Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows

We show that certain mechanical systems, including a geodesic flow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy.The assumptions we make in the case of geodesic flows are:

(a)

The metric and the external perturbation are smooth enough.

(b)

The geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection.

(c)

The frequency of the external perturbation is Diophantine.

(d)

The external potential satisfies a generic condition depending on the periodic orbit considered in (b).

The assumptions on the metric are C² open and are known to be dense on many manifolds. The assumptions on the potential fail only in infinite codimension spaces of potentials.The proof is based on geometric considerations of invariant manifolds and their intersections. The main tools include the scattering map of normally hyperbolic invariant manifolds, as well as standard perturbation theories (averaging, KAM and Melnikov techniques).We do not need to assume that the metric is Riemannian and we obtain results for Finsler or Lorentz metrics. Indeed, there is a formulation for Hamiltonian systems satisfying scaling hypotheses. We do not need to make assumptions on the global topology of the manifold nor on its dimension.     

Stability Functions

In this article we discuss the role of stability functions in geometric invariant theory and apply stability function techniques to various types of asymptotic problems in the Kähler geometry of GIT quotients. We discuss several particular classes of examples, namely, toric varieties, spherical varieties and the symplectic version of quiver varieties.     

Preservation and destruction of periodic orbits by symplectic integrators

We investigate what happens to periodic orbits and lower-dimensional tori of Hamiltonian systems under discretisation by a symplectic one-step method where the system may have more than one degree of freedom. We use an embedding of a symplectic map in a quasi-periodic non-autonomous flow and a KAM result of Jorba and Villaneuva (J Nonlinear Sci 7:427–473, 1997) to show that periodic orbits persist in the new flow, but with slightly perturbed period and an additional degree of freedom when the map is non-resonant with the periodic orbit. The same result holds for lower-dimensional tori with more degrees of freedom. Numerical experiments with the two degree of freedom Hénon–Heiles system are used to show that in the case where the method is resonant with the periodic orbit, the orbit is destroyed and replaced by two invariant sets of periodic points—analogous to what is understood for one degree of freedom systems.     

Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach

In this paper, we present and illustrate a general methodology to apply KAM theory in particular problems, based on an a posteriori approach. We focus on the existence of real analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: Given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities, we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer-assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to \(\varepsilon =0.9716\)), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map.     

Localized Stable Manifolds for Whiskered Tori in Coupled Map Lattices with Decaying Interaction

In this paper we consider lattice systems coupled by local interactions. We prove invariant manifold theorems for whiskered tori (we recall that whiskered tori are quasi-periodic solutions with exponentially contracting and expanding directions in the linearized system). The invariant manifolds we construct generalize the usual (strong) (un)stable manifolds and allow us to consider also non-resonant manifolds. We show that if the whiskered tori are localized near a collection of specific sites, then so are the invariant manifolds. We recall that the existence of localized whiskered tori has recently been proven for symplectic maps and flows in Fontich et al. (J Diff Equ, 2012), but our results do not need that the systems are symplectic. For simplicity we will present first the main results for maps, but we will show that the result for maps imply the results for flows. It is also true that the results for flows can be proved directly following the same ideas.     

Induced Hamiltonian maps on the symplectic quotient

Let (M,ω) be a symplectic manifold and G a compact Lie group that acts on M. Assume that the action of G on M is Hamiltonian. Then a G-equivariant Hamiltonian map on M induces a map on the symplectic quotient of M by G. Consider an autonomous Hamiltonian H with compact support on M, with no non-constant closed trajectory in time less than 1 and time-1 map fH. If the map fH descends to the symplectic quotient to a map Φ(fH) and the symplectic manifold M is exact and Ham(M,ω) has no short loops, we prove that the Hofer norm of the induced map Φ(fH) is bounded above by the Hofer norm of fH.     

Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions

We present theorems which provide the existence of invariant whiskered tori in finite-dimensional exact symplectic maps and flows. The method is based on the study of a functional equation expressing that there is an invariant torus.We show that, given an approximate solution of the invariance equation which satisfies some non-degeneracy conditions, there is a true solution nearby. We call this an a posteriori approach.The proof of the main theorems is based on an iterative method to solve the functional equation.The theorems do not assume that the system is close to integrable nor that it is written in action-angle variables (hence we can deal in a unified way with primary and secondary tori). It also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant linear map.The a posteriori formulation allows us to justify approximate solutions produced by many non-rigorous methods (e.g. formal series expansions, numerical methods). The iterative method is not based on transformation theory, but rather on successive corrections. This makes it possible to adapt the method almost verbatim to several infinite-dimensional situations, which we will discuss in a forthcoming paper. We also note that the method leads to fast and efficient algorithms. We plan to develop these improvements in forthcoming papers.     

Symplectic method for the construction of ergodic measures on invariant submanifolds of nonautonomous hamiltonian systems: Lagrangian manifolds, their structure, and mather homologies

We develop a new approach to the study of properties of ergodic measures for nonautonomous periodic Hamiltonian flows on symplectic manifolds, which are used in many problems of mechanics and mathematical physics. Using Mather’s results on homologies of invariant probability measures that minimize some Lagrangian functionals and the symplectic theory developed by Floer and others for the investigation of symplectic actions and transversal intersections of Lagrangian manifolds, we propose an analog of a Mather-type β-function for the study of ergodic measures associated with nonautonomous Hamiltonian systems on weakly exact symplectic manifolds. Within the framework of the Gromov-Salamon-Zehnder elliptic methods in symplectic geometry, we establish some results on stable and unstable manifolds for hyperbolic invariant sets, which are used in the theory of adiabatic invariants of slowly perturbed integrable Hamiltonian systems. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 675–691, May, 2006.     

Complex dynamics in physical pendulum equation with suspension axis vibrations

The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov＇s method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov＇s method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Ω = nω ＋ εv, n = 1 - 4, where ν is not rational to ω. We are not able to prove the existence of chaos for n = 5 - 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters α, δ, f0 and Ω; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from period- one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven＇t find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.     

Partially hyperbolic geodesic flows are Anosov1Les flots géodésiques partiellement hyperboliques sont Anosov

We prove that if a $\text{Z}$ or $\text{R}$-action by symplectic linear maps on a symplectic vector bundle E has a weakly dominated invariant splitting E=S⊕U with dimU=dimS, then the action is hyperbolic. In particular, contact and geodesic flows with a dominated splitting with dimS=dimU are Anosov. To cite this article: G. Contreras, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 585–590.     

Asymptotic arc-components of unimodal inverse limit spaces

We consider the inverse limit space (I,f) of a unimodal bonding map f as fixed bonding map. If f has a periodic turning point, then (I,f) has a finite non-empty set of asymptotic arc-components. We show how asymptotic arc-components can be determined from the kneading sequence of f. This gives an alternative to the substitution tiling space approach taken by Barge and Diamond [Ergodic Theory Dynamical Systems 21 (2001) 1333].     

On the n-body problem on surfaces of revolution

We explore the n-body problem, $n\ge 3$, on a surface of revolution with a general interaction depending on the pairwise geodesic distance. Using the geometric methods of classical mechanics we determine a large set of properties. In particular, we show that Saari's conjecture fails on surfaces of revolution admitting a geodesic circle. We define homographic motions and, using the discrete symmetries, prove that when the masses are equal, they form an invariant manifold. On this manifold the dynamics are reducible to a one-degree of freedom system. We also find that for attractive interactions, regular n-gon shaped relative equilibria with trajectories located on geodesic circles typically experience a pitchfork bifurcation. Some applications are included.     

Base manifolds for fibrations of projective irreducible symplectic manifolds

Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M→X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.     

Quantization of complex Lagrangian submanifolds

Let Λ be a smooth Lagrangian submanifold of a complex symplectic manifold X. We construct twisted simple holonomic modules along Λ in the stack of deformation-quantization modules on X.     

Rigidity of harmonic maps of maximum rank

Thehomotopical rank of a mapf:M →N is, by definition, min{dimg(M) ¦g homotopic tof}. We give upper bounds for this invariant whenM is compact Kähler andN is a compact discrete quotient of a classical symmetric space, e.g., the space of positive definite matrices. In many cases the upper bound is sharp and is attained by geodesic immersions of locally hermitian symmetric spaces. An example is constructed (Section 9) to show that there do, in addition, exist harmonic maps of quite a different character. A byproduct is construction of an algebraic surface with large and interesting fundamental group. Finally, a criterion for lifting harmonic maps to holomorphic ones is given, as is a factorization theorem for representations of the fundamental group of a compact Kähler manifold. The technique for the main result is a combination of harmonic map theory, algebra, and combinatorics; it follows the path pioneered by Siu in his ridigity theorem and later extended by Sampson.     

On the geometry and quantization of symplectic Howe pairs

We study the orbit structure and the geometric quantization of a pair of mutually commuting hamiltonian actions on a symplectic manifold. If the pair of actions fulfils a symplectic Howe condition, we show that there is a canonical correspondence between the orbit spaces of the respective moment images. Furthermore, we show that reduced spaces with respect to the action of one group are symplectomorphic to coadjoint orbits of the other group. In the Kähler case we show that the linear representation of a pair of compact connected Lie groups on the geometric quantization of the manifold is then equipped with a representation-theoretic Howe duality.     

The geometric Hopf invariant and double points

The geometric Hopf invariant of a stable map F is a stable _boxclose/2{{\mathbb Z}/2} -equivariant map h(F) such that the stable \mathbb Z/2{{\mathbb Z}/2} -equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper, we express the geometric Hopf invariant of the Umkehr map F of an immersion f : M^m \looparrowright Nⁿ{f : M^m \looparrowright N^n} in terms of the double point set of f. We interpret the Smale–Hirsch–Haefliger regular homotopy classification of immersions f in the metastable dimension range 3m <  2n – 1 (when a generic f has no triple points) in terms of the geometric Hopf invariant.     

Applications of the odd symplectic group in Hamiltonian systems

In this paper we give two applications of the odd symplectic group to the study of the linear Poincaré maps of a periodic orbits of a Hamiltonian vector field, which cannot be obtained using the standard symplectic theory. First we look at the geodesic flow. We show that the period of the geodesic is a noneigenvalue modulus of the conjugacy class in the odd symplectic group of the linear Poincaré map. Second, we study an example of a family of periodic orbits, which forms a folded Robinson cylinder. The stability of this family uses the fact that the unipotent odd symplectic Poincaré map at the fold has a noneigenvalue modulus.     

Semisimple Symplectic Symmetric Spaces

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure which is invariant under the geodesic symmetries. When the transvection group G⁰ of such a symmetric space M is semisimple, its action on (M,) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G⁰-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra of G⁰. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.