Symplectic Maps Research Articles

Geometry of transit orbits in the periodically-perturbed restricted three-body problem

In the circular restricted three-body problem, low energy transit orbits are revealed by linearizing the governing differential equations about the collinear Lagrange points. This procedure fails when time-periodic perturbations are considered, such as perturbation due to the sun (i.e., the bicircular problem) or orbital eccentricity of the primaries. For the case of a time-periodic perturbation, the Lagrange point is replaced by a periodic orbit, equivalently viewed as a hyperbolic-elliptic fixed point of a symplectic map (the stroboscopic Poincaré map). Transit and non-transit orbits can be identified in the discrete map about the fixed point, in analogy with the geometric construction of Conley and McGehee about the index-1 saddle equilibrium point in the continuous dynamical system. Furthermore, though the continuous time system does not conserve the Hamiltonian energy (which is time-varying), the linearized map locally conserves a time-independent effective Hamiltonian function. We demonstrate that the phase space geometry of transit and non-transit orbits is preserved in going from the unperturbed to a periodically-perturbed situation, which carries over to the full nonlinear equations.

An Arnold-type principle for non-smooth objects

In this article, we study the Arnold conjecture in settings where objects under consideration are no longer smooth but only continuous. The example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point shows that the conjecture does not admit a direct generalization to continuous settings. However, it appears that the following Arnold-type principle continues to hold in \(C^0\) settings: suppose that X is a non-smooth object for which one can define spectral invariants. If the number of spectral invariants associated to X is smaller than the number predicted by the (homological) Arnold conjecture, then the set of fixed/intersection points of X is homologically non-trivial, hence it is infinite. We recently proved that the above principle holds for Hamiltonian homeomorphisms of closed and aspherical symplectic manifolds. In this article, we verify this principle in two new settings: \(C^0\) Lagrangians in cotangent bundles and Hausdorff limits of Legendrians in 1-jet bundles which are isotopic to 0-section. An unexpected consequence of the result on Legendrians is that the classical Arnold conjecture does hold for Hausdorff limits of Legendrians in 1-jet bundles.

Symplectic mapping class groups of K3 surfaces and Seiberg–Witten invariants

The purpose of this note is to prove that the symplectic mapping class groups of many K3 surfaces are infinitely generated. Our proof makes no use of any Floer-theoretic machinery but instead follows the approach of Kronheimer and uses invariants derived from the Seiberg–Witten equations.

On symplectic stabilisations and mapping classes

We are interested in comparing properties of symplectic mapping class groups of symplectic manifolds of dimension four or higher with properties of classical mapping class groups of surfaces. For n ⩾ 2 $n \geqslant 2$ , consider a configuration of Lagrangian S n $S^n$ s in a Weinstein domain M 2 n $M^{2n}$ . If it is analogous, in some sense that we make precise, to a configuration of exact Lagrangian S 1 $S^1$ s on a surface Σ $\Sigma$ , we show that any relation between Dehn twists in the S n $S^n$ s must also hold between the S 1 $S^1$ s. Such analogous pairs of configurations include plumbings of T * S 1 $T^\ast S^1$ s and T * S n $T^\ast S^n$ s with the same plumbing graph, and vanishing cycles for a two-variable singularity and for its stabilisation. We give a number of corollaries for subgroups of symplectic mapping class groups.

On the Gauss map of equivariant immersions in hyperbolic space

Given an oriented immersed hypersurface in hyperbolic space H n + 1 $\mathbb {H}^{n+1}$ , its Gauss map is defined with values in the space of oriented geodesics of H n + 1 $\mathbb {H}^{n+1}$ , which is endowed with a natural para-Kähler structure. In this paper, we address the question of whether an immersion G $G$ of the universal cover of an n $n$ -manifold M $M$ , equivariant for some group representation of π 1 ( M ) $\pi _1(M)$ in Isom ( H n + 1 ) $\mathrm{Isom}(\mathbb {H}^{n+1})$ , is the Gauss map of an equivariant immersion in H n + 1 $\mathbb {H}^{n+1}$ . We fully answer this question for immersions with principal curvatures in ( − 1 , 1 ) $(-1,1)$ : while the only local obstructions are the conditions that G $G$ is Lagrangian and Riemannian, the global obstruction is more subtle, and we provide two characterizations, the first in terms of the Maslov class, and the second (for M $M$ compact) in terms of the action of the group of compactly supported Hamiltonian symplectomorphisms.

On simultaneous linearization of certain commuting nearly integrable diffeomorphisms of the cylinder

AbstractLet $$\mathcal {F}$$ F and $$\mathcal {K}$$ K be commuting $$C^\infty $$ C ∞ diffeomorphisms of the cylinder $$\mathbb {T}\times \mathbb {R}$$ T × R that are, respectively, close to $$\mathcal {F}_0 (x, y)=(x+\omega (y), y)$$ F 0 ( x , y ) = ( x + ω ( y ) , y ) and $$T_\alpha (x, y)=(x+\alpha , y)$$ T α ( x , y ) = ( x + α , y ) , where $$\omega (y)$$ ω ( y ) is non-degenerate and $$\alpha $$ α is Diophantine. Using the KAM iterative scheme for the group action we show that $$\mathcal {F}$$ F and $$\mathcal {K}$$ K are simultaneously $$C^\infty $$ C ∞ -linearizable if $$\mathcal {F}$$ F has the intersection property (including the exact symplectic maps) and $$\mathcal {K}$$ K satisfies a semi-conjugacy condition. We also provide examples showing necessity of these conditions. As a consequence, we get local rigidity of certain class of $$\mathbb {Z}^2$$ Z 2 -actions on the cylinder, generated by commuting twist maps.

Breakup of transport barriers in plasmas with flow described by symplectic maps

Transport barriers associated with the last KAM surfaces to break up are studied in the context of turbulent transport produced by a discrete spectrum of waves in two dimensions using a 2D symplectic map. It is based on the guiding center motion of test particles but finite Larmor radius (FLR) effects are included taking an average over the gyroperiod. Poloidal sheared flows are included which are responsible for the creation of the transport barrier. For large wave amplitudes, widespread chaos dominates the phase space interrupted only by the most stable KAM surfaces. The torus breakup depends on the values of the parameters of the map: wave amplitude A, flow velocity C and the Larmor radius ρ. For two types of sheared poloidal flows, having monotonic and non-monotonic shear, phase diagrams for barrier breakup are obtained by following the iterations of two points on either side of the barrier until the trajectories get mixed. The threshold that delimits the barrier breakup has the typical feature of a fractal curve for all phase diagrams.

Non-Linear Effects in Optical Systems by Lie Algebra and Symplectic Mapping

The use of signals of different frequencies determines the geometrical deviation with respect to the optical axes of a given beam. This angle can be determined by Sympletic Map (SM), a powerful and simple mathematical tool for the characterization and construction of images in Geometrical Optics. The Sympletic Map constitutes a Lie Group, with an algebra associated: the Lie Algebra. In general, the SM can be expressed as an infinite series, where each term corresponds to different contributions produced by the optical devices that constitute the optical system (lenses, apertures, bandwidth cutoff, etc.). The level of correction to be performed on the image to recover the original object is clear and controllable by SM. This formalism can be extended easily to physical optics to describe diffraction and interference phenomena.

Epsilon-non-squeezing and $C^0$-rigidity of epsilon-symplectic embeddings

An embedding $\varphi \colon (M_1, \omega_1) \to (M_2, \omega_2)$ (of symplectic manifolds of the same dimension) is called $\epsilon$-symplectic if the difference $\varphi^* \omega_2 - \omega_1$ is $\epsilon$-small with respect to a fixed Riemannian metric on $M_1$. We prove that if a sequence of $\epsilon$-symplectic embeddings converges uniformly (on compact subsets) to another embedding, then the limit is $E$-symplectic, where the number $E$ depends only on $\epsilon$ and $E (\epsilon) \to 0$ as $\epsilon \to 0$. This generalizes $C^0$-rigidity of symplectic embeddings, and answers a question in topological quantum computing by Michael Freedman. As in the symplectic case, this rigidity theorem can be deduced from the existence and properties of symplectic capacities. An $\epsilon$-symplectic embedding preserves capacity up to an $\epsilon$-small error, and linear $\epsilon$-symplectic maps can be characterized by the property that they preserve the symplectic spectrum of ellipsoids (centered at the origin) up to an error that is $\epsilon$-small. We sketch an alternative proof using the shape invariant, which gives rise to an analogous characterization and rigidity theorem for $\epsilon$-contact embeddings.

Measure, dimension, and complexity of the transient motion in Hamiltonian systems

Hamiltonian systems that are either open, leaking, or contain holes in the phase space possess solutions that eventually escape the system’s domain. The motion described by such escape orbits before crossing the escape threshold can be understood as a transient behavior. In this work, we introduce a numerical method to visually illustrate and quantify the transient motion in Hamiltonian systems based on the transient measure, a finite-time version of the natural measure. We apply this method to two physical systems: the single-null divertor tokamak, described by a symplectic map, and the Earth–Moon system, as modeled by the planar circular restricted three-body problem. Our results portray how different locations for the ensemble of initial conditions may lead to different transient dynamical scenarios in both systems. We show that these scenarios can be properly quantified from a geometrical aspect, the transient correlation dimension, and a dynamical aspect, the transient complexity coefficient.

Stepsize errors in the N-body problem: discerning Mercury’s true possible long-term orbits

ABSTRACT Numerical integrations of the Solar system have been carried out for decades. Their results have been used, for example to determine whether the Solar system is chaotic, whether Mercury’s orbit is stable, or to help discern Earth’s climate history. We argue that all of the past studies we consider in this work are affected by numerical chaos to different degrees, affecting the possible orbits and instability probability of Mercury, sometimes significantly. We show how to eliminate the effects of numerical chaos by resolving Mercury’s pericentre passage. We also show that several higher order symplectic maps do not exhibit significant differences in resolving pericentre passage of Mercury (at fixed time-step), making their advantages suspect for calculating long-term orbits. Resolving pericentre passage affects a wide array of orbital numerical studies, like exoplanet studies, studies of the galactic centre, and other N-body problems.

Non-differentiable irrational curves for twist map

AbstractWe construct a $C^1$ symplectic twist map g of the annulus that has an essential invariant curve $\Gamma $ such that $\Gamma $ is not differentiable and g restricted to $\Gamma $ is minimal.

A polyfold proof of the Arnold conjecture

We give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds M via a direct Piunikhin–Salamon–Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from {{mathbb {C}}{mathbb {P}}}^1times M to {{mathbb {C}}}^+ times M and {{mathbb {C}}}^-times M, as developed by Fish–Hofer–Wysocki–Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.

Revisiting the relation between the Lyapunov time and the instability time

In this effort we focus on the so-called TL−Tinst relationship i.e., any relation among the Lyapunov time and a characteristic instability time of a given dynamical system. By means of extensive numerical simulations with a high-dimensional dynamical system, a 4D symplectic map, we investigate a possible correlation between both time-scales. Herein the instability time is the one associated to diffusion along the homoclinic tangle of the resonances of the system. We found that different laws could fit the computed values, depending mostly on the dynamics of the system when varying the involved parameters; in some small domain of the parameter space a power law appears while in a larger one an exponential relation fits quite well the computed values of TL and Tinst. We compare the obtained functional forms of the relationships with those known for lower-dimensional systems and identify typical functional dependences, confirmed analytically.

Lie Groupoids, Deformation of Unstable Curves, and Construction of Equivariant Kuranishi Charts

In this paper we give the detailed construction of a G -equivariant Kuranishi chart of moduli spaces of pseudo-holomorphic curves to a symplectic manifold with G -action, for an arbitrary compact Lie group G . The proof is based on the deformation theory of unstable marked curves using the language of Lie groupoids (which is not necessarily étale) and the Riemannian center of mass technique. This proof is actually similar to Fukaya and Ono (Arnold conjecture and Gromov–Witten invariant, Topology 38 (1999), 933–1048, Sects. 13 and 15), except that the usage of the language of Lie groupoids makes the argument more transparent.

Visualizing the phase space of the HeI[formula omitted] van der Waals complex using Lagrangian descriptors

In this paper we demonstrate the capability of the method of Lagrangian descriptors to unveil the phase space structures that characterize transport in high-dimensional symplectic maps. In order to illustrate its use, we apply it to a four-dimensional symplectic map model that is used in chemistry to explore the nonlinear dynamics of van der Waals complexes. The advantage of this technique is that it allows us to easily and effectively extract the invariant manifolds that determine the dynamics of the system under study by means of examining the intersections of the underlying phase space structures with low-dimensional slices. With this approach, one can perform a full computational phase space tomography from which three-dimensional representations of the higher-dimensional phase space can be systematically reconstructed. This analysis may be of much help for the visualization and understanding of the nonlinear dynamical mechanisms that take place in high-dimensional systems. In this context, we demonstrate how this tool can be used to detect whether the stable and unstable manifolds of the system intersect forming turnstile lobes that enclose a certain phase space volume, and the nature of their intersection.

On C0-Continuity of the Spectral Norm for Symplectically Non-Aspherical Manifolds

Abstract The purpose of this paper is to study the relation between the $C^0$-topology and the topology induced by the spectral norm on the group of Hamiltonian diffeomorphisms of a closed symplectic manifold. Following the approach of Buhovsky–Humilière–Seyfaddini, we prove the $C^0$-continuity of the spectral norm for complex projective spaces and negative monotone symplectic manifolds. The case of complex projective spaces provides an alternative approach to the $C^0$-continuity of the spectral norm proven by Shelukhin. We also prove a partial $C^0$-continuity of the spectral norm for rational symplectic manifolds. Some applications such as the Arnold conjecture in the context of $C^0$-symplectic topology are also discussed.

Dynamical invariants of mapping torus categories

This paper describes constructions in homological algebra that are part of a strategy whose goal is to understand and classify symplectic mapping tori. More precisely, given a dg category and an auto-equivalence, satisfying certain assumptions, we introduce a category Mϕ-called the mapping torus category- that describes the wrapped Fukaya category of an open symplectic mapping torus. Then we define a family of bimodules on a natural deformation of Mϕ, uniquely characterize it and using this, we distinguish Mϕ from the mapping torus category of the identity. The proof of the equivalence of Mϕ with wrapped Fukaya category is proven in a different paper ([17]).

Integrable symplectic maps via reduction of Bäcklund transformation

В качестве иллюстративных примеров рассмотрены стационарные потенциальные уравнения, целью анализа которых является объяснение того, как c помощью преобразования Беклунда можно построить интегрируемые симплектические отображения. Сначала дан краткий обзор преобразований Беклунда потенциального уравнения КдФ и потенциального уравнения КдФ пятого порядка. Далее в координатах Якоби-Остроградского получены канонические гамильтоновы формы стационарных потенциальных уравнений. На основе редукции преобразования Беклунда построены симплектические отображения и показана их интегрируемость.

Geometric quantization of symplectic maps and Witten's asymptotic conjecture

We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the spaces of holomorphic sections of a prequantizing line bundle over compact Kähler manifolds under deformations of the complex structure. We show that the parallel transport in the induced vector bundle over the deformation space behaves like a Toeplitz operator, and compute its first coefficient. We then use this result to establish a semi-classical trace formula for the induced quantization of symplectic maps, and give an application to Witten's asymptotic expansion conjecture for the quantum representations of the mapping class group.