Vassili Gelfreich

Fermi acceleration in non-autonomous billiards

Fermi acceleration can be modelled by a classical particle moving inside a time-dependent domain and elastically reflecting from its boundary. In this paper, we describe how the results from the dynamical system theory can be used to explain the existence of trajectories with unbounded energy. In particular, we show for slowly oscillating boundaries that the energy of the particle may increase exponentially fast in time.

Long-periodic orbits and invariant tori in a singularly perturbed Hamiltonian system

In this paper, we study a singularly perturbed, two-degree-of-freedom Hamiltonian system with a normally elliptic slow manifold. We prove that the slow manifold persists but can have a large number (∼e −1 ) of exponentially small (≤e −c/e ) gaps. We demonstrate the existence of KAM tori in a neighborhood of the slow manifold. In addition, we investigate a bifurcation which describes the creation of a gap in the slow manifold and derive its normal form.

Fermi acceleration and adiabatic invariants for non-autonomous billiards

Recent results concerned with the energy growth of particles inside a container with slowly moving walls are summarized, augmented, and discussed. For breathing bounded domains with smooth boundaries, it is proved that for all initial conditions the acceleration is at most exponential. Anosov-Kasuga averaging theory is reviewed in the application to the non-autonomous billiards, and the results are corroborated by numerical simulations. A stochastic description is proposed which implies that for periodically perturbed ergodic and mixing billiards averaged particle energy grows quadratically in time (e.g., exponential acceleration has zero probability). Then, a proof that in non-integrable breathing billiards some trajectories do accelerate exponentially is reviewed. Finally, a unified view on the recently constructed families of non-ergodic billiards that robustly admit a large set of exponentially accelerating particles is presented.

Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift-Hohenberg equation as an example

We study homoclinic orbits of the Swift-Hohenberg equation near a Hamiltonian-Hopf bifurcation. It is well known that in this case the normal form of the equation is integrable at all orders. Therefore the difference between the stable and unstable manifolds is exponentially small and the study requires a method capable of detecting phenomena beyond all algebraic orders provided by the normal form theory. We propose an asymptotic expansion for a homoclinic invariant which quantitatively describes the transversality of the invariant manifolds. We perform high-precision numerical experiments to support the validity of the asymptotic expansion and evaluate a Stokes constant numerically using two independent methods.

Drift of slow variables in slow-fast Hamiltonian systems

Abstract We study the drift of slow variables in a slow-fast Hamiltonian system with several fast and slow degrees of freedom. Keeping the slow variables frozen, for any periodic trajectory of the fast subsystem we define an action. For a family of periodic orbits, the action is a scalar function of the slow variables and can be considered as a Hamiltonian function which generates some slow dynamics. These dynamics depend on the family of periodic orbits. Assuming that for the frozen slow variables the fast system has a pair of hyperbolic periodic orbits connected by two transversal heteroclinic trajectories, we prove that for any path composed of a finite sequence of slow trajectories generated by action Hamiltonians, there is a trajectory of the full system whose slow component shadows the path.

Near strongly resonant periodic orbits in a Hamiltonian system

We study an analytic Hamiltonian system near a strongly resonant periodic orbit. We introduce a modulus of local analytic classification. We provide asymptotic formulae for the exponentially small splitting of separatrices for bifurcating hyperbolic periodic orbits. These formulae confirm a conjecture formulated by V. I. Arnold in the early 1970s.

Universal dynamics in a neighborhood of a generic elliptic periodic point

We show that a generic area-preserving two-dimensional map with an elliptic periodic point is Cω-universal, i.e., its renormalized iterates are dense in the set of all real-analytic symplectic maps of a two-dimensional disk. The results naturally extend onto Hamiltonian and volume-preserving flows.

Unique resonant normal forms for area-preserving maps at an elliptic fixed point

We construct a resonant normal form for an area-preserving map near a generic resonant elliptic fixed point. The normal form is obtained by a simplification of a formal interpolating Hamiltonian. The resonant normal form is unique and therefore provides the formal local classification for area-preserving maps with the elliptic fixed point. The total number of formal invariants is infinite. We consider the cases of weak (of order

Width of the Homoclinic Zone in the Parameter Space for Quadratic Maps

n\ge5

Analytic invariants associated with a parabolic fixed point in C2

) and strong (of order