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Register a new userOn curvature and hyperbolicity of monotone Hamiltonian systems
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Authors
Paul W.Y. Lee
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Assume that a Hamiltonian system is monotone. In this paper, we give several characterizations on when such a system is Anosov. Assuming that a monotone Hamiltonian system has no conjugate point, we show that there are two distributions which are invariant under the Hamiltonian flow. We show that a monotone Hamiltonian flow without conjugate point is Anosov if and only if these distributions are transversal. We also show that if the reduced curvature of the Hamiltonian system is non-positive, then the flow is Anosov if and only if the reduced curvature is negative somewhere along each trajectory.
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Two types of dynamics, chaotic and monotone, are compared. It is shown that monotone maps in strongly ordered spaces do not have chaotic attracting sets.
In this paper, we study the Poisson stability (in particular, stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, almost automorphy, recurrence in the sense of Birkhoff, Levitan almost periodicity, pseudo periodicity, almost recurrence in the sense of Bebutov, pseudo recurrence, Poisson stability) of motions for monotone nonautonomous dynamical systems and of solutions for some classes of monotone nonautonomous evolution equations (ODEs, FDEs and parabolic PDEs). As a byproduct, some of our results indicate that all the trajectories of monotone systems converge to the above mentioned Poisson stable trajectories under some suitable conditions, which is interesting in its own right for monotone dynamics.
The dynamics of a beam held on a horizontal frame by springs and bouncing off a step is described by a separable two degrees of freedom Hamiltonian system with impacts that respect, point wise, the separability symmetry. The energy in each degree of freedom is preserved, and the motion along each level set is conjugated, via action angle coordinates, to a geodesic flow on a flat two-dimensional surface in the four dimensional phase space. Yet, for a range of energies, these surfaces are not the simple Liouville-Arnold tori - these are tori of genus two, thus the motion on them is not conjugated to simple rotations. Namely, even though energy is not transferred between the two degrees of freedom, the impact system is quasi-integrable and is not of the Liouville-Arnold type. In fact, for each level set in this range, the motion is conjugated to the well studied and highly non-trivial dynamics of directional motion in L-shaped billiards, where the billiard area and shape as well as the direction of motion vary continuously on iso-energetic level sets. Return maps to Poincare section of the flow are shown to be conjugated, on each level set, to interval exchange maps which are computed, up to quadratures, in the general nonlinear case and explicitly for the case of two linear oscillators bouncing off a step. It is established that for any such oscillator-step system there exist step locations for which some of the level sets exhibit motion which is neither periodic nor ergodic. Changing the impact surface by introducing additional steps, staircases, strips and blocks from which the particle is reflected, leads to iso-energy surfaces that are foliated by families of genus-k level set surfaces, where the number and order of families of genus k depend on the energy.
We prove that every sectional-Anosov flow of a compact 3-manifold $M$ exhibits a finite collection of hyperbolic attractors and singularities whose basins form a dense subset of $M$. Applications to the dynamics of sectional-Anosov flows on compact 3-manifolds include a characterization of essential hyperbolicity, sensitivity to the initial conditions (improving cite{ams}) and a relationship between the topology of the ambient manifold and the denseness of the basin of the singularities.
Let X be a subset of R^n whose interior is connected and dense in X, ordered by a polyhedral cone in R^n with nonempty interior. Let T be a monotone homeomorphism of X whose periodic points are dense. Then T is periodic.
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