164 research outputs found
Smooth Hamiltonian systems with soft impacts
In a Hamiltonian system with impacts (or "billiard with potential"), a point
particle moves about the interior of a bounded domain according to a background
potential, and undergoes elastic collisions at the boundaries. When the
background potential is identically zero, this is the hard-wall billiard model.
Previous results on smooth billiard models (where the hard-wall boundary is
replaced by a steep smooth billiard-like potential) have clarified how the
approximation of a smooth billiard with a hard-wall billiard may be utilized
rigorously. These results are extended here to models with smooth background
potential satisfying some natural conditions. This generalization is then
applied to geometric models of collinear triatomic chemical reactions (the
models are far from integrable -degree of freedom systems with ).
The application demonstrates that the simpler analytical calculations for the
hard-wall system may be used to obtain qualitative information with regard to
the solution structure of the smooth system and to quantitatively assist in
finding solutions of the soft impact system by continuation methods. In
particular, stable periodic triatomic configurations are easily located for the
smooth highly-nonlinear two and three degree of freedom geometric models.Comment: 33 pages, 8 figure
Frequency spanning homoclinic families
A family of maps or flows depending on a parameter which varies in an
interval, spans a certain property if along the interval this property depends
continuously on the parameter and achieves some asymptotic values along it. We
consider families of periodically forced Hamiltonian systems for which the
appropriately scaled frequency is spanned, namely it covers
the semi-infinite line Under some natural assumptions on the
family of flows and its adiabatic limit, we construct a convenient labelling
scheme for the primary homoclinic orbits which may undergo a countable number
of bifurcations along this interval. Using this scheme we prove that a properly
defined flux function is in Combining this proof with previous
results of RK and Poje, immediately establishes that the flux function and the
size of the chaotic zone depend on the frequency in a non-monotone fashion for
a large class of Hamiltonian flows
Oscillating mushrooms: adiabatic theory for a non-ergodic system
Can elliptic islands contribute to sustained energy growth as parameters of a
Hamiltonian system slowly vary with time? In this paper we show that a mushroom
billiard with a periodically oscillating boundary accelerates the particle
inside it exponentially fast. We provide an estimate for the rate of
acceleration. Our numerical experiments confirms the theory. We suggest that a
similar mechanism applies to general systems with mixed phase space.Comment: final revisio
An analytical study of transport, mixing and chaos in an unsteady vortical flow
We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field induced by a vortex pair plus an oscillating strainrate field. In the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically; fluid is entrained and detrained from the neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikov's technique to investigate the behaviour of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate
Leaky Fermi accelerators
A Fermi accelerator is a billiard with oscillating walls. A leaky accelerator
interacts with an environment of an ideal gas at equilibrium by exchange of
particles through a small hole on its boundary. Such interaction may heat the
gas: we estimate the net energy flow through the hole under the assumption that
the particles inside the billiard do not collide with each other and remain in
the accelerator for sufficiently long time. The heat production is found to
depend strongly on the type of the Fermi accelerator. An ergodic accelerator,
i.e. one which has a single ergodic component, produces a weaker energy flow
than a multi-component accelerator. Specifically, in the ergodic case the
energy gain is independent of the hole size, whereas in the multi-component
case the energy flow may be significantly increased by shrinking the hole size.Comment: 7 pages, 5 figure
Stable motions of high energy particles interacting via a repelling potential
The motion of N particles interacting by a smooth repelling potential and confined to a compact d-dimensional region is proved to be, under mild conditions, non-ergodic for all sufficiently large energies. Specifically, choreographic solutions, for which all particles follow approximately the same path close to an elliptic periodic orbit of the single-particle system, are proved to be KAM stable in the high energy limit. Finally, it is proved that the motion of N repelling particles in a rectangular box is non-ergodic at high energies for a generic choice of interacting potential: there exists a KAM-stable periodic motion by which the particles move fast only in one direction, each on its own path, yet in synchrony with all the other parallel moving particles. Thus, we prove that for smooth interaction potentials the Boltzmann ergodic hypothesis fails for a finite number of particles even in the high energy limit at which the smooth system appears to be very close to the Boltzmann hard-sphere gas
Soft billiards with corners
We develop a framework for dealing with smooth approximations to billiards
with corners in the two-dimensional setting. Let a polygonal trajectory in a
billiard start and end up at the same billiard's corner point. We prove that
smooth Hamiltonian flows which limit to this billiard have a nearby periodic
orbit if and only if the polygon angles at the corner are ''acceptable''. The
criterion for a corner polygon to be acceptable depends on the smooth potential
behavior at the corners, which is expressed in terms of a {scattering
function}. We define such an asymptotic scattering function and prove the
existence of it, explain how it can be calculated and predict some of its
properties. In particular, we show that it is non-monotone for some potentials
in some phase space regions. We prove that when the smooth system has a
limiting periodic orbit it is hyperbolic provided the scattering function is
not extremal there. We then prove that if the scattering function is extremal,
the smooth system has elliptic periodic orbits limiting to the corner polygon,
and, furthermore, that the return map near these periodic orbits is conjugate
to a small perturbation of the Henon map and therefore has elliptic islands. We
find from the scaling that the island size is typically algebraic in the
smoothing parameter and exponentially small in the number of reflections of the
polygon orbit
Equilibration of energy in slow-fast systems
Do partial energies in slow–fast Hamiltonian systems equilibrate? This is a long-standing problem related to the foundation of statistical mechanics. Altering the traditional ergodic assumption, we propose that nonergodicity in the fast subsystem leads to equilibration of the whole system. To show this principle, we introduce a set of mechanical toy models—the springy billiards—and describe stochastic processes corresponding to their adiabatic behavior. We expect that these models and this principle will play an important role in the quest to establish and study the underlying postulates of statistical mechanics, one of the long-standing scientific grails
Symmetry breaking perturbations and strange attractors
The asymmetrically forced, damped Duffing oscillator is introduced as a
prototype model for analyzing the homoclinic tangle of symmetric dissipative
systems with \textit{symmetry breaking} disturbances. Even a slight fixed
asymmetry in the perturbation may cause a substantial change in the asymptotic
behavior of the system, e.g. transitions from two sided to one sided strange
attractors as the other parameters are varied. Moreover, slight asymmetries may
cause substantial asymmetries in the relative size of the basins of attraction
of the unforced nearly symmetric attracting regions. These changes seems to be
associated with homoclinic bifurcations. Numerical evidence indicates that
\textit{strange attractors} appear near curves corresponding to specific
secondary homoclinic bifurcations. These curves are found using analytical
perturbational tools
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