Symplectic Maps Research Articles

Approximation of nearly-periodic symplectic maps via structure-preserving neural networks

A continuous-time dynamical system with parameter varepsilon is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as varepsilon approaches 0. Nearly-periodic maps are discrete-time analogues of nearly-periodic systems, defined as parameter-dependent diffeomorphisms that limit to rotations along a circle action, and they admit formal U(1) symmetries to all orders when the limiting rotation is non-resonant. For Hamiltonian nearly-periodic maps on exact presymplectic manifolds, the formal U(1) symmetry gives rise to a discrete-time adiabatic invariant. In this paper, we construct a novel structure-preserving neural network to approximate nearly-periodic symplectic maps. This neural network architecture, which we call symplectic gyroceptron, ensures that the resulting surrogate map is nearly-periodic and symplectic, and that it gives rise to a discrete-time adiabatic invariant and a long-time stability. This new structure-preserving neural network provides a promising architecture for surrogate modeling of non-dissipative dynamical systems that automatically steps over short timescales without introducing spurious instabilities.

Study of turbulent transport in magnetized plasmas with flow using symplectic maps

Turbulent transport in a magnetized plasma of the kind found in tokamaks is modeled by a 2D wave spectrum that allows reduction to a symplectic map. The properties of particle transport when chaos sets in are analyzed in various circumstances including finite Larmor radius (FLR) effects and a background plasma flow. For large wave amplitudes, regular particle orbits become chaotic, which represents a type of Lagrangian turbulence. When chaos becomes global, it leads to the loss of particle confinement. Poloidal flows tend to decrease the chaos in some regions, and they can give rise to the formation of transport barriers. FLR effects not only reduce chaos but also give rise to non-local behavior. Thus, when the particles have a thermal distribution of Larmor radii, a non-Gaussian particle distribution function in space is obtained. However, the transport preserves its diffusive scaling when there is no flow. Previous results about the dependence of the diffusion coefficient with amplitude are re-derived analytically and numerically taking into account FLR effects. In the presence of general poloidal flows, the transport has to be described by a two-step map. They modify the nature of transport in the direction of the flow from diffusive to ballistic to super-ballistic, depending on the type of flow. The transverse transport, in turn, shows suppression of the oscillations with wave amplitude that are present in the absence of flow. When the plasma flow varies linearly with radius, the transport can be studied with a similar single-step map, and the transverse diffusion coefficient is reduced while parallel transport can become super-ballistic. For non-monotonic flows, there are accelerating modes that can produce ballistic-like particles while the bulk of the particles behaves diffusively.

On the integrable symplectic map and the $$N$$-soliton solution of the Toda lattice

On the integrable symplectic map and the $$N$$-soliton solution of the Toda lattice

On the integrable symplectic map and the $N$-soliton solution of the Toda lattice

Используются три типа полиномиальных разложений спектральных функций, чтобы ввести гамильтонову систему и симплектическое отображение для решетки Тоды. Обсуждаются интегрируемость симплектического отображения и координаты Дарбу. Симплектическое отображение линеаризуется с помощью координат Дарбу, что приводит к задаче обращения. Обращение используется для построения $N$-солитонного решения решетки Тоды.

Density of a thin film billiard reflection pseudogroup in a Hamiltonian symplectomorphism pseudogroup

Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary C∞-smooth hypersurface γ ⊂ ℝn+1 that is either a global strictly convex closed hypersurface, or a germ of a hypersurface. We deal with the pseudogroup generated by compositional ratios of reflections from γ and of reflections from its small deformations. In the case when γ is a global convex hypersurface, we show that the C∞-closure of the latter pseudogroup contains the pseudogroup of Hamiltonian diffeomorphisms between domains in the phase cylinder: the space of oriented lines intersecting γ transversally. We prove an analogous local result in the case when γ is a germ. The derivatives of the above compositional differences in the deformation parameter are Hamiltonian vector fields calculated by Ron Perline. To prove the main results, we find the Lie algebra generated by them and prove its C∞-density in the Lie algebra of Hamiltonian vector fields. We also prove analogues of the above results for hypersurfaces in Riemannian manifolds.

Global Fukaya category I

Abstract Let $Ham (M,\omega ) $ denote the Frechet Lie group of Hamiltonian symplectomorphisms of a monotone symplectic manifold $(M, \omega ) $. Let $NFuk (M, \omega )$ be the $A _{\infty } $-nerve of the Fukaya category $Fuk (M, \omega )$, and let $(|\mathbb {S}|, NFuk (M, \omega ))$ denote the $NFuk (M, \omega )$ component of the “space of $\infty $-categories” $|\mathbb {S}| $. Using Floer-Fukaya theory for a monotone $(M, \omega ),$ we construct a natural up to homotopy classifying map $$ \begin{align*}& BHam (M, \omega) \to (|\mathbb{S}|, NFuk (M, \omega)). \end{align*}$$This verifies one sense of a conjecture of Teleman on existence of action of $Ham (M, \omega )$ on the Fukaya category of $(M, \omega ) $. This construction is very closely related to the theory of the Seidel homomorphism and the quantum characteristic classes of the author, and this map is intended to be the deepest expression of their underlying geometric theory. In part II, the above map is shown to be nontrivial by an explicit calculation. In particular, we arrive at a new non-trivial “quantum” invariant of any smooth manifold, which motives the statement of a kind of “quantum” Novikov conjecture.

Locally maximising orbits for the non-standard generating function of convex billiards and applications

Given an exact symplectic map T of a cylinder with a generating function H satisfying the so-called negative twist condition, , we study the locally maximising orbits of T, that is, configurations which are local maxima of the action functional . We provide a necessary and sufficient condition for a configuration to be locally maximising. Using it, we consider a situation where T has two generating functions with respect to two different sets of symplectic coordinates. We suggest a simple geometric condition which guarantees that the set of locally maximising orbits with respect to both of these generating functions coincide. As the main application we show that the two generating functions for planar Birkhoff billiards satisfy this geometric condition. We apply it to get the following result: consider a centrally symmetric curve γ, for which the Birkhoff billiard map has a rotational invariant curve α of four-periodic orbits. We prove that a certain L 2-distance between γ and its ‘best approximating’ ellipse can be bounded from above in terms of the measure of the complement of the set filled by locally maximising orbits lying between α and the boundary of the phase cylinder. Moreover, this estimate is sharp, giving an effective version of a recent result on Birkhoff conjecture for centrally symmetric curves (Bialy and Mironov 2022 Ann. Math. 196 389–413). We also get a similar bound for arbitrary curves γ, that relates the measure of the complement of the set of locally maximising orbits with the L 2-distance between γ and its ‘best approximating’ circle.

Shearless bifurcations in particle transport for reversed-shear tokamaks

Some internal transport barriers in tokamaks have been related to the vicinity of extrema of the plasma equilibrium profiles. This effect is numerically investigated by considering the guiding-centre trajectories of plasma particles undergoing $\boldsymbol {E}\times \boldsymbol {B}$ drift motion, considering that the electric field has a stationary non-monotonic radial profile and an electrostatic fluctuation. In addition, the equilibrium configuration has a non-monotonic safety factor profile. The numerical integration of the equations of motion yields a symplectic map with shearless barriers. By changing the safety factor profile parameters, the appearance and breakup of these shearless curves are observed. The shearless curve's successive breakup and recovery are explained using concepts from bifurcation theory. We also present bifurcation sequences associated with the creation of multiple shearless curves. Physical consequences of scenarios with multiple shearless curves are discussed.

A short proof of cuplength estimates on Lagrangian intersections

<abstract><p>In this note we give a short proof of Arnold's conjecture for the zero section of a cotangent bundle of a closed manifold. The proof is based on some basic properties of Lagrangian spectral invariants from Floer theory.</p></abstract>

Mapas padrão twist e não-twist

Symplectic maps can provide a straightforward and accurate way to visualize and quantify the dynamics of conservative systems with two degrees of freedom. These maps can be easily iterated from the simplest computers to obtain trajectories with great accuracy. Their usage arises in many fields, including celeste mechanics, plasma physics, chemistry, and so on. In this paper we introduce two examples of symplectic maps, the standard map and the standard non-twist map, exploring the phase space transformation as their control parameters are varied. Furthermore, for the non-twist map, we also present, that its chaotic motion is separated in phase space by the non-twist transport barrier.

Partial barriers to chaotic transport in 4D symplectic maps.

Chaotic transport in Hamiltonian systems is often restricted due to the presence of partial barriers, leading to a limited flux between different regions in phase space. Typically, the most restrictive partial barrier in a 2D symplectic map is based on a cantorus, the Cantor set remnants of a broken 1D torus. For a 4D symplectic map, we establish a partial barrier based on what we call a cantorus-NHIM-a normally hyperbolic invariant manifold with the structure of a cantorus. Using a flux formula, we determine the global 4D flux across a partial barrier based on a cantorus-NHIM by approximating it with high-order periodic NHIMs. In addition, we introduce a local 3D flux depending on the position along a resonance channel, which is relevant in the presence of slow Arnold diffusion. Moreover, for a partial barrier composed of stable and unstable manifolds of a NHIM, we utilize periodic NHIMs to quantify the corresponding flux.

Nonlinear Orbital Stability of Periodic Motions in the Planar Restricted Four-Body Problem

In this work we study the nonlinear orbital stability problem for periodic motions emanating from the stable relative equilibrium. To describe motions of the small body in a neighborhood of its periodic orbit, we introduce the so-called local variables. Then we reduce the orbital stability problem to the stability problem of a stationary point of symplectic mapping generated by the system phase flow on the energy level corresponding to the unperturbed periodic motion. This allows rigorous conclusions to be drawn on orbital stability for both the nonresonant and the resonant cases. We apply this method to investigate orbital stability in the case of third- and fourth-order resonances as well as in the nonresonant case. The results of the study are presented in the form of a stability diagram.

Deformations of cluster mutations and invariant presymplectic forms

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A_2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A_3 and A_4, we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions in the discrete sine-Gordon equation.

Quantifying chaos using Lagrangian descriptors.

We present and validate simple and efficient methods to estimate the chaoticity of orbits in low-dimensional conservative dynamical systems, namely, autonomous Hamiltonian systems and area-preserving symplectic maps, from computations of Lagrangian descriptors (LDs) on short time scales. Two quantities are proposed for determining the chaotic or regular nature of orbits in a system's phase space, which are based on the values of the LDs of these orbits and of nearby ones: The difference and ratio of neighboring orbits' LDs. Using as generic test models the prototypical two degree of freedom Hénon-Heiles system and the two-dimensional standard map, we find that these indicators are able to correctly characterize the chaotic or regular nature of orbits to better than 90% agreement with results obtained by implementing the Smaller Alignment Index (SALI) method, which is a well-established chaos detection technique. Further investigating the performance of the two introduced quantities, we discuss the effects of the total integration time and of the spacing between the used neighboring orbits on the accuracy of the methods, finding that even typical short time, coarse-grid LD computations are sufficient to provide reliable quantification of the systems' chaotic component, using less CPU time than the SALI. In addition to quantifying chaos, the introduced indicators have the ability to reveal details about the systems' local and global chaotic phase space structure. Our findings clearly suggest that LDs can also be used to quantify and investigate chaos in continuous and discrete low-dimensional conservative dynamical systems.

Hamiltonian theory of the crossing of the 2Qx−2Qy=0 nonlinear coupling resonance

In a recent paper, the adiabatic theory of Hamiltonian systems was successfully applied to study the crossing of the linear coupling resonance, ${Q}_{x}\ensuremath{-}{Q}_{y}=0$. A detailed explanation of the well-known phenomena that occur during the resonance-crossing process, such as emittance exchange and its dependence on the adiabaticity of the process, was obtained. In this paper, we consider the crossing of the resonance of nonlinear coupling $2{Q}_{x}\ensuremath{-}2{Q}_{y}=0$ using the same theoretical framework. We perform the analysis using a Hamiltonian model in which the nonlinear coupling resonance is excited and the corresponding dynamics is studied in detail, in particular looking at the phase-space topology and its evolution, in view of characterizing the emittance exchange phenomena. The theoretical results are then tested using a symplectic map. Thanks to this approach, scaling laws of general interest for applications are derived.

Algebro-geometric solutions to the lattice potential modified Kadomtsev–Petviashvili equation

Algebro-geometric solutions of the lattice potential modified Kadomtsev–Petviashvili (lpmKP) equation are constructed. A Darboux transformation of the Kaup–Newell spectral problem is employed to generate a Lax triad for the lpmKP equation, as well as to define commutative integrable symplectic maps which generate discrete flows of eigenfunctions. These maps share the same integrals with the finite-dimensional Hamiltonian system associated to the Kaup–Newell spectral problem. We investigate asymptotic behaviors of the Baker–Akhiezer functions and obtain their expression in terms of Riemann theta function. Finally, algebro-geometric solutions for the lpmKP equation are reconstructed from these Baker–Akhiezer functions.

Hochschild Entropy and Categorical Entropy

We study the categorical entropy and counterexamples to Gromov–Yomdin type conjecture via homological mirror symmetry of K3 surfaces established by Sheridan–Smith. We introduce asymptotic invariants of quasi-endofunctors of dg categories, called the Hochschild entropy. It is proved that the categorical entropy is lower bounded by the Hochschild entropy. Furthermore, motivated by Thurston’s classical result, we prove the existence of a symplectic Torelli mapping class of positive categorical entropy. We also consider relations to the Floer-theoretic entropy.

Gevrey estimates for asymptotic expansions of Tori in weakly dissipative systems* *Both authors have been supported by NSF Grant DMS 1800241.

We consider a singular perturbation for a family of analytic symplectic maps of the annulus possessing a KAM torus. The perturbation introduces dissipation and contains an adjustable parameter. By choosing the adjustable parameter, one can ensure that the torus persists under perturbation. Such models are common in celestial mechanics. In field theory, the adjustable parameter is called the counterterm and in celestial mechanics, the drift. It is known that there are formal expansions in powers of the perturbation both for the quasi-periodic solution and the counterterm. We prove that the asymptotic expansions for the quasiperiodic solutions and the counterterm satisfy Gevrey estimates. That is, the nth term of the expansion is bounded by a power of n!. The Gevrey class (the power of n!) depends only on the Diophantine condition of the frequency and the order of the friction coefficient in powers of the perturbative parameter. The method of proof we introduce may be of interest beyond the problem considered here. We consider a modified Newton method in a space of power expansions. As is custumary in KAM theory, each step of the method is estimated in a smaller domain. In contrast with the KAM results, the domains where we control the Newton method shrink very fast and the Newton method does not prove that the solutions are analytic. On the other hand, by examining carefully the process, we can obtain estimates on the coefficients of the expansions and conclude the series are Gevrey.

Fractal Structures and Magnetic Footprints in a Divertor Tokamak

Fractal structures appear very often in open Hamiltonian systems, and can be identified in the deposition of chaotic magnetic field line on the plates of a tokamak divertor. Indeed, tokamaks with divertors are used to control the magnetic confinement of plasmas, such that the field lines created by electric currents have escape channels, through which plasma particles can be diverted out of the tokamak wall and redirected to divertor plates. In this work, we use a symplectic map to investigate the deposition patterns on the plates of a divertor. We show that the pattern of magnetic footprints on divertor plates (deposition patterns) underlying the chaotic orbits involves a number of fractal structures related to the existence of a nonattractive invariant chaotic set. In order to investigate the fractal characteristic of magnetic footprints, we analyze quantitatively the degree of stability of the fractal structure by calculating the entropy of the basin. These numerical analyses indicate that the fractal pattern on the divertor plates depends sensibly on the magnetic field structure. We show qualitative evidences of the Wada property.

Manipulation of transverse emittances in circular accelerators by crossing nonlinear 2D resonances

Controlling nonlinear effects in the transverse dynamics of charged particle beams in circular accelerators opens new possibilities for controlling some of the beam properties. Beam splitting by crossing a stable 1D nonlinear resonance is part of the routine operation of the CERN Proton Synchrotron. The beam undergoes trapping and transport inside stable islands created in the horizontal plane to allow multi-turn extraction toward the Super Proton Synchrotron, where the beam is used for fixed-target experiments. This process acts only on the horizontal beam emittance, inducing a reduction of its initial value. In this paper, we present a generalization of this approach, in which both transverse planes are affected by the proposed technique. We will discuss in detail how to manipulate the transverse emittances by means of a controlled crossing of a 2D nonlinear resonance. The novel technique will be presented by discussing the theoretical analysis of a Hamiltonian model, as well as simulating the performance of the proposed manipulation using a more realistic nonlinear symplectic map.