转化为具有广义Hamilton结构的周期参数扰动的慢变系统, 运用Melnikov方法对系统的同宿轨道进行了分析计算, 并给出了系统的同宿轨道参数分支条件. 同时, 通过数值实验, 对周期参数扰动控制策略及同宿轨道进行了仿真, 验证了文中理论分析的正确性. 关键词： Hamilton系统 Melnikov方法 同宿轨道 周期参数扰动     

TRANSIENT LIFETIME SPECTRUM OF TWO CHARGED PARTICLES IN A PAUL TRAP

The transient properties of laser-cooled two charged particles in a Paul trap are studied numerically. We find the existence of characteristic lifetime of an attractor, which is thought to be an important feature of the transient dynamics of the weakly dissipative system. The theoretical analysis shows that it is caused by pseudo-periodic orbit which is the residual sign of periodic orbit of the focus-saddle bifurcation. Study of dissipative coupled standard maps shows that this is a general conclusion for weakly dissipative system.     

用直线稳定方法控制耦合哈密顿系统的混沌运动

讨论了不稳定不动点邻域的不稳定轨道的稳定问题．通过对系统施加外部的控制信号，将直线稳定方法推广到控制高维保守系统一耦合标准映象的混沌运动．通过对外加控制信号的调整，使系统不稳定不动点邻域的不稳定轨道沿着连接任意时刻轨道点和该不动点的直线趋向不动点，从而使难于控制的高维保守系统的不稳定轨道趋于稳定．这种方法不需要事先掌握系统动力行为，而且具有较强的抗干扰能力。     

Dynamics of a three DOF mechanical system with dry friction

The dynamics of a three-block mechanical system is investigated: each block is pulled by a belt and is subjected to linear elastic and nonlinear frictional forces which induce oscillations in the system. The study of the full dynamics of the system is partially reduced to the study of a two-dimensional map; its attractors, their basins of attraction and their Lyapunov exponents provide a powerful tool to understand the dynamic behaviour of the full mechanical system which possesses rich dynamics characterised by periodic, quasi-periodic, chaotic and hyper-chaotic attractors.     

Directed transient long-range transport in a slowly driven Hamiltonian system of interacting particles

We study the Hamiltonian dynamics of a one-dimensional chain of linearly coupled particles in a spatially periodic potential which is subjected to a time-periodic mono-frequency external field. The average over time and space of the related force vanishes and hence, the system is effectively without bias which excludes any ratchet effect. We pay special attention to the escape of the entire chain when initially all of its units are distributed in a potential well. Moreover for an escaping chain we explore the possibility of the successive generation of a directed flow based on large accelerations. We find that for adiabatic slope-modulations due to the ac-field transient long-range transport dynamics arises whose direction is governed by the initial phase of the modulation. Most strikingly, that for the driven many particle Hamiltonian system directed collective motion is observed provides evidence for the existence of families of transporting invariant tori confining orbits in ballistic channels in the high-dimensional phase spaces.     

一种数值方法定义Hénon映射符号动力学的讨论

一种能方便地求解Ｈéｎｏｎ映射周期轨道的数值方法可以定义相应的符号动力学，本文讨论了它在（ａ，ｂ）参数平面上的有效区间，同时分析了它在周期轨道分叉临界曲线附近个别区域出现的特殊现象。     

AN ANALYTICAL AND NUMERICAL STUDY OF A MODIFIED VAN DER POL OSCILLATOR

A three-dimensional system of differential equations that models an electronic oscillator is considered. The equations allow a variety of periodic orbits that originate from a degenerate Hopf bifurcation, which is analytically studied. Numerical results are presented that show the existence of saddle-node cusps of periodic orbits, as well as period-doubling bifurcations, that result in the coexistence of multiple “canard” orbits if one of the parameters is small. The presence of chaotic attractors is also detected.