Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems

The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions (ICs) and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models, the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case, the initial conditions are the injection angles. For higher-dimensional systems and small nonlinearities, the chaotic stripes are the initial condition inside which Arnold diffusion occurs.     

KAM theory analysis of the dynamics of three coaxial vortex rings

The equations of motion of three coaxial vortex rings in Euclidean 3-space are formulated as a Hamiltonian system. It is shown that the Hamiltonian function for this system can be written as the sum of a completely integrable part H₀ (related to the motion of three point vortices in the plane) and a non-integrable perturbation H₁. Then it is proved that when the vortex strengths all have the same sign and the ratio of the mean distances among the rings is very small in comparison to the mean radius of the rings, H₁/H₀1. Moreover, it is shown that H₁/H₀ is very small for all time for certain initial positions of the rings under the same assumptions. It is proved that the decomposition of the Hamiltonian and the estimates carry over to a reduced form of the system in coordinates moving with the center of vorticity and having one less degree of freedom. Then KAM theory is applied to prove the existence of invariant two-dimensional tori containing quasiperiodic motions. The existence of periodic solutions is also demonstrated. Several examples are solved numerically to show transitions from quasiperiodic and periodic to chaotic regimes in accordance with the theoretical results.     

Chaotic capture of vortices by a moving body. I. The single point vortex case

The study of the dynamical properties of vortex systems is an important and topical research area, and is becoming of ever increasing usefulness to a variety of physical applications. In this paper, we present a study of a model of a rotational singularity which obeys a logarithmic potential interacting with a bluff body in a uniform inviscid laminar flow, e.g., a line vortex interacting with a cylinder in three dimensions or a point vortex with a circular boundary in two dimensions. We show that this system is Hamiltonian and simple enough to be solved analytically for the stagnation points and separatrices of the flow, and a bifurcation diagram for the relevant parameters and classification of the various types of motion is given. We also show that, by introducing a periodic perturbation to the body, chaotic motion of the vortex can be readily generated, and we present analytic criteria for the generation of chaos using the Poincare-Melnikov-Arnold method. This leads to an important dynamical effect for the model, i.e., that the possibility exists for the vortex to be chaotically captured around the body for periods of time which are extremely sensitive to initial conditions. The basic mechanism for this capture is due to the chaotic dynamics and is similar to that of other chaotic scattering phenomena. We show numerically that cases exist where the vortex can be captured around an elliptic point external to (and possibly far from) the body, and the existence of other very complicated motions are also demonstrated. Finally, generalizations of the problem of the vortex-body interaction are indicated, and some possible applications are postulated such as the interaction of line vortices with aircraft wings.     

The Spatiotemporal Evolution of Wave Packets under Chaotic Condition

Using the minimum uncertainty state of quantum integrable system H0 as initial state,the spatiotemporal evolution of the wave packet under the action of perturbed Hamiltonian is studied causally as in classical mechanics,Due to the existence of the avoided energy level crossing in the spectrum there exist nonlinear resonances between some paris of neighboring components of the wave packet,the deterministic dynamical evolution becomes very complicated and appears to be chaotic.It is proposed to use expectation values for the whole set of basic dynamical variables and the corresponding spreading widths to describe the topological features concisely such that the quantum chaotic motion can be studied in contrast with the quantum regular motion and well characterized with the asymptotic behaviors .It has been demonstrated with numerical results that such a wave packet has indeed quantum behaviors of ergodicity as in corresponding classical case.     

On the motion of a heavy rigid body in an ideal fluid with circulation

We consider Chaplygin's equations [Izd. Akad. Nauk SSSR 3, 3 (1933)] describing the planar motion of a rigid body in an unbounded volume of an ideal fluid while circulation around the body is not zero. Hamiltonian structures and new integrable cases are revealed; certain remarkable partial solutions are found and their stability is examined. The nonintegrability of the system describing the motion of a body in the field of gravity is proved and the chaotic behavior of the system is illustrated.     

Fluid particle advection in the vicinity of the Föppl vortex system

Fluid particle advection in the vicinity of the Föppl vortex system is considered. Due to periodic motion of vortices about the Föppl equilibrium, fluid particles within the vortex atmosphere, the fluid region with a velocity field being induced by the vortices, can move chaotic in the sense of exponential divergence of near trajectories. This chaotic motion leads to the vortex atmosphere particles to be carried away from the atmosphere to the exterior flow. In this Letter, the part of the carried away fluid particles is numerically assessed and the dynamics of the fluid release from the vortex atmosphere is demonstrated.     

Parabolic resonances and instabilities

A parabolic resonance is formed when an integrable two-degrees-of-freedom (d.o.f.) Hamiltonian system possessing a circle of parabolic fixed points is perturbed. It is proved that its occurrence is generic for one parameter families (co-dimension one phenomenon) of near-integrable, two d.o.f. Hamiltonian systems. Numerical experiments indicate that the motion near a parabolic resonance exhibits a new type of chaotic behavior which includes instabilities in some directions and long trapping times in others. Moreover, in a degenerate case, near a flat parabolic resonance, large scale instabilities appear. A model arising from an atmospherical study is shown to exhibit flat parabolic resonance. This supplies a simple mechanism for the transport of particles with small (i.e. atmospherically relevant) initial velocities from the vicinity of the equator to high latitudes. A modification of the model which allows the development of atmospherical jets unfolds the degeneracy, yet traces of the flat instabilities are clearly observed. (c) 1997 American Institute of Physics.     

A study on drag reduction of a rotationally oscillating circular cylinder at low Reynolds number

The flow field around a rotationally oscillating circular cylinder in a uniform flow is studied by using a particle image velocimetry to understand the mechanism of drag reduction and the corresponding suppression of vortex shedding in the cylinder wake at low Reynolds number. Experiments are conducted on the flow around the circular cylinder under rotational oscillation at forcing Strouhal number 1, rotational amplitude 2 and Reynolds number 2,000. It is found from the flow measurement by PIV that the width of the wake is narrowed and the velocity fluctuations are reduced by the rotational oscillation of the cylinder, which results in the drag reduction rate of 30%. The mechanism of drag reduction is studied by phase-averaged PIV measurement, which indicates the formation of periodic small-scale vortices from both sides of the cylinder. It is found from the cross-correlation measurement between the velocity fluctuations that the large-scale structure of vortex shedding is almost removed in the cylinder wake, when the small-scale vortices are generated at the unstable frequency of shear layer by the influence of rotational oscillation.     

Ehrenfest Time at the Transition from Integrable Motion to Chaotic Motion

Ehrenfest time depends differently on the Planck constant in integrable motion and chaotic motion. We study how its dependence on the Planck constant changes when there is a continuous transition from regular motion to chaotic motion. We find that the dependence is a weighted compromise between its two distinct dependences in regular and chaotic motions. The study is carried out with the system of periodically driven anharmonic oscillator. As the system is quite typical, the result may apply generally.     

Symmetries and regular behavior of Hamiltonian systems

The behavior of the phase trajectories of the Hamilton equations is commonly classified as regular and chaotic. Regularity is usually related to the condition for complete integrability, i.e., a Hamiltonian system with n degrees of freedom has n independent integrals in involution. If at the same time the simultaneous integral manifolds are compact, the solutions of the Hamilton equations are quasiperiodic. In particular, the entropy of the Hamiltonian phase flow of a completely integrable system is zero. It is found that there is a broader class of Hamiltonian systems that do not show signs of chaotic behavior. These are systems that allow n commuting "Lagrangian" vector fields, i.e., the symplectic 2-form on each pair of such fields is zero. They include, in particular, Hamiltonian systems with multivalued integrals. (c) 1996 American Institute of Physics.     

Exponential decay and scaling laws in noisy chaotic scattering

In this Letter we present a numerical study of the effect of noise on a chaotic scattering problem in open Hamiltonian systems. We use the second order Heun method for stochastic differential equations in order to integrate the equations of motion of a two-dimensional flow with additive white Gaussian noise. We use as a prototype model the paradigmatic Hénon-Heiles Hamiltonian with weak dissipation which is a well-known example of a system with escapes. We study the behavior of the scattering particles in the scattering region, finding an abrupt change of the decay law from algebraic to exponential due to the effects of noise. Moreover, we find a linear scaling law between the coefficient of the exponential law and the intensity of noise. These results are of a general nature in the sense that the same behavior appears when we choose as a model a two-dimensional discrete map with uniform noise (bounded in a particular interval and zero otherwise), showing the validity of the algorithm used. We believe the results of this work be useful for a better understanding of chaotic scattering in more realistic situations, where noise is presented.     

Transition to chaos in the conservative four-wave parametric interactions

In this work, we analyze the transition from regular to chaotic states in the parametric four-wave interactions. The temporal evolution describing the coupling of two sets of three-waves with quadratic nonlinearity is considered. This system is shown to undergo a chaotic transition via the separatrix chaos scenario, where a soliton-like solution (separatrix) that is found for the integrable (perfect matched) case becomes irregular as a small mismatch is turned on. As the mismatch is increased the separatrix chaotic layer spreads along the phase space, eventually engrossing most part of it. This scenario is typical of low-dimensional Hamiltonian systems.     

Infinite Number of Integrals of Motion in Classically Integrable System With Boundary (Ⅱ)

In Affine Toda field theory, links among three generating functions for integrals of motion derived from P. (Ⅰ) are studied, and some classically integrable boundary conditions are obtained. An infinite number of integrals of motion are calculated in ZMS model with quasi-periodic condition. We find the classically integrable boundary conditions and K± matrices of ZMS model with independent boundary conditions on each end. It is identified that an infinite number of integrals of motion does exist and one of them is the Hamiltonian, so this system is completely integrable.     

Painlevé analysis and integrability of the trapped ionic system

Integrability in the Painlevé sense of the trapped ionic system in the quadrupole field with superpositions of rotationally symmetric hexapole and octopole fields is studied. Five integrable cases of the system are reported. First Integrals of the planar motion are founded. Confirming three-dimensional integrability of the equations of motion, the third explicit integrals of motion are constructed directly for each case. We carried out a numerical study to observe the regularity and chaotic regions via the Poincaré surface of sections, and corroborate the analytical results.     

Evaluation of an algebraic model for the vibrations of water,effects of a discrete symmetry

We evaluate the dynamics of an algebraic model Hamiltonian for the vibrational motion of the water molecule. We pay special attention to the effects of the discrete symmetry of order 2 of the model. For a comparison between the quantum dynamics and the classical dynamics it is necessary to desymmetrize such quantum states which are based on types of motion which come in symmetry related pairs. For the other states based on motion invariant under the symmetry operation a desymmetrization would be meaningless. The desymmetrized quantum states show a simple connection to the guiding motions of the classical dynamics which can be used for a complete assignment of the states even though the system is not integrable in the sense of Liouville and shows chaotic behaviour in large parts of the classical phase space.     

Geometry of a chaotic layer

An analysis is made of the dependence of the geometric shape of the chaotic layer near the separatrix of a nonlinear resonance of a Hamiltonian system on the parameters of this system. A separatrix algorithmic mapping, which describes the motion near the separatrix in the presence of an asymmetric perturbation having an arbitrary degree of asymmetry. The separatrix algorithmic mapping is an algorithm containing conditional transfer instructions, is considered. An analytic procedure is derived to reduce the separatrix algorithmic mapping to the unified surface of the cross section of the initial Hamiltonian system (mapping synchronization procedure). It is observed that in the case of the high-frequency perturbation λ → +∞ (where λ is the ratio of the perturbation frequency to the frequency of small phase oscillations at resonance), the chaotic layer is subjected to strong bending in the sense that during motion near the separatrix theamplitude of the energy deviations relative to the unperturbed separatrix value is much larger than the layer width. However, the synchronized separatrix algorithmic mapping ensures an accurate representation of the phase portrait of the layer for both low and high values of the parameter λ provided that the amplitude of the perturbation is fairly small. This is demonstrated by comparing the phase portraits obtained using the synchronized separatrix algorithmic mapping with the results of direct numerical integrations of the initial Hamiltonian system.     

Irregular mixing due to a vortex pair interacting with a fixed vortex

The paper examines scalar advection caused by a point–vortex pair encountering a fixed point vortex in a uniform flow. The interaction produces two types of vortex motion. First is unbounded as the pair moves unrestrictedly after encountering the fixed vortex. The scalar exchanging between the pair's bubble and fixed vortex's neighbourhood is numerically estimated. Second is bounded as the pair's vortices periodically oscillate about the fixed vortex. The pair's periodic motion perturbs scalar motion causing a portion of scalar trajectories to manifest chaotic behaviour. We analyse scalar transport using Poincaré sections, which reveal regular and chaotic transport regions.     

Coupled magnetoplasma and phonon systems

A long-wavelength Hamiltonian approach for coupled plasmons and polar phonons in a degenerate semiconductor with and without a uniform d.c. magnetic field is presented. Neglecting ionic motion, and with a uniform magnetic field present, a Hamiltonian is formulated for the magnetoplasma oscillations and a Hamiltonian is presented which includes the coupling between longitudinal optical phonons and collective hybrid-cyclotron plasma resonances. Canonical transformations, which put the latter Hamiltonian into second quantized diagonal form, are given. The normal modes, obtained by diagonalizing the appropriate Hamiltonian, correspond to the zeros and the infinities of the appropriate dielectric function. In analogy with the electron-phonon case, the resonances for a compensated system of coupled electrons and holes, in the presence of a uniform magnetic field, are treated by means of a Hamiltonian. The coupling in the above problems is represented by a product of two sets of plasma frequencies, the electron and ion or the electron and hole in the latter case. In all cases the coupling is independent of the magnetic field.