Paul阱中共线三离子体系的经典动力学

研究了在Paul阱囚禁场赝势作用下共线构形的三离子体系经典动力学特性.尽管这是一个非线性体系，但不存在混沌，即体系在任何能量下运动都是规则的，而相空间则由两个轨迹为对称和反对称周期(或准周期)轨道的KAM不变环面构成.体系的两条最简单的周期轨道S和A的周期随能量E的下降而增大，并在E趋于体系的最小值E_min=3.0时分别为反对称和对称谐振动. 关键词：     

Chaotic scattering,transport, and fractals in a simple hydrodynamic flow

Advection of passive tracers in an unsteady hydrodynamic flow consisting of a background stream and a vortex is analyzed as an example of chaotic particle scattering and transport. A numerical analysis reveals a nonattracting chaotic invariant set Λ that determines the scattering and trapping of particles from the incoming flow. The set has a hyperbolic component consisting of unstable periodic and aperiodic orbits and a nonhyperbolic component represented by marginally unstable orbits in the particle-trapping regions in the neighborhoods of the boundaries of outer invariant tori. The geometry and topology of chaotic scattering are examined. It is shown that both the trapping time for particles in the mixing region and the number of times their trajectories wind around the vortex have hierarchical fractal structure as functions of the initial particle coordinates. The hierarchy is found to have certain properties due to an infinite number of intersections of the stable manifold in Λ with a material line consisting of particles from the incoming flow. Scattering functions are singular on a Cantor set of initial conditions, and this property must manifest itself by strong fluctuations of quantities measured in experiments.     

Basin boundaries in asymmetric vibrations of a circular plate

In order to investigate further nonlinear asymmetric vibrations of a clamped circular plate under a harmonic excitation, we reexamine a primary resonance, studied by Yeo and Lee [Corrected solvability conditions for non-linear asymmetric vibrations of a circular plate, Journal of Sound and Vibration 257 (2002) 653-665] in which at most three stable steady-state responses (one standing wave and two traveling waves) are observed to exist. Further examination, however, tells that there exist at most five stable steady-state responses: one standing wave and four traveling waves. Two of the traveling waves lose their stability by Hopf bifurcation and have a sequence of period-doubling bifurcations leading to chaos. When the system has five attractors: three equilibrium solutions (one standing wave and two traveling waves) and two chaotic attractors (two modulated traveling waves), the basin boundaries of the attractors on the principal plane are obtained. Also examined is how basin boundaries of the modulated motions (quasi-periodic and chaotic motions) evolve as a system parameter varies. The basin boundaries of the modulated motions turn out to have the fractal nature.     

Dynamics of chaotic driving: rotation in the restricted three-body problem

We investigate the rotation of a small nonspherical body in the planar restricted three-body problem along periodic, quasi-periodic, and chaotic orbits of the small body's center of mass. The rotation dynamics is chaotic in all three cases, but a systematic overview of it via stroboscopic mappings is possible only in the periodic case. We propose to explore the structured phase space patterns by following an ensemble of trajectories, a droplet, in the phase space. The temporal evolution of the pattern can be characterized by a time-dependent fractal dimension. It is shown to converge exponentially to a time-independent value for long times. In the presence of dissipation, the droplet typically converges to a so-called snapshot chaotic attractor whose shape might change chaotically in time, but whose asymptotic fractal dimension is constant.     

Orbital and epicyclic motion of charged test particles around non-rotating Einstein-Æther black holes

The study of charged test particle dynamics in the combined black hole gravitational field and magnetic field around it could provide important theoretical insight into astrophysical processes around such compact object. We have explored the orbital and epicyclic motion of charged test particles in the background of non-rotating Einstein-Æther black holes in the presence of external uniform magnetic field. We numerically integrate the equations of motion and analyze the trajectories of the charged test particles. We examined the stability of circular orbits using effective potential technique and study the characteristics of innermost stable circular orbits. We analyze the key features of quasi-harmonic oscillations of charged test particles nearby the stable circular orbits in an equatorial plane of the black hole, and investigate the radial profiles of the frequencies of latitudinal as well as radial harmonic oscillations in dependence on the strength of magnetic field, mass of the black hole and dimensionless coupling constants of the theory. We demonstrate that the magnetic field and dimensionless parameters of the theory have strong influence on charged particle motion around Einstein-Æther black holes.     

Semiclassical theory of chaotic conductors

We calculate the Landauer conductance through chaotic ballistic devices in the semiclassical limit, to all orders in the inverse number of scattering channels without and with a magnetic field. Families of pairs of entrance-to-exit trajectories contribute, similarly to the pairs of periodic orbits making up the small-time expansion of the spectral form factor of chaotic dynamics. As a clue to the exact result we find that close self-encounters slightly hinder the escape of trajectories into leads. Our result explains why the energy-averaged conductance of individual chaotic cavities, with disorder or "clean," agrees with predictions of random-matrix theory.     

Local control of chaotic motion

On the basis of the Ott, Grebogi and Yorke method (OGY) of controlling chaotic motion by stabilizing unstable periodic orbits we propose a control method which allows a nearly continuous adjusting of the control parameter and which therefore is capable also for controlling noisy systems. Any motion which is a solution of the system's equation of motion can be stabilized, unstable periodic orbits as well as chaotic trajectories. We demonstrate the feasibility of the method by stabilizing experimentally arbitrarily chosen chaotic trajectories of a driven damped pendulum affected by noise.     

Dynamics of extended bodies with spin-induced quadrupole in Kerr spacetime: generic orbits

We discuss motions of extended bodies in Kerr spacetime by using Mathisson–Papapetrou–Dixon equations. We firstly solve the conditions for circular orbits, and calculate the orbital frequency shift due to the mass quadrupoles. The results show that we need not consider the spin-induced quadrupoles in extreme-mass-ratio inspirals for space-based gravitational wave detectors. We quantitatively investigate the temporal variation of rotational velocity of the extended body due to the coupling of quadrupole and background gravitational field. For generic orbits, we numerically integrate the Mathisson–Papapetrou–Dixon equations for evolving the motion of an extended body orbiting a Kerr black hole. By comparing with the monopole–dipole approximation, we reveal the influences of quadrupole moments of extended bodies on the orbital motion and chaotic dynamics of extreme-mass-ratio systems. We do not find any chaotic orbits for the extended bodies with physical spins and spin-induced quadrupoles. Possible implications for gravitational wave detection and pulsar timing observation are outlined.     

Geodesic motions of test particles in a relativistic core–shell spacetime

In this paper, we discuss the geodesic motions of test particles in the intermediate vacuum between a monopolar core and an exterior shell of dipoles, quadrupoles and octopoles. The radii of the innermost stable circular orbits at the equatorial plane depend only on the quadrupoles. A given oblate quadrupolar leads to the existence of two innermost stable circular orbits, and their radii are larger than in the Schwarzschild spacetime. However, a given prolate quadrupolar corresponds to only one innermost stable circular orbit, and its radius is smaller than in the Schwarzschild spacetime. As to the general geodesic orbits, one of the recently developed extended phase space fourth order explicit symplectic-like methods is efficiently applicable to them although the Hamiltonian of the relativistic core–shell system is not separable. With the aid of both this fast integrator without secular growth in the energy errors and gauge invariant chaotic indicators, the effect of these shell multipoles on the geodesic dynamics of order and chaos is estimated numerically.     

Chaotic electron trajectories in a circular free electron laser

The motion of a relativistic electron is analyzed in the field configuration consisting of a circular wiggler magnetic field, an axial magnetic field, and the equilibrium self-electric and self-magnetic fields produced by the non-neutral electron ring. By generating Poincare surface-of-section maps, it is shown that when the equilibrium self-fields is strong enough, the electron motions become chaotic. Although the realistic circular wiggler magnetic field destroys the integrability of the electron motion as the equilibrium self-fields do, the role the latter plays to make the motions become chaotic is stronger than the former. In addition, the axial magnetic field can restrain the occurrence of the chaoticity.     

Gravitational Waves from a Pseudo-Newtonian Kerr Field with Halos

A close relation between gravitational waveforms and the types of trajectories in a superposed field between a pseudo-Newtonian Kerr black hole and quadrupolar halos is shown in detail. The gravitational waveforms emitted from circular, KAM tori and chaotic orbits must be periodic, quasiperiodic and stochastic, respectively. The chaotic motion can maximally enhance both the amplitudes and the energy emission rates of the waves.     

Chaotic Dynamics of Test Particle in the Gravitational Field with Magnetic Dipoles

We investigate the dynamics of the test particle in the gravitational field with magnetic dipoles in thispaper. At first we study the gravitational potential by numerical simulations. We find, for appropriate parameters, thatthere are two different cases in the potential curve, one of which is the one-well case with a stable critical point, and theother is the three-well case with three stable critical points and two unstable ones. As a consequence, the chaotic motionwill rise. By performing the evolution of the orbits of the test particle in the phase space, we find that the orbits of thetest particle randomly oscillate without any periods, even sensitively depending on the initial conditions and parameters.chaotic motion of the test particle in the field with magnetic dipoles becomes even obvious as the value of the magneticdipoles increases.     

平行电磁场中锂原子自电离的半经典分析

采用包含组合回归的扩展的闭合轨道理论计算了平行电磁场中锂原子依赖于时间的自电离谱,并用半经典的方法解释了电离过程中的混沌现象.讨论了电离电子逃逸时间谱分形结构中隐含的各韵律段的电离轨迹,并得到了轨迹的一般规律,其中特别关注由核散射产生的特殊的逃逸轨迹的性质.具体研究了磁场对锂原子自电离混沌脉冲阵列中电子逃逸轨道和逃逸时间谱的影响.结果发现随着外加磁场的增大,电离脉冲越来越复杂,混沌现象也越明显.这显示了逃逸轨道对初始条件的敏感依赖性.     

Nonlinearity associated with the motion of a Bloch wall-onset of a chaotic regime and analysis of the Barkhausen noise

The noise V(t) generated by the displacement of a magnetic domain wall is investigated. As the velocity of the wall increases. the trajectories in the (V, V) plane exhibit a transition from quasi-periodic to chaotic motion.     

Characterizing plastic depinning dynamics with the fluctuation theorem

We demonstrate that the fluctuation theorem can be used to characterize plastic flow phases in collectively interacting particle assemblies driven over quenched disorder when strong fluctuations and crackling noise with 1/f ^α character occur. By measuring the frequency of entropy-destroying trajectories and the diffusivity near the threshold for motion, we map out the different dynamic phases and demonstrate that the fluctuation theorem holds in the strongly fluctuating plastic flow regime which was previously shown to be chaotic. For different driving rates and disorder strength, we find that it is possible to define an effective temperature which decreases with increasing drive, as expected for this type of system. When the size of the pinning sites is large, we identify specific regimes where the fluctuation theorem holds only at long times due to an excess of negative entropy events that occur when particles undergo circular motions within the traps. We discuss how the fluctuation theorem could be applied to plastic flow in other driven nonthermal systems with quenched disorder such as superconducting vortices, magnetic domain walls, Coulomb glasses, and earthquake models.     

On the Stability of Realistic Three-Body Problems

We consider the system Sun—Jupiter—Ceres as an example of a planar, circular, restricted three-body problem and, after substituting the mass ratio of Jupiter/Sun (which is approximately 10^-3) with a parameter , we prove the existence of stable quasi-periodic motions with frequencies close to the observed (average) frequencies reported in “The Astronomical Almanac” for . The proof is “computer-assisted”. Received: 1 April 1996 / Accepted: 25 October 1996     

2D chaotic quantum states in superlattices

The semiclassical motion of an electron along the axis of a superlattice may be calculated from the miniband dispersion curve. Under a weak electric field the electron executes Bloch oscillations which confines the motion along the superlattice axis. When a magnetic field, tilted with respect to the superlattice axis, is applied the electron orbits become chaotic. The onset of chaos is characterised by a complex mixed stable-chaotic phase space and an extension of the orbital trajectories along the superlattice axis. This delocalisation of the orbits is also reflected in the quantum eigenstates of the system some of which show well-defined patterns of high probability density whose shapes resemble certain semiclassical orbits. This suggests that the onset of chaos will be manifest in electron transport through a finite superlattice. We also propose that these phenomena may be observable in the motion of trapped ultra-cold atoms in an optically induced superlattice potential and magnetic quadrupole potential whose axis is tilted relative to the superlattice axis.     

The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states

We present a rigorous study of the classical ground-states under boundary conditions of a class of one-dimensional models generalizing the discrete Frenkel-Kontorova model. The extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic, quasi-periodic and chaotic orbits. For all boundary conditions, we select among all the extremum solutions of the energy of the model, those which correspond to the ground-states of the infinite system. We prove that these ground-states are either periodic (commensurate) or quasi-periodic (incommensurate) but are never chaotic. We also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions. The topological structure of the whole set of ground-states is described in details. In addition to physical applications, consequences for twist map homeomorphisms are mentioned. Part II (S. Aubry, P.Y. LeDaeron and G. Andre) will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map.     

From collective periodic running states to completely chaotic synchronised states in coupled particle dynamics

We consider the damped and driven dynamics of two interacting particles evolving in a symmetric and spatially periodic potential. The latter is exerted to a time-periodic modulation of its inclination. Our interest is twofold: First, we deal with the issue of chaotic motion in the higher-dimensional phase space. To this end, a homoclinic Melnikov analysis is utilised assuring the presence of transverse homoclinic orbits and homoclinic bifurcations for weak coupling allowing also for the emergence of hyperchaos. In contrast, we also prove that the time evolution of the two coupled particles attains a completely synchronised (chaotic) state for strong enough coupling between them. The resulting "freezing of dimensionality" rules out the occurrence of hyperchaos. Second, we address coherent collective particle transport provided by regular periodic motion. A subharmonic Melnikov analysis is utilised to investigate persistence of periodic orbits. For directed particle transport mediated by rotating periodic motion, we present exact results regarding the collective character of the running solutions entailing the emergence of a current. We show that coordinated energy exchange between the particles takes place in such a manner that they are enabled to overcome--one particle followed by the other--consecutive barriers of the periodic potential resulting in collective directed motion.     

Integrability lost: Chaotic dynamics of classical strings on a confining holographic background

It is known that classical string dynamics on pure AdS₅×S⁵${\mathit{AdS}}_5\times S^5$ is integrable and plays an important role in solvability. This is a deep and central issue in holography. Here we investigate similar classical integrability for a more realistic confining background and provide a negative answer. The dynamics of a class of simple string configurations on AdS soliton background can be mapped to the dynamics of a set of non-linearly coupled oscillators. In a suitable limit of small fluctuations we discuss a quasi-periodic analytic solution of the system. Numerics indicates chaotic behavior as the fluctuations are not small. Integrability implies the existence of a regular foliation of the phase space by invariant manifolds. Our numerics shows how this nice foliation structure is eventually lost due to chaotic motion. We also verify a positive Lyapunov index for chaotic orbits. Our dynamics is roughly similar to other known non-integrable coupled oscillator systems like Hénon–Heiles equations.