Symplectic Maps Research Articles

A new integrable symplectic map by the binary nonlinearization to the super AKNS system

Based on the constructed new Lie super-algebra from OSP(2,2), the super bi-Hamiltonian structure of a new super AKNS hierarchy is obtained by making use of super-trace identity. For the new super AKNS system, an explicit symmetry constraint between the potentials and the eigenfunctions is proposed. Moreover, the super AKNS system is decomposed into two compatible finite-dimensional super integrable systems and the obtained super systems are proved to be finite-dimensional super integrable Hamiltonian systems in the super-symmetry manifold R 4 N ∣ 4 N .

On the contact mapping class group of the contactization of the $$A_m$$ A m -Milnor fiber

We construct an embedding of the full braid group on $$m+1$$ strands $$B_{m+1}$$ , $$m \ge 1$$ , into the contact mapping class group of the contactization $$Q \times S^1$$ of the $$A_m$$ -Milnor fiber Q. The construction uses the embedding of $$B_{m+1}$$ into the symplectic mapping class group of Q due to Khovanov and Seidel, and a natural lifting homomorphism. In order to show that the composed homomorphism is still injective, we use a partially linearized variant of the Chekanov–Eliashberg dga for Legendrians which lie above one another in $$Q \times {\mathbb {R}}$$ , reducing the proof to Floer homology. As corollaries we obtain a contribution to the contact isotopy problem for $$Q \times S^1$$ , as well as the fact that in dimension 4, the lifting homomorphism embeds the symplectic mapping class group of Q into the contact mapping class group of $$Q \times S^1$$ .

Domains of analyticity and Lindstedt expansions of KAM tori in some dissipative perturbations of Hamiltonian systems

Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. The limit of small dissipation, which is the object of the present study, is particularly interesting.We consider a family of conformally symplectic maps (very simple modifications, which we present, give results for differential equations) defined on a 2d-dimensional symplectic manifold with exact symplectic form ; we assume that satisfies . The d-dimensional parameter μ is called drift. We assume that , where , .We study the perturbative expansions and the domains of analyticity in near of the parameterization of the quasi-periodic orbits of frequency ω (assumed to be Diophantine) and of the parameter μ. Notice that this is a singular perturbation, since any friction (no matter how small) reduces the set of quasi-periodic solutions in the system. We prove that the tori are analytic in a domain in the complex plane, obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The radii of the excluded balls decrease faster than any power of the distance of the center to the origin. We state also a conjecture on the optimality of our results. The boundary of the domain is very thin, so that we can perform unique analytic continuation of the invariant tori along closed circles enclosing the points in the complement of the analyticity. We show that there is no monodromy of these continuations, either for the tori, for the invariant manifolds or for the drift.The proof is based on the following procedure. To find a quasi-periodic solution, one solves an invariance equation for the embedding of the torus, depending on the parameters of the family. Assuming that the frequency of the torus satisfies a Diophantine condition, under mild non-degeneracy assumptions, using a Lindstedt procedure we construct an approximate solution to all orders of the invariance equation describing the KAM torus. Starting from such approximate solution, we use an a posteriori KAM theorem to get the true solution of the invariance equation. This allows also the study of monogenic and Whitney differentiability properties of the extensions as well as the monodromy.

Nonlinearizations of Darboux transformations of Kaup-Newell equation based on therestricted Kaup-Newell flow

Nonlinearizations of Darboux transformations of Kaup-Newell equation are investigated.Based on the assumption that the maps obtained by nonlinearizing of Darboux transformations of Kaup-Newell equationare just the Backlund transformations of the restricted Kaup-Newell flows, constraints betweenthe potentials in the Darboux matrices and the eigenfunctions can be simply established.Nonlinearizations of four fundamental Darboux transformations are carried out andfour integrable symplectic maps which share the same set of invariants are obtained.

Arnold Diffusion in A Priori Chaotic Symplectic Maps

We assume that a symplectic real-analytic map has an invariant normally hyperbolic cylinder and an associated transverse homoclinic cylinder. We prove that generically in the real-analytic category the boundaries of the invariant cylinder are connected by trajectories of the map.

The degenerate special Lagrangian equation

This article introduces the degenerate special Lagrangian equation (DSL) and develops the basic analytic tools to construct and study its solutions. The DSL governs geodesics in the space of positive graph Lagrangians in Cn. Existence of geodesics in the space of positive Lagrangians is an important step in a program for proving existence and uniqueness of special Lagrangians. Moreover, it would imply certain cases of the strong Arnold conjecture from Hamiltonian dynamics.We show the DSL is degenerate elliptic. We introduce a space–time Lagrangian angle for one-parameter families of graph Lagrangians, and construct its regularized lift. The superlevel sets of the regularized lift define subequations for the DSL in the sense of Harvey–Lawson. We extend the existence theory of Harvey–Lawson for subequations to the setting of domains with corners, and thus obtain solutions to the Dirichlet problem for the DSL in all branches. Moreover, we introduce the calibration measure, which plays a rôle similar to that of the Monge–Ampère measure in convex and complex geometry. The existence of this measure and regularity estimates allow us to prove that the solutions we obtain in the outer branches of the DSL have a well-defined length in the space of positive Lagrangians.

A study of symplectic integrators for planetary system problems: error analysis and comparisons

The symplectic Wisdom-Holman map revolutionized long-term integrations of planetary systems. There is freedom in such methods of how to split the Hamiltonian and which coordinate system to employ, and several options have been proposed in the literature. These choices lead to different integration errors, which we study analytically and numerically. The Wisdom-Holman method in Jacobi coordinates and the method of Hernandez, H16, compare favorably and avoid problems of some of the other maps, such as incorrect center-of-mass position or truncation errors even in the one-planet case. We use H16 to compute the evolution of Pluto's orbital elements over 500 million years in a new calculation.

Gevrey-smoothness of lower dimensional hyperbolic invariant tori for nearly integrable symplectic mappings

This paper provides a normal form for a class of lower dimensional hyperbolic invariant tori of nearly integrable symplectic mappings with generating functions. We prove the persistence and the Gevrey-smoothness of the invariant tori under some conditions.

A construction of a large family of commuting pairs of integrable symplectic birational four-dimensional maps.

We give a construction of completely integrable four-dimensional Hamiltonian systems with cubic Hamilton functions. Applying to the corresponding pairs of commuting quadratic Hamiltonian vector fields the so called Kahan-Hirota-Kimura discretization scheme, we arrive at pairs of birational four-dimensional maps. We show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on [Formula: see text], and possess two independent integrals of motion, which are perturbations of the original Hamilton functions and which are in involution with respect to the perturbed symplectic structure. Thus, these maps are completely integrable in the Liouville-Arnold sense. Moreover, under a suitable normalization of the original pairs of vector fields, the pairs of maps commute and share the invariant symplectic structure and the two integrals of motion.

V.I. Arnold's conjecture on avoidance of intersection, Hausdorff codimension, and a structure theorem for finite index maps

The functional analytic approach taken in this work gives a geometric description of the image of a smooth finite index map between Hilbert spaces, and solves avoidance of intersection problems. An application of this work extends from finite to infinite dimensions the results of Von Neumann and Wigner in finite dimensions on the avoidance of intersection problems in physics.The present work solves V.I. Arnold's conjecture on avoidance of intersection. The solution depends on determining the size of the image of a finite index map between infinite dimensional separable Hilbert spaces. We define the Hausdorff codimension of subsets of infinite dimensional linear space. Hausdorff codimension measures the metric size of subsets of an infinite dimensional linear space, and includes previous descriptions by B. White and others on the size of sets. We apply Hausdorff codimension to finite index maps and establish a structure theorem for these maps that includes Smale's extension of Sard's Theorem. An application of the structure theorem to the avoidance of intersection problem determines the metric size of a set of functions whose images intersect a given countable union of smooth submanifolds each with codimension at least two.

A Family of Hierarchical Symplectic Maps for N-body Simulations with Post-Newtonian Corrections

Symplectic maps are well known for preserving the phase-space volume in Hamiltonian dynamics and are particularly suited for problems that require long integration times, such as the [Formula: see text]-body problem. However, when combined with a varying time-step scheme, they end up losing its symplecticity and become numerically inefficient. We address this problem by using a recursive Hamiltonian splitting based on the time-symmetric value of the individual time-steps required by the particles in the system. We present a family of 48 quasi-symplectic maps with different orders of convergence (2nd-, 4th- & 6th-order) and three time-stepping schemes: i) 16 using constant time-steps, ii) 16 using shared adaptive time-steps, and iii) 16 using hierarchical (individual) time-steps. All maps include post-Newtonian corrections up to order 3.5PN. We describe the method and present some details of the implementation.

Bifurcations of Cubic Homoclinic Tangencies in Two-dimensional Symplectic Maps

We study bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps. We distinguish two types of cubic homoclinic tangencies, and each type gives different first return maps derived to diverse conservative cubic Henon maps with quite different bifurcation diagrams. In this way, we establish the structure of bifurcations of periodic orbits in two parameter general unfoldings generalizing to the conservative case the results previously obtained for the dissipative case. We also consider the problem of 1:4 resonance for the conservative cubic Henon maps.

Floer homology of automorphisms of Liouville domains

We introduce a combination of fixed point Floer homology and symplectic homology for Liouville domains. As an application, we detect non-trivial elements in the symplectic mapping class group of a Liouville domain.

A KAM Theorem for Lower Dimensional Elliptic Invariant Tori of Nearly Integrable Symplectic Mappings

This paper develops a new KAM theorem for a class of lower dimensional elliptic invariant tori of nearly integrable symplectic mappings with generating functions but without assuming any nondegenerate condition.

Relaxation Oscillations and Chaos in a Duffing Type Equation: A Case Study

Results of numerical simulations of a Duffing type Hamiltonian system with a slow periodically varying parameter are presented. Using theory of adiabatic invariants, reversibility of the system and theory of symplectic maps, along with thorough numerical experiments, we present many details of the orbit behavior for the system. In particular, we found many symmetric mixed mode periodic orbits, both being hyperbolic and elliptic, the regions with a perpetual adiabatic invariant and chaotic regions. For the latter region we present details of chaotic behavior: calculation of homoclinic tangles and Lyapunov exponents.

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 Froeschle map.

Mukherjee–Choudhury–Chowdhury spectral problem and the semi-discrete integrable system

Starting from the Mukherjee–Choudhury–Chowdhury spectral problem, we derive a semi-discrete integrable system by a proper time spectral problem. A Bäcklund transformation of Darboux type of this system is established with the help of gauge transformation of the Lax pairs. By means of the obtained Bäcklund transformation, an exact solution is given. Moreover, Hamiltonian form of this system is constructed. Further, through a constraint of potentials and eigenfunctions, the Lax pair and the adjoint Lax pair of the obtained semi-discrete integrable system are nonlinearized as an integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system in the Liouville sense. Finally, the involutive representation of solution of the obtained semi-discrete integrable system is presented.

On the cohomology of almost-complex and symplectic manifolds and proper surjective maps

Let [Formula: see text] be an almost-complex manifold. In [Comparing tamed and compatible symplectic cones and cohomological properties of almost-complex manifolds, Comm. Anal. Geom. 17 (2009) 651–683], Li and Zhang introduce [Formula: see text] as the cohomology subgroups of the [Formula: see text]th de Rham cohomology group formed by classes represented by real pure-type forms. Given a proper, surjective, pseudo-holomorphic map between two almost-complex manifolds, we study the relationship among such cohomology groups. Similar results are proven in the symplectic setting for the cohomology groups introduced in [Cohomology and Hodge Theory on Symplectic manifolds: I, J. Differ. Geom. 91(3) (2012) 383–416] by Tseng and Yau and a new characterization of the hard Lefschetz condition in dimension [Formula: see text] is provided.

Symplectic mapping class group relations generalizing the chain relation

In this paper, we examine mapping class group relations of some symplectic manifolds. For each [Formula: see text] and [Formula: see text], we show that the [Formula: see text]-dimensional Weinstein domain [Formula: see text], determined by the degree [Formula: see text] homogeneous polynomial [Formula: see text], has a Boothby–Wang type boundary and a right-handed fibered Dehn twist along the boundary that is symplectically isotopic to a product of right-handed Dehn twists along Lagrangian spheres. We also present explicit descriptions of the symplectomorphisms in the case [Formula: see text] recovering the classical chain relation for the torus with two boundary components.

What is the mechanism of power-law distributed Poincaré recurrences in higher-dimensional systems?

The statistics of Poincaré recurrence times in Hamiltonian systems typically shows a power-law decay with chaotic trajectories sticking to some phase-space regions for long times. For higher-dimensional systems the mechanism of this power-law trapping is still unknown. We investigate trapped orbits of a generic 4D symplectic map in phase space and frequency space and find that, in contrast to 2D maps, the trapping is i) not due to a hierarchy in phase space. Instead, it occurs at the surface of the regular region, ii) outside of the Arnold web. The chaotic dynamics in this sticky region is iii) dominated by resonance channels which reach far into the chaotic region: We observe iii.a) clear signatures of some kind of partial transport barriers and conjecture iii.b) a stochastic process with an effective drift along resonance channels. These two processes lay the basis for a future understanding of the mechanism of power-law trapping in higher-dimensional systems.