Slow-fast Hamiltonian Systems Research Articles

On Phase at a Resonance in Slow-Fast Hamiltonian Systems

On Phase at a Resonance in Slow-Fast Hamiltonian Systems

Emergence of quasiequilibrium state and energy distribution for the beads-spring molecule interacting with a solvent.

We study the energy distribution during the emergence of a quasiequilibrium (QE) state in the course of relaxation to equipartition in slow-fast Hamiltonian systems. A bead-spring model where beads (masses) are connected by springs is considered. The QE lasts for a long time because the energy exchange between the high-frequency vibrational and other motions is prevented when springs in the molecule become stiff. We numerically calculated the time-averaged kinetic energy and found that the kinetic energy of the solvent particles was always higher than that of the bead in a molecule. This is explained by adopting the equipartition theorem in QE, and it agrees well with the numerical results. The energy difference can help determine how far the system is from achieving equilibrium, and it can be used as an indicator of the number of frozen or inactive degrees existing in the molecule.

Theoretical model of the nonlinear resonant interaction of whistler-mode waves and field-aligned electrons

The nonlinear resonant interaction of intense whistler-mode waves and energetic electrons in the Earth's radiation belts is traditionally described by theoretical models based on the consideration of slow–fast resonant systems. Such models reduce the electron dynamics around the resonance to the single pendulum equation that provides solutions for the electron nonlinear scattering (phase bunching) and phase trapping. Applicability of this approach is limited to not-too-small electron pitch-angles (i.e., sufficiently large electron magnetic moments), whereas model predictions contradict to the test particle results for small pitch-angle electrons. This study is focused on such field-aligned (small pitch-angle) electron resonances. We show that the nonlinear resonant interaction can be described by the slow–fast Hamiltonian system with the separatrix crossing. For the first cyclotron resonance, this interaction results in the electron pitch-angle increase for all resonant electrons, contrast to the pitch-angle decrease predicted by the pendulum equation for scattered electrons. We derive the threshold value of the magnetic moment of the transition to a new regime of the nonlinear resonant interaction. For field-aligned electrons, the proposed model provides the magnitude of magnetic moment changes in the nonlinear resonance. This model supplements existing models for not-too-small pitch-angles and contributes to the theory of the nonlinear resonant electron interaction with intense whistler-mode waves.

On the Geometry of Slow-Fast Phase Spaces and the Semiclassical Quantization

In the context of the averaging method for Poisson and symplectic structures and the theory of Hannay–Berry connections, we discuss some aspects of the semiclassical quantization for a class of slow-fast Hamiltonian systems with two degrees of freedom. For a pseudodifferential Weyl operator with two small parameters corresponding to the semiclassical and adiabatic limits, we show how to construct some series of quasimodes associated to a family of Lagrangian 2-tori which are almost invariant with respect to the classical dynamics.

Jumps of energy near a separatrix in slow-fast Hamiltonian systems

Jumps of energy near a separatrix in slow-fast Hamiltonian systems

Jumps of energy near a separatrix in slow-fast Hamiltonian systems

Jumps of energy near a separatrix in slow-fast Hamiltonian systems

Deformations of Poisson structures on fibered manifolds and adiabatic slow–fast systems

In the context of normal forms, we study a class of slow–fast Hamiltonian systems on general Poisson fiber bundles with symmetry. Our geometric approach is motivated by a link between the deformation theory for Poisson structures on fibered manifolds and the adiabatic perturbation theory. We present some normalization results which are based on the averaging theorem for horizontal 2-cocycles on Poisson fiber bundles.

Geometry of slow–fast Hamiltonian systems and Painlevé equations

In the first part of the paper we introduce some geometric tools needed to describe slow–fast Hamiltonian systems on smooth manifolds. We start with a smooth bundle p:M→B where (M,ω) is a C∞-smooth presymplectic manifold with a closed constant rank 2-form ω and (B,λ) is a smooth symplectic manifold. The 2-form ω is supposed to be compatible with the structure of the bundle, that is the bundle fibers are symplectic manifolds with respect to the 2-form ω and the distribution on M generated by kernels of ω is transverse to the tangent spaces of the leaves and the dimensions of the kernels and of the leaves are supplementary. This allows one to define a symplectic structure Ωε=ω+ε−1p∗λ on M for any positive small ε, where p∗λ is the lift of the 2-form λ to M. Given a smooth Hamiltonian H on M one gets a slow–fast Hamiltonian system with respect to Ωε. We define a slow manifold SM for this system. Assuming SM is a smooth submanifold, we define a slow Hamiltonian flow on SM. The second part of the paper deals with singularities of the restriction of p to SM. We show that if dimM=4,dimB=2 and Hamilton function H is generic, then the behavior of the system near a singularity of fold type is described, to the main order, by the equation Painlevé-I, and if this singularity is a cusp, then the related equation is Painlevé-II.

Adiabatics using phase space translations and small parameter “dynamics”

For slow-fast Hamiltonian systems with one fast degree of freedom, we describe the construction of the complete adiabatic invariant and the complete adiabatic term at once in all asymptotic orders by using the small parameter “dynamics” and parallel translations in the phase space.

Rapid geometrical chaotization in slow-fast Hamiltonian systems.

In this Rapid Communication we demonstrate effects of a new mechanism of adiabaticity destruction in Hamiltonian systems with a separatrix in the phase space. In contrast to the slow diffusive-like destruction typical for many systems, this new mechanism is responsible for very fast chaotization in a large phase volume. To investigate this mechanism we consider a Hamiltonian system with two degrees of freedom and with a separatrix in the phase plane of fast variables. The fast chaotization is due to an asymmetry of the separatrix and corresponding geometrical jumps of an adiabatic invariant. This system describes the motion of charged particles in a inhomogeneous electromagnetic field with a specific configuration. We show that geometrical jumps of the adiabatic invariant result in a very fast chaotization of particle motion.

Capture into resonance and escape from it in a forced nonlinear pendulum

We study dynamics of a nonlinear pendulum under a periodic force with small amplitude and slowly decreasing frequency. It is well known that when the frequency of the external force passes through the value of the frequency of the unperturbed pendulum's oscillations, the pendulum can be captured into the resonance. The captured pendulum oscillates in such a way that the resonance is preserved, and the amplitude of the oscillations accordingly grows. We consider this problem in the frames of a standard Hamiltonian approach to resonant phenomena in slow-fast Hamiltonian systems developed earlier, and evaluate the probability of capture into the resonance. If the system passes the resonance at small enough initial amplitudes of the pendulum, the capture occurs with necessity (so-called autoresonance). In general, the probability of capture varies between one and zero, depending on the initial amplitude. We demonstrate that a pendulum captured at small values of its amplitude escapes from the resonance in the domain of oscillations close to the separatrix of the pendulum, and evaluate the amplitude of the oscillations at the escape.

Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems

We present a coordinate-free approach for constructing approximate first integrals of generalized slow-fast Hamiltonian systems, based on the global averaging method on parameter-dependent phase spaces with \documentclass[12pt]{minimal}\begin{document}$\mathbb {S}^1$\end{document}S1-symmetry. Explicit global formulas for approximate second-order first integrals are derived. As examples, we analyze the case quadratic in the fast variables (in particular, the elastic pendulum), and the charged particle in a slowly-varying magnetic field.

On the global structure of normal forms for slow-fast Hamiltonian systems

In the framework of Lie transforms and the global method of averaging, the normal forms of a multidimensional slow-fast Hamiltonian system are studied in the case when the flow of the unperturbed (fast) system is periodic and the induced \(\mathbb{S}^1 \)1-action is not necessarily free and trivial. An intrinsic splitting of the second term in a \(\mathbb{S}^1 \)1-invariant normal form of first order is derived in terms of the Hannay-Berry connection assigned to the periodic flow.

Dynamical Features in a Slow-fast Piecewise Linear Hamiltonian System

We demonstrate that a piecewise linear slow-fast Hamiltonian system with an equi- librium of the saddle-center type can have a sequence of small parameter values for which a one-round homoclinic orbit to this equilibrium exists. This contrasts with the well-known find- ings by Amick and McLeod and others that solutions of such type do not exist in analytic Hamiltonian systems, and that the separatrices are split by the exponentially small quantity. We also discuss existence of homoclinic trajectories to small periodic orbits of the Lyapunov family as well as symmetric periodic orbits near the homoclinic connection. Our further re- sult, illustrated by simulations, concerns the complicated structure of orbits related to passage through a non-smooth bifurcation of a periodic orbit.

On the accuracy of conservation of adiabatic invariants in slow-fast Hamiltonian systems

Let the adiabatic invariant of action variable in slow-fast Hamiltonian system with two degrees of freedom have two limiting values along the trajectories as time tends to infinity. The difference of two limits is exponentially small in analytic systems. An iso-energetic reduction and canonical transformations are applied to transform the slow-fast systems to form of systems depending on slowly varying parameters in a complexified phase space. On the basis of this method an estimate for the accuracy of conservation of adiabatic invariant is given for such systems.

Jump of the adiabatic invariant at a separatrix crossing: Degenerate cases

An expression for the quasi-random jump of the adiabatic invariant at a separatrix crossing is obtained for a slow–fast Hamiltonian system with two degrees of freedom in the case when the separatrix passes through a degenerate saddle point in the phase plane of the fast variables. The general case with an arbitrary degree of degeneracy was considered, and this degree is assumed to remain fixed in the process of evolution of the slow variables. The typical value of the jump is larger than in the non-degenerate case studied earlier. Though strongly degenerate, such a setting can be relevant for physical problems. The influence of the asymmetry of a phase portrait on the magnitude of adiabatic invariant jumps was considered as well. An example of this kind is studied, namely the motion of ions in current sheets with complex inner structure.

Jumps of adiabatic invariant at the separatrix of a degenerate saddle point

We consider a slow-fast Hamiltonian system with two degrees of freedom. One degree of freedom corresponds to slow variables, and the other one corresponds to fast variables. A characteristic ratio of the rates of change of slow and fast variables is a small parameter κ. For every fixed value of the slow variables, in the phase portrait of the fast variables there are a saddle point and separatrices passing through it. When the slow variables change, phase points may cross the separatrices. The action variable of the fast motion is an adiabatic invariant of the full system as long as a trajectory is far from the separatrices: value of the adiabatic invariant is conserved with an accuracy of order of κ on time intervals of order of 1/κ. A passage through a narrow neighborhood of the separatrices results in a jump of the adiabatic invariant. We consider a case when the saddle point is degenerate. We derive an asymptotic formula for the jump of the adiabatic invariant which turns out to be a value of order of κ(3/4) (in the case of a non-degenarate saddle point a similar jump is known to be a value of order of κ). Accumulation of these jumps after many consecutive separatrix crossings leads to the "diffusion" of the adiabatic invariant and chaotic dynamics. We verify the analytical expression for the jump of the adiabatic invariant by numerical simulations. We discuss application of the obtained results to the description of charged particle dynamics in the Earth magnetosphere.

Nonlinear dynamics of charged particles in an oblique electromagnetic wave

We study dynamics of a charged particle under action of an electromagnetic wave that propagates obliquely to a background uniform magnetic field. The dynamics is described by a slow–fast Hamiltonian system. We show that long-term dynamics is dominated by phenomena of capture of particle into resonance with the wave and escape from this resonance, as well as of scattering on resonance. We find that the variation of the particleʼs kinetic energy on the time interval between capture and escape is bounded and accumulated in the motion along the background field. We discuss possible applications of the obtained results.

Heteroclinic Orbits in Slow–Fast Hamiltonian Systems with Slow Manifold Bifurcations

Motivated by a problem in which a heteroclinic orbit represents a moving inter- face between ordered and disordered crystalline states, we consider a class of slow-fast Hamiltonian systems in which the slow manifold loses normal hyperbolicity due to a trans- critical or pitchfork bifurcation as a slow variable changes. We show that under assumptions appropriate to the motivating problem, a singular heteroclinic solution gives rise to a true heteroclinic solution. In contrast to previous approaches to such problems, our approach uses blow-up of the bifurcation manifold, which allows geometric matching of inner and outer solutions.

Geometric shadowing in slow–fast Hamiltonian systems

We study a class of slow–fast Hamiltonian systems with any finite number of degrees of freedom, but with at least one slow one and two fast ones. At ε = 0 the slow dynamics is frozen. We assume that the frozen system (i.e. the unperturbed fast dynamics) has families of hyperbolic periodic orbits with transversal heteroclinics.For each periodic orbit we define an action J. This action may be viewed as an action Hamiltonian (in the slow variables). It has been shown in Brännström and Gelfreich (2008 Physica D 237 2913–21) that there are orbits of the full dynamics which shadow any finite combination of forward orbits of J for a time t = O(ε−1).We introduce an assumption on the actions of periodic orbits which enables us to shadow any continuous curve (of arbitrary length) in the slow phase space for any time. The slow dynamics shadows the curve as a purely geometrical object, thus the time on the slow dynamics has to be reparametrized.