Symplectic Maps Research Articles

Return Maps, Dynamical Consequences and Applications

After reviewing some general settings for return maps in problems reducible to 2D symplectic maps, details on the construction of return maps are presented. Different forms of such maps close to splitted separatrices (separatrix maps) are introduced, taking into account the size and shape of the splitting function and also the return time to the domains of interest. Then it is shown how to derive approximations by suitable standard-like maps. Dynamical consequences concerning the existence of invariant rotational curves (IRC) are derived. An application is made to theoretically estimate the location of the outermost IRC in the Sitnikov problem, which is in good agreement with numerical data. To compare with the cases which are approximated by the classical standard map, some details on the properties of the standard-like map with two harmonic terms are included. Finally a method to estimate the amount of chaos depending on the form of the separatrix map is introduced. Except otherwise stated all the systems we consider are assumed to be analytic, despite several of the properties we study are no longer analytic.

Coupled symplectic maps as models for subdiffusive processes in disordered Hamiltonian lattices

We investigate dynamically and statistically diffusive motion in a chain of linearly coupled 2-dimensional symplectic McMillan maps and find evidence of subdiffusion in weakly and strongly chaotic regimes when all maps of the chain possess a saddle point at the origin and the central map is initially excited. In the case of weak coupling, there is either absence of diffusion or subdiffusion with q>1-Gaussian probability distributions, characterizing weak chaos. However, for large enough coupling and already moderate number of maps, the system exhibits strongly chaotic (q≈1) subdiffusive behavior, reminiscent of the subdiffusive energy spreading observed in a disordered Klein–Gordon Hamiltonian. Our results provide evidence that coupled symplectic maps can exhibit physical properties similar to those of disordered Hamiltonian systems, even though the local dynamics in the two cases is significantly different.

Symplectic maps and chromatic optics in particle accelerators

We have applied the nonlinear map method to comprehensively characterize the chromatic optics in particle accelerators. Our approach is built on the foundation of symplectic transfer maps of magnetic elements. The chromatic lattice parameters can be transported from one element to another by the maps. We introduce a Jacobian operator that provides an intrinsic linkage between the maps and the matrix with parameter dependence. The link allows us to directly apply the formulation of the linear optics to compute the chromatic lattice parameters. As an illustration, we analyze an alternating-gradient cell with nonlinear sextupoles, octupoles, and decapoles and derive analytically their settings for the local chromatic compensation. As a result, the cell becomes nearly perfect up to the third-order of the momentum deviation.

Whfast: a fast and unbiased implementation of a symplectic Wisdom–Holman integrator for long-term gravitational simulations

We present WHFast, a fast and accurate implementation of a Wisdom-Holman symplectic integrator for long-term orbit integrations of planetary systems. WHFast is significantly faster and conserves energy better than all other Wisdom-Holman integrators tested. We achieve this by significantly improving the Kepler-solver and ensuring numerical stability of coordinate transformations to and from Jacobi coordinates. These refinements allow us to remove the linear secular trend in the energy error that is present in other implementations. For small enough timesteps we achieve Brouwer's law, i.e. the energy error is dominated by an unbiased random walk due to floating-point round-off errors. We implement symplectic correctors up to order eleven that significantly reduce the energy error. We also implement a symplectic tangent map for the variational equations. This allows us to efficiently calculate two widely used chaos indicators the Lyapunov characteristic number (LCN) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). WHFast is freely available as a flexible C package, as a shared library, and as an easy-to-use python module.

Invariant curves for exact symplectic twist maps of the cylinder with Bryuno rotation numbers

Since Moser's seminal work it is well known that the invariant curves of smooth nearly integrable twist maps of the cylinder with Diophantine rotation number are preserved under perturbation. In this paper we show that, in the analytic class, the result extends to Bryuno rotation numbers. First, we will show that the series expansion for the invariant curves in powers of the perturbation parameter can be formally defined, then we shall prove that the series converges absolutely in a neighbourhood of the origin. This will be achieved using multiscale analysis and renormalisation group techniques to express the coefficients of the series as sums of values which are represented graphically as tree diagrams and then exploit cancellations between terms contributing to the same perturbation order. As a byproduct we shall see that, when perturbing linear maps, the series expansion for an analytic invariant curve converges for all perturbations if and only if the corresponding rotation number satisfies the Bryuno condition.

Dynamics of the birational maps arising from [formula omitted] and [formula omitted] quivers

The dynamics of the maps associated to F0 and dP3 quivers is studied in detail. We show that the corresponding reduced symplectic maps are conjugate to globally periodic maps by providing explicit conjugations. The dynamics in R+N of the original maps is obtained by lifting the dynamics of these globally periodic maps and the solution of the discrete dynamical systems generated by each map is given. A better understanding of the dynamics is achieved by considering first integrals. The relationship between the complete integrability of the globally periodic maps and the dynamics of the original maps is explored.

Impact of punctual flat magnetic shear on the field line transport

We investigate the magnetic field line transport for tokamak equilibria with monotonic magnetic shear perturbed by resonant fields. We show that when the local profile is flat at the plasma edge a transport barrier can be created leading to a field line transport reduction. This transport reduction is due to the field lines topological modifications, caused by a local flattened profile that reduces the global field lines escape pattern. The results are obtained by applying a symplectic map that describes perturbed magnetic field lines in large aspect ratio tokamaks.

Exotic spheres and the topology of symplectomorphism groups

We show that, for certain families $\phi_{\mathbf{s}}$ of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of $T^*S^n$ with the family of pullbacks $\phi^*_{\mathbf{s}}$ gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find examples: where the Dehn twist along a parametrised Lagrangian sphere depends up to Hamiltonian isotopy on its parametrisation; where the symplectomorphism group is not simply-connected, and where the symplectomorphism group does not have the homotopy-type of a finite CW-complex. We show that these phenomena persist for Dehn twists along the standard matching spheres of the $A_m$-Milnor fibre. The nontriviality is detected by considering the action of symplectomorphisms on the space of parametrised Lagrangian submanifolds. We find related examples of symplectic mapping classes for $T^*(S^n\times S^1)$ and of an exotic symplectic structure on $T^*(S^n\times S^1)$ standard at infinity.

A Franks' lemma for convex planar billiards

Let γ be an orbit of the billiard flow on a convex planar billiard table; then the perpendicular part of the derivative of the billiard flow along γ is a symplectic linear map DP. This paper contains a proof of the following Franks’ lemma for a residual set of convex planar billiard tables: for any closed orbit, the map DP can be perturbed freely within a neighbourhood in Sp(1) by a C2-small perturbation in the space of convex planar billiard tables.

Pinball Dynamics: Unlimited Energy Growth in Switching Hamiltonian Systems

A family of discontinuous symplectic maps arising naturally in the study of nonsmooth switched Hamiltonian systems is considered. This family depends on two parameters and is a canonical model for the study of bounded and unbounded behavior in discontinuous area-preserving transformations due to nonlinear resonances. This paper provides a general description of the map and a construction of nontrivial unbounded solutions for the special case of the pinball transformation. An asymptotic expansion of the pinball map in the limit of large energy is derived and used for the construction of unbounded solutions. For the generic values of the parameters, in the large energy limit, the map behaves similarly to another one considered earlier by Kesten (Acta Arith 1966).

Chaotic enhancement of dark matter density in binary systems

We study the capture of galactic dark matter particles (DMP) in two-body and few-body systems with a symplectic map description. This approach allows modeling the scattering of $10^{16}$ DMPs after following the time evolution of the captured particle on about $10^9$ orbital periods of the binary system. We obtain the DMP density distribution inside such systems and determine the enhancement factor of their density in a center vicinity compared to its galactic value as a function of the mass ratio of the bodies and the ratio of the body velocity to the velocity of the galactic DMP wind. We find that the enhancement factor can be on the order of tens of thousands.

A KAM Theorem for a Class of Nearly Integrable Symplectic Mappings

This paper considers a class of nearly integrable twist symplectic mappings with generating functions and proves the persistence of lower dimensional elliptic invariant tori under Russmann’s non-degeneracy conditions.

Symplectic map description of Halley's comet dynamics

The main features of 1P/Halley chaotic dynamics can be described by a two dimensional symplectic map. Using Mel'nikov integral we semi-analytically determine such a map for 1P/Halley taking into account gravitational interactions from the Sun and the eight planets. We determine the Solar system kick function ie the energy transfer to 1P/Halley along one passage through the Solar system. Our procedure allows to compute for each planet its contribution to the Solar system kick function which appears to be the sum of the Keplerian potential of the planet and of a rotating circular gravitational dipole potential due to the Sun movement around Solar system barycenter. We test the robustness of the symplectic Halley map by directly integrating Newton's equations over $\sim 2.4\cdot 10^4$ yr around Y2K and by reconstructing the Solar system kick function. Our results show that the Halley map with fixed parameters gives a reliable description of comet dynamics on time scales of $10^4$ yr while on a larger scales the parameters of the map are slowly changing due to slow oscillations of orbital momentum.

The Force-Gradient Symplectic Finite-Difference Time-Domain Scheme for Maxwell's Equations

In this paper, a novel force-gradient explicit symplectic finite-difference time-domain (SFDTD) algorithm based on the T + V type Hamiltonian decomposition is derived to investigate the propagation characteristics of electromagnetic waves, in which Maxwell's equations are numerically integrated by third-order symplectic map combined with force gradient in the time direction. The numerical stability and dispersion analyses are presented. The numerical results effectively confirm that the force-gradient SFDTD (F-SFDTD) method is reliable for electromagnetic wave computation; its accuracy is superior to the non-force-gradient SFDTD method and the traditional FDTD method. The implementation of F-SFDTD method is easily applied to compute the back scattered radar cross sections (RCS) using 2-D TM mode for a perfectly conducting cylinder.

Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics

This work proposes a model-reduction methodology that preserves Lagrangian structure and achieves computational efficiency in the presence of high-order nonlinearities and arbitrary parameter dependence. As such, the resulting reduced-order model retains key properties such as energy conservation and symplectic time-evolution maps. We focus on parameterized simple mechanical systems subjected to Rayleigh damping and external forces, and consider an application to nonlinear structural dynamics. To preserve structure, the method first approximates the system's “Lagrangian ingredients''---the Riemannian metric, the potential-energy function, the dissipation function, and the external force---and subsequently derives reduced-order equations of motion by applying the (forced) Euler--Lagrange equation with these quantities. From the algebraic perspective, key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness: matrix gappy proper orthogonal decomposition and reduced-basis sparsification. Results for a parameterized truss-structure problem demonstrate the practical importance of preserving Lagrangian structure and illustrate the proposed method's merits: it reduces computation time while maintaining high accuracy and stability, in contrast to existing nonlinear model-reduction techniques that do not preserve structure.

On Complexified Analytic Hamiltonian Flows and Geodesics on the Space of Kähler Metrics

In the case of a compact real analytic symplectic manifold M we describe an approach to the complexification of Hamiltonian flows [Se, Do1, Th1] and corresponding geodesics on the space of Kahler metrics. In this approach, motivated by recent work on quantization, the complexified Hamiltonian flows act, through the Grobner theory of Lie series, on the sheaf of complex valued real analytic functions, changing the sheaves of holomorphic functions. This defines an action on the space of (equivalent) complex structures on M and also a direct action on M. This description is related to the approach of [BLU] where one has an action on a complexification M_C of M followed by projection to M. Our approach allows for the study of some Hamiltonian functions which are not real analytic. It also leads naturally to the consideration of continuous degenerations of diffeomorphisms and of Kahler structures of M. Hence, one can link continuously (geometric quantization) real, and more general non-Kahler, polarizations with Kahler polarizations. This corresponds to the extension of the geodesics to the boundary of the space of Kahler metrics. Three illustrative examples are considered. We find an explicit formula for the complex time evolution of the Kahler potential under the flow. For integral symplectic forms, this formula corresponds to the complexification of the prequantization of Hamiltonian symplectomorphisms. We verify that certain families of Kahler structures, which have been studied in geometric quantization, are geodesic families.

Solving negative and mixed Toda hierarchies by Jacobi inversion problems

Abstract The Jacobi inversion problems of negative and mixed Toda hierarchies are investigated through a symplectic map and some finite-dimensional Hamiltonian systems. Each negative equation is decomposed into the symplectic flow and a negative Hamiltonian flow, each mixed equation is decomposed into the symplectic flow and a mixed Hamiltonian flow. The separated variables are introduced to study these Hamiltonian systems. Based on the Hamilton–Jacobi theory, the relationship between the action-angle coordinates and the Jacobi-inversion problems is established.

Homoclinic tangle of the ideal separatrix in the DIII-D tokamak from (30, 10) + (40, 10) perturbation

Trajectories of magnetic field lines are a 1½ degree of freedom Hamiltonian system. The perturbed separatrix in a divertor tokamak is radically different from the unperturbed one. This is because magnetic asymmetries cause the separatrix to form extremely complicated structures called homoclinic tangles. The shape of flux surfaces in the edge region of divertor tokamaks such as the DIII (J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441 (1985)) is fundamentally different from near-circular. Recently, a new method is developed to calculate the homoclinic tangle and lobes of the separatrix of divertor tokamaks in physical space (A. Punjabi and A. Boozer, Phys. Lett. A 378, 2410 (2014)). This method is based on three elements: preservation of the two invariants—symplectic and topological neighborhood—and a new set of canonical coordinates called the natural canonical coordinates. The very complicated shape of edge surfaces can be represented very accurately and very realistically in these new coordinates (A. Punjabi and H. Ali, Phys. Plasmas 15, 122502 (2008); A. Punjabi, Nucl. Fusion 49, 115020 (2009)). A symplectic map in the new coordinates can advance the separatrix manifold forward and backward in time. Every time the two manifolds meet in a fixed poloidal plane, they intersect and form homoclinic tangle to preserve the two invariants. The new coordinates can be mapped to physical space and the dynamical evolution of the homoclinic tangle can be seen and pictured in physical space. Here, the new method is applied to the DIII-D tokamak to study the basic features of the homoclinic tangle of the unperturbed separatrix from two Fourier components, which represent the peeling-ballooning modes of equal amplitude and no radial dependence, and the results are analyzed. Homoclinic tangle has a very complicated structure and becomes extremely complicated for as the lines take more toroidal turns, especially near the X-point. Homoclinic tangle is the most complicated near the X-point and forms the largest lobes there. On average, the field lines cover a distance of about 9 m per turn. Poloidal rotation of the lines has large gradients in the poloidal direction. The average normal displacement of the lines on the separatrix varies from 5 mm to 7 cm. Average outward displacement of the lines is considerably larger than the inward displacement; however, on the average more lines are displaced inside than outside of the separatrix. The field line diffusion normal to the separatrix has extremely wide variation and very large poloidal gradients. Half of all the lines are lost in less than 6 turns. Complicated electric potentials will be required to maintain the neutrality of the plasma, and the E × B drifts from these fields can modify plasma confinement and influence the edge physics (A. Punjabi and A. Boozer, Phys. Lett. A 378, 2410 (2014)).

Stochastically symplectic maps and their applications to the Navier–Stokes equation

Poincaré's invariance principle for Hamiltonian flows implies Kelvin's principle for solution to Incompressible Euler equation. Constantin–Iyer Circulation Theorem offers a stochastic analog of Kelvin's principle for Navier–Stokes equation. Weakly symplectic diffusions are defined to produce stochastically symplectic flows in a systematic way. With the aid of symplectic diffusions, we produce a family of martigales associated with solutions to Navier–Stokes equation that in turn can be used to prove Constantin–Iyer Circulation Theorem. We also review some basic facts in symplectic and contact geometry and their applications to Euler equation.

Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class

It is well known that the matrix of a metaplectic operator with respect to phase-space shifts is concentrated along the graph of a linear symplectic map. We show that the algebra generated by metaplectic operators and by pseudodifferential operators in a Sjöstrand class enjoys the same decay properties. We study the behavior of these generalized metaplectic operators and represent them by Fourier integral operators. Our main result shows that the one-parameter group generated by a Hamiltonian operator with a potential in the Sjöstrand class consists of generalized metaplectic operators. As a consequence, the Schrödinger equation preserves the phase-space concentration, as measured by modulation space norms.