Existence Of Invariant Curves Research Articles

Existence of invariant curves with prescribed frequency for degenerate area preserving mappings

Existence of invariant curves with prescribed frequency for degenerate area preserving mappings

Existence of invariant curves for area-preserving mappings under weaker non-degeneracy conditions

We consider a class of analytic area-preserving mappings Cm-smoothly depending on a parameter. Without imposing on any non-degeneracy assumption, we prove a formal KAM theorem for the mappings, which implies many previous KAM-type results under some non-degeneracy conditions. Moreover, by this formal KAM theorem, we can also obtain some new interesting results under some weaker non-degeneracy conditions. Thus, the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.

Folded chaotic whispering-gallery modes in nonconvex, waveguide-coupled planar optical microresonators

Chaotic whispering-gallery modes have significance both for optical applications and for our understanding of the interplay between wave phenomena and the classical ray limit in the presence of chaotic dynamics and openness. In strongly non-convex geometries, a theorem by Mather rules out the existence of invariant curves in phase space corresponding to rays circulating in whispering-gallery patterns, so that no corresponding modes of this type are expected. Here we discuss numerical computations of the electromagnetic fields in planar dielectric cavities that are strongly non-convex becausethey are coupled to waveguides. We find a family of special states which retains many features of the chaotic whispering-gallery modes known from convex shapes: an intensity pattern corresponding to near-grazing incidence along extended parts of the boundary, and comparatively high cavity Q factors. The modes are folded into a figure-eight pattern, so overlap with the boundary is reduced in the region of self-intersection. The modes combine the phenomenology of chaotic whispering-gallery modes with an important technological advantage: the ability to directly attach waveguides without spoiling the Q factor of the folded mode. Using both a boundary-integral method and the FDTD technique, we explore the dependence of the phenomenon on wavelength in relation to cavity size, refractive-index contrast to the surrounding medium, and the degree of shape deformation. A novel feature that distinguishes folded from regular whispering gallery modes is that a given shape will support high-Q folded chaotic whispering gallery modes only in certain wavelength windows.

Invariant curves for planar competitive and cooperative maps

In this paper we present results on the existence of invariant curves for planar maps that are monotone with respect to either the south-east or north-east ordering. Some of these curves are the stable or unstable manifolds of hyperbolic fixed points (saddle points) or non-hyperbolic fixed points, and are also the boundary of basins of attraction of such points.

Invariant curves for a second-order difference equation modelled from macroeconomics

In this paper we investigate invariant curves of a planar mapping, which equivalently formulates a second-order difference equation modelled from macroeconomics. We construct all invariant curves in the linear case and prove the existence of invariant curves in the nonlinear case. Furthermore, we discuss the continuous dependence of invariant curves in the nonlinear case.

A Unification of Models of Tethered Satellites

In this paper, different conservative models of tethered satellites are related mathematically, and it is established in what limit they may provide useful insight into the underlying dynamics. An infinite dimensional model is linked to a finite dimensional model, the slack-spring model, through a conjecture on the singular perturbation of tether thickness. The slack-spring model is then naturally related to a billiard model in the limit of an inextensible spring. Next, the motion of a dumbbell model, which is lowest in the hierarchy of models, is identified within the motion of the billiard model through a theorem on the existence of invariant curves by exploiting Moser's twist map theorem. Finally, numerical computations provide insight into the dynamics of the billiard model.

The Lagrangian Stability for a Class of Second-Order Quasi-Periodic Reversible Systems

We study the following two-order differential equation,(Φp(x'))'+f(x,t)Φp(x')+g(x,t)=0,whereΦp(s)=|s|(p-2)s,p>0.f(x,t)andg(x,t)are real analytic functions inxandt,2aπp-periodic inx, and quasi-periodic intwith frequencies(ω1,…,ωm). Under some odd-even property off(x,t)andg(x,t), we obtain the existence of invariant curves for the above equations by a variant of small twist theorem. Then all solutions for the above equations are bounded in the sense ofsupt∈R|x′(t)|<+∞.

Global invariant manifolds

Abstract The main result of our article is an analog of the local manifold theorem for a hyperbolic point. We give a set of sufficient conditions that ensure the existence of the global invariant manifold. We will prove the existence of invariant curves which lie in the quaternionic Julia set of the map f c ( X ) = X 2 + c .

Normal Form Invariants Around Spin-Orbit Periodic Orbits

We consider a model of spin-orbit interaction, describing the motion of an oblate satellite rotating about an internal spin-axis and orbiting about a central planet. The resulting second order differential equation depends upon the parameters provided by the equatorial oblateness of the satellite and its orbital eccentricity. Normal form transformations around the main spin-orbit resonances are carried out explicitly. As an outcome, one can compute some invariants; the fact that these quantities are not identically zero is a necessary condition to prove the existence of nearby periodic orbits (Birkhoff fixed point theorem). Moreover, the nonvanishing of the invariants provides also the stability of the spin-orbit resonances, since it guarantees the existence of invariant curves surrounding the periodic orbit.

Chaotic, regular and unbounded behaviour in the elastic impact oscillator

We study the elastic impact oscillator which is a periodically-forced mass on a spring impacting against a rigid wall. This gives rise to an area-preserving mapping that typically displays chaos at low energies and regular behaviour at high energies. The boundary between the two types of behaviour is complicated by the existence of discontinuities. The high energy behaviour is strongly influenced by resonances and the existence of invariant curves that bound the energy gain of the mass has an interesting dependence upon the system parameters.

EXISTENCE OF INVARIANT CURVES FOR MAPS CLOSE TO DEGENERATE MAPS, AND A SOLUTION OF THE FERMI-ULAM PROBLEM

The Ulam model is studied in this paper: a small elastic ball moves vertically between two infinitely heavy horizontal walls, each of which moves in the vertical direction according to a periodic law. It is proved that the velocity of the ball is always bounded. The proof is based on a generalization of Moser's theorem on the existence of invariant curves under an area preserving mapping of an annulus. Bibliography: 16 titles.

General linear methods: connection to one step methods and invariant curves

We generalize a result of Kirchgraber (1986) on multistep methods. We show that every strictly stable general linear method is essentially conjugate to a one step method of the same order. This result may be used to show that general properties of one step methods carry over to general linear methods. As examples we treat the existence of invariant curves and the construction of attracting sets.

Invariant curves around a parabolic fixed point at infinity

AbstractThe stability of a fixed point of an area-preserving transformation in the plane is characterized by the invariant curves which surround it. The existence of invariant curves had been extensively studied for elliptic fixed points. Here we study the similar problem for parabolic fixed points. In particular we are interested in the case where the fixed point is at infinity.