Arnold Diffusion Research Articles

Instability and diffusion in the elliptic restricted three-body problem

The importance of the stability characteristics of the planar elliptic restricted three-body problem is that they offer insight about the general dynamical mechanisms causing instability in celestial mechanics. To analyze these concerns, elliptic–elliptic and hyperbolic–elliptic resonance orbits (periodic solutions with lower period) are numerically discovered by use of Newton's differential correction method. We find indications of stability for the elliptic–elliptic resonance orbits because slightly perturbed orbits define a corresponding two-dimensional invariant manifold on the Poincare surface-section. For the resonance orbit of the hyperbolic–elliptic type, we show numerically that its stable and unstable manifolds intersect transversally in phase-space to induce instability. Then, we find indications that there are orbits which jump from one resonance zone to the next before escaping to infinity. This phenomenon is related to the so-called Arnold diffusion.

Arnold diffusion in large systems

A new regime of Arnold diffusion in which the diffusion rate has a power-law dependence on the perturbation strength is studied theoretically and in numerical experiments. The theory developed predicts this new regime to be universal in the perturbation intermediate asymptotics, the width of the latter increasing with the dimensionality of the perturbation frequency space, particularly in large systems with many degrees of freedom. The results of numerical experiments agree satisfactorily with the theoretical estimates.

Quantization of the Stochastic Pump Model of Arnold Diffusion

A semiclassical quantization of the stochastic pump model of Arnold diffusion is presented. The semiclassical model is found to be equivalent to a one-dimensional disordered wire, yielding localized states limiting the extent of diffusion. {copyright} {ital 1997} {ital The American Physical Society}

Arnold's Diffusion with Two Resonances

Arnold's Diffusion with Two Resonances

Gravitational ionization: a chaotic net in the Kepler system

The long-term nonlinear dynamics of a Keplerian binary system under the combined influences of gravitational radiation damping and external tidal perturbations is analysed. Gravitational radiation reaction leads the binary system towards eventual collapse, while the external periodic perturbations could lead to the ionization of the system via Arnold diffusion. When these two opposing tendencies nearly balance each other, interesting chaotic behaviour occurs which is briefly studied in this paper. It is possible to show that periodic orbits can exist in this system for sufficiently small damping. Moreover, we employ the method of averaging to investigate the phenomenon of capture into resonance.

ANALYTICAL EXPRESSIONS FOR STABLE AND UNSTABLE MANIFOLDS IN HIGHER DEGREE OF FREEDOM HAMILTONIAN SYSTEMS

Perturbations of completely integrable Hamiltonian systems with three or more degrees of freedom are studied. In particular, the unperturbed systems are assumed to be separable into a product of simple oscillator-type systems and a system containing homo- or heteroclinic connections consisting of stable and unstable manifolds of saddle points. Under a perturbation, the manifolds persist but separate and may no longer intersect. In this paper we show how, with proper choices for initial conditions, one may solve the variational equations to obtain analytical expressions for orbits on the perturbed manifolds in the form of expansions in the small parameter characterizing the perturbation. The derivation also shows how the distance between the manifolds can be uniquely defined, and thus provides an alternative to the traditional higher dimensional Melnikov method. It is finally argued that the approximate knowledge of the shape and position of the perturbed manifolds could be utilized for the study of large-scale phase-space motions, such as those associated with Arnold diffusion. The theory is further illuminated in two example problems.

Energy diffusion due to nonlinear perturbation on linear Hamiltonians.

In nonintegrable Hamiltonian systems, energy initially localized in a few degrees of freedom tends to disperse through nonlinear couplings. We analyze such processes in systems of many degrees of freedom. As a complement to the well-known Arnold diffusion, which describes energy diffusion by chaotic motion near separatrices, our analysis treats another universal case: coupled small oscillations near stable equilibrium points. Because we are concerned with the low-energy regime, where the nonlinearity of the unperturbed Hamiltonian is negligibly small, existing theories of Arnold diffusion cannot apply. Using probability theories we show that resonances of small detuning, which are ubiquitous in systems of many degrees of freedom, make energy diffusion possible. These resonances are the cause of energy equipartition in the low-energy limit. From our analysis, simple analytic equations that relate the energy, the degrees of freedom, the strength of nonlinear coupling, and the time scale for equipartition emerge naturally. These equations reproduce results from large-scale numerical simulations with remarkable accuracy. \textcopyright{} 1996 The American Physical Society.

On the ionization of a Keplerian binary system by periodic gravitational radiation

The gravitational ionization of a Keplerian binary system via normally incident periodic gravitational radiation of definite helicity is discussed. The periodic orbits of the planar tidal equation are investigated on the basis of degenerate continuation theory. The relevance of the Kolmogorov-Arnold-Moser theory to the question of gravitational ionization is elucidated, and it is conjectured that the process of ionization is closely related to the Arnold diffusion of the perturbed system.

Evaporation of Petroleum Products from Contaminated Soils

Bioremediation can remove petroleum products from soil that has been contaminated by leaking underground storage tanks, but abiotic processes such as evaporation can contribute significantly to the overall removal process. The mathematical model described in this paper was developed to predict the evaporation rate of volatile liquids from petroleum-contaminated sand. The model is based on simple concepts relating to molecular diffusion embodied in the theory underlying the estimation of binary diffusivities using measurements made with an Arnold diffusion cell. The model in its simplified form indicates that the rate of evaporation for a particular volatile liquid is proportional to the square root of the product of diffusivity and partial pressure divided by the molecular weight of the liquid. This in part explains why evaporative losses from sand are so much higher for gasoline than for diesel fuel. The model also shows that the time for evaporation is directly proportional to the square of the depth dried out and inversely proportional to the vapor pressure of the volatile liquid. The model was tested using gravimetric measurements of the evaporation of n-heptane, unleaded gasoline, and diesel fuel from sand under laboratory conditions.

Some remarks on a variational approach to Arnold's diffusion

In this paper we consider the following class of Lagrangian systems:$\qquad\qquad \qquad\qquad L_{\epsilon,\mu}(q,\dot{q},Q,\dot{Q},t) = \frac{\dot{Q}^2}2+\frac{\dot{q}^2}2 + \epsilon(1-\cos q)+ \mu f(q,\dot{q},Q,\dot{Q},t,\mu) $ which has been studied by many authors in connection with Arnold's diffusion. Extending [2] prove, by variational means, that, for suitable perturbations including for example:$\qquad\qquad \qquad\qquad f(q,\dot{q},Q,\dot{Q}, t,\mu)=(1-\cos q)(\cos Q+\cos t) + \mu^{p-1} \sin(q+Q) \quad (p>2) $if $\mu$ is small enough, exists a diffusion orbit of $L_{\epsilon, \mu}$ such that $\dot{Q}(t)$ undergoes a variation of order $1$ in a time $t_d$ polinomial in $\mu$, $t_d\approx \frac 1{\mu^2}$.

Arnold Diffusion and Equipartition in an Oscillator Chaina

Annals of the New York Academy of SciencesVolume 773, Issue 1 p. 205-220 Arnold Diffusion and Equipartition in an Oscillator Chain† ALLAN J. LICHTENBERG, ALLAN J. LICHTENBERG Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory University of California, Berkeley, California 94720Search for more papers by this author ALLAN J. LICHTENBERG, ALLAN J. LICHTENBERG Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory University of California, Berkeley, California 94720Search for more papers by this author First published: December 1995 https://doi.org/10.1111/j.1749-6632.1995.tb12170.x † The author would like to acknowledge the support of Office of Naval Research Grant N00014–89-J-1097. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume773, Issue1Waves in AstrophysicsDecember 1995Pages 205-220 RelatedInformation

Nonlinear instability and chaos in plasma wave–wave interactions. II. Numerical methods and results

In Part I of this work [Phys. Plasmas 2, 1926 (1995)], the behavior of linearly stable, integrable systems of waves in a simple plasma model was described using a Hamiltonian formulation. Explosive instability arose from nonlinear coupling between positive and negative energy modes, with well-defined threshold amplitudes depending on the physical parameters. In this concluding paper, the nonintegrable case is treated numerically. The time evolution is modeled with an explicit symplectic integrator derived using Lie algebraic methods. For amplitudes large enough to support two-wave decay interactions, strongly chaotic motion destroys the separatrix bounding the stable region in phase space. Diffusive growth then leads to explosive instability, effectively reducing the threshold amplitude. For initial amplitudes too small to drive decay instability, slow growth via Arnold diffusion might still lead to instability; however, this was not observed in numerical experiments. The diffusion rate is probably underestimated in this simple model.

Self-consistent chaos and Arnold diffusion in a cyclotron-maser wave-particle system.

Self-consistent chaos and Arnold diffusion in a cyclotron-maser wave-particle system.

The fractal structure of quasi-periodic solutions of hamiltonian systems

After a simple example giving an outlook of the whole complexity of solutions, the classical Kolmogorov-Arnold-Moser (KAM) demonstration is outlined with some useful extensions. The usual Arnold tori have a dimensionality equal to the number of degrees of freedom and a generalization of the KAM demonstration shows the very general existence of quasi-periodic motions on invariant tori with a smaller dimensionality. These latter invariant tori corresponds to resonances and lead to a well-ordered picture of solutions. The solutions are classified into three main types: (a) the periodic and quasi-periodic orbits that are ‘first-order stable’; (b) the chaotic orbits; (c) the open orbits coming from infinity and going back to infinity. The solutions of the first type separate the two other types, but they fill a nowhere dense set when the Hamiltonian problem of interest is non-integrable. This set has a structure looking like a fractal but without an exact scale property. It is conjectured (‘Arnold diffusion conjecture’) that all non-integrable analytic autonomous Hamiltonian systems lead to these same types of picture and classification.

The four-element discrete nonlinear Schrodinger equation-non-integrability and Arnold diffusion

We study the integrability of the discrete nonlinear Schrodinger equation with four elements. Using the two-element system as the underlying integrable subsystem we treat the coupling to the additional oscillators perturbationally. A symmetry reduction method is used to map the coupled system of four elements on a three-degree of freedom Hamiltonian system in the reduced phase space. We analyse the geometrical structure of the phase space of the resulting reduced system focusing on the homoclinic manifold. By means of Wiggins' generalized Melnikov method we prove non-integrability through the existence of chaotic dynamics in demonstrating the presence of Arnold diffusion for the coupled dynamics.

Chaotic particle motion and halo formation induced by charge nonuniformities in an intense ion beam propagating through a periodic quadrupole focusing field

A test-particle model is used to investigate the charged-particle dynamics in an intense matched ion beam with nonuniform density profile propagating through an alternating-gradient quadrupole focusing field in the space-charge-dominated regime. It is shown that self-field nonlinearities due to the transverse nonuniformity in the beam density profile not only can result in chaotic ion motion but also can cause halo formation by the ejection of particles from the beam core. The structure of the particle phase space is studied. It is shown that the process of halo formation can occur on a fast time scale—on the order of a few lattice periods. The halo size is found to be determined by a Kolmogorov–Arnold–Moser (KAM) surface on a time scale much shorter than that of Arnold diffusion.

Diffusion at intersecting resonances in Hamiltonian systems

We study n-degree-of-freedom, nearly integrable Hamiltonian systems near the intersection of a stronger and a weaker resonance. We construct motions that cross the weaker resonance along the stronger one at a speed larger than that of Arnold diffusion. We also derive estimates for the measure of such solutions.

Energy transitions and time scales to equipartition in the Fermi-Pasta-Ulam oscillator chain.

We study the energy transitions and time scales, in the Fermi-Pasta-Ulam oscillator chain, at which the energy E, initially in a single or small group of low-frequency modes, is distributed among modes. The energy transitions, with increasing energy, are classified. At low energy the linear parts of the energies are distributed in a geometrically decreasing series ${\mathit{E}}_{\mathit{h}}$=${\mathrm{\ensuremath{\rho}}}^{2}$${\mathit{E}}_{\mathit{h}\mathrm{\ensuremath{-}}2\ensuremath{\gamma}}$, with \ensuremath{\gamma} the mode in which most of the initial energy is placed and \ensuremath{\rho}=(3\ensuremath{\beta}${\mathit{E}}_{\ensuremath{\gamma}}$)/(4\ensuremath{\pi}\ensuremath{\gamma}). A transition occurs at R\ensuremath{\equiv}6\ensuremath{\beta}${\mathit{E}}_{\ensuremath{\gamma}}$(N+1)/${\mathrm{\ensuremath{\pi}}}^{2}$\ensuremath{\sim}1, with N the number of oscillators and \ensuremath{\beta} the quartic coupling constant. Above this transition there is strong local coupling among neighboring modes with a characteristic resonant frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathit{b}}$\ensuremath{\sim}4\ensuremath{\beta}\ensuremath{\gamma}${\mathit{E}}_{\ensuremath{\gamma}}$/${\mathit{N}}^{2}$. There is a second transition at a critial energy \ensuremath{\beta}${\mathit{E}}_{\mathit{c}}$\ensuremath{\sim}0.3, above which stochasticity among low-frequency resonances transfers energy into high-frequency resonances by the Arnold diffusion mechanism. Above this transition we numerically determine a universal scaling for the time scale to approach equipartition among the modes. The universal time scale is qualitatively explained in terms of the driving time scale ${\mathrm{\ensuremath{\tau}}}_{\mathit{b}}$=2\ensuremath{\pi}/${\mathrm{\ensuremath{\Omega}}}_{\mathit{b}}$ and a diffusive filling time.

Codimension-one partitioning and phase space transport in multi-degree-of-freedom Hamiltonian systems with non-toroidal invariant manifold intersections

Invariant manifold formulations of phase-space transport theory offer a well-known means of prediction for the chaotic dynamics associated with 1 1 2 - and 2-degree-of-freedom systems. Whether such a means of prediction extends to systems with more than 2 degrees-of-freedom has thus far received little attention. We derive a multi-degree-of-freedom turnstile-based phase space transport theory for Hamiltonian flows with non-toroidal intersections of global stable and unstable manifolds of invariant normally hyperbolic sets of a Poincaré section on the energy surface. These non-toroidal manifold intersections are the typical case for flows with more than 2 degrees-of-freedom, and they disallow a partitioning of phase space via segments of stable and unstable manifolds alone. We show it is nevertheless possible for these intersection geometries to construct codimension-one partitions (dividing surfaces) of the Poincaré section that respect asymptotic behavior, and hence divide between regions of qualitatively different dynamics (bounded versus unbounded, rotation versus libration, and so on). These partitioning surfaces thus provide a geometrically exact criterion for capture into, and escape from, regions in the Poincaré section corresponding to qualitatively different dynamics. A turnstile-based phase space transport theory is then derived to describe transitions across the partitioning surfaces. We use Melnikov theory to study partition geometry, and to search through multi-dimensional phase space and parameter space to explore variation of O( ε) flux across the partition. Searches of this kind would be helpful in the control and optimization of any number of nonlinear physical processes associated with transitions across broken separatrices, such as molecular dissociation and atomic ionization. We observe significant variation in the effectiveness of separatrix crossing with differing phase and action values, and hence note the motivation to retain phase information, and the role of Arnold diffusion in sending points to regions of enhanced or diminished separatrix crossing, referred to as Arnold enhancement and Arnold suppression of transport across codimension-one partitions.

NUMERICAL MODELING OF MAGNETIZATION DYNAMICS IN SUPERFLUID HELIUM-3

This is mainly a review article on magnetization dynamics in superfluid helium-3 within the framework of the Leggett–Takagi model. The mathematical setting of the latter is formulated. The problem of resonance relaxation, important for the physics of superfluid helium-3, is discussed from the point of view of the KAM theory. Numerical modeling is employed for studying the Arnold diffusion which, as is shown, could be the cause of catastrophic relaxation, observed experimentally. We suggest that similar methods may be employed for studying topology of invariant KAM tori.