Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems
Anatoly Neishtadt Carles Simó Dmitry Treschev Alexei Vasiliev
We consider a 2 d.o.f. natural Hamiltonian system with one degree of freedom corresponding to fast motion and the other one corresponding to slow motion. The Hamiltonian function is the sum of potential and kinetic energies, the kinetic energy being a weighted sum of squared momenta. The ratio of time derivatives of slow and fast variables is of order $\epsilon $«$ 1$. At frozen values of the slow variables there is a separatrix on the phase plane of the fast variables and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition we prove the existence of many, of order $1/\epsilon$, stable periodic trajectories in the domain of the separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order $\epsilon$. Thus, the total measure of the stability islands is estimated from below by a value independent of $\epsilon$. We find the location of stable periodic trajectories and an asymptotic formula for the number of these trajectories. As an example, we consider the problem of motion of a charged particle in the parabolic model of magnetic field in the Earth magnetotail.
keywords: slow-fast systems stability islands. chaotic seas separatrix crossings
Oscillator and thermostat
Dmitry Treschev
We study the problem of a potential interaction of a finite- dimensional Lagrangian system (an oscillator) with a linear infinite-dimensional one (a thermostat). In spite of the energy preservation and the Lagrangian (Hamiltonian) nature of the total system, under some natural assumptions the final dynamics of the finite-dimensional component turns out to be simple while the thermostat produces an effective dissipation.
keywords: Hamiltonian systems final dynamics. Lagrangian systems
Travelling waves in FPU lattices
Dmitry Treschev
Fermi-Pasta-Ulam lattice is a classical mechanical system of an infinite number of discrete particles on a line. Each particle is assumed to interact with the nearest left and right neighbors only. We construct travelling waves in the system assuming that the potential has a singularity at zero. The waves appear near the hard ball limit.
keywords: asynchronous exponential growth. Semigroups eventual and essential compactness growth bounds integrated semigroups
Stability islands in the vicinity of separatrices of near-integrable symplectic maps
Carles Simó Dmitry Treschev
We discuss the problem of existence of elliptic periodic trajectories inside lobes bounded by segments of stable and unstable separatrices of a hyperbolic fixed point. We show that such trajectories generically exist in symplectic maps arbitrary close to integrable ones. Elliptic periodic trajectories as a rule, generate stability islands. The area of such an island is of the same order as the lobe area, but the quotient of areas can be very small. Numerical examples are included.
keywords: lobes defined by invariant manifold area preserving maps separatrix maps. stability islands
A locally integrable multi-dimensional billiard system
Dmitry Treschev

We consider a multi-dimensional billiard system in an $(n+1)$-dimensional Euclidean space, the direct product of the "horizontal" hyperplane and the "vertical" line. The hypersurface that determines the system is assumed to be smooth and symmetric in all coordinate hyperplanes. Hence there exists a periodic orbit $γ$ of period 2 moving along the "vertical" coordinate axis. The question we ask is as follows. Is it possible to choose such a system to have the dynamics locally (near $γ$) conjugated to the dynamics of a linear map?

Since the problem is local, the billiard hypersurface can be determined as the graphs of the functions $± f$, where $f$ is even and defined in a neighborhood of the origin on the "horizontal" coordinate hyperplane. We prove that $f$ exists as a formal Taylor series in the non-resonant case and give numerical evidence for convergence of the series.

keywords: Billiard system integrable billiard system multi-dimensional billiard system

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