Symplectic Maps Research Articles

A family of Liouville integrable lattice equations with a parameter and its two symmetry constraints

A two-by-two matrix spectral problem with an arbitrary constant is introduced, a family of integrable lattice equations involving a parameter is derived by the technique of Lax pairs. The typical lattice equation in the obtained family is a deformed Toda lattice system. The bi-Hamiltonian structure of the obtained family of integrable lattice equations is established by using the discrete trace identity. And we prove the Liouville integrability of obtained family of integrable lattice equations. Then, Bargmann symmetry constraint and an implicit symmetry constraint of the obtained family are presented by binary nonlinearization method. Under these two symmetry constraints, we obtain two series of integrable symplectic maps and two sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense, and every integrable lattice equation in the obtained family is decomposed by an integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system. As a special case, two integrable decompositions of the deformed Toda lattice equation are given.

A finite genus solution of the H1 model

The H1 model in the Adler–Bobenko–Suris list, i.e. the lattice potential KdV equation and the closely related lattice KdV equation with Nijhoff’s discretization, are investigated. A new Lax pair of the H1 model is given, by which integrable symplectic maps are constructed through a non-linearization procedure. Resorting to these maps, finite genus solutions of the H1 model as well as the lattice KdV equation are calculated.

Local path fitting: A new approach to variational integrators

A new approach for constructing variational integrators is presented. In the general case, the estimation of the action integral in a time interval [tk,tk+1] is used to construct a symplectic map (qk,qk+1)→(qk+1,qk+2). The basic idea, is that only the partial derivatives of the estimated action integral of the Lagrangian are needed in the general theory. The analytic calculation of these derivatives, gives rise to a new integral that depends on the Euler–Lagrange vector itself (which in the continuous and exact case vanishes) and not on the Lagrangian. Since this new integral can only be computed through a numerical method based on some internal grid points, we can locally fit the exact curve by demanding the Euler–Lagrange vector to vanish at these grid points. Thus, the integral vanishes, and the process dramatically simplifies the calculation of high order approximations. The new technique is tested in high order solutions of the two-body problem with high eccentricity (up to 0.99) and of the outer planets of the solar system.

Generic Properties of Closed Orbits of Hamiltonian Flows from Mañé's Viewpoint

We show the genericity from the viewpoint of of generic properties (as symplectic linear maps) of the differential of Poincare maps of periodic orbits of Hamiltonians. Combining this result and the work of Oliveira we get Kupka-Smale type theorem a la Mane for regular energy levels of Tonelli Hamiltonians in compact manifolds. Our proof relies on techniques from geometric control theory.

A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map

We construct a $C^1$ symplectic twist map $f$ of the annulus that has an essential invariant curve $\Gamma$ such that: •$\Gamma$ is not differentiable; •$f$ ↾ $\Gamma$ is conjugate to a Denjoy counterexample.

Collisional effects in the tokamap

Plasmas confined in tokamaks with non-symmetric perturbations are surrounded by a chaotic layer of magnetic field lines that guide charged particles to the tokamak wall. We use an analytical two-dimensional symplectic mapping to study the resulting fractal patterns of field line escape. However, particles may experience several collisions before escaping toward the tokamaks wall. We add a random collisional term to the field line mapping to investigate how the particle collisions modify their escape patterns.

Hamiltonian control used to improve the beam stability in particle accelerator models

We derive a Hamiltonian control theory which can be applied to a 4D symplectic map that models a ring particle accelerator composed of elements with sextupole nonlinearity. The controlled system is designed to exhibit a more regular orbital behavior than the uncontrolled one. Using the Smaller Alignment Index (SALI) chaos indicator, we are able to show that the controlled system has a dynamical aperture up to 1.7 times larger than the original model.

Comparison of inboard and outboard magnetic footprints from topological noise and field errors in the DIII-D

Any canonical transformation of Hamiltonian equations is symplectic, and any area-preserving transformation in 2D is a symplectomorphism. Based on these, a discrete symplectic map and its continuous symplectic analog are derived for forward magnetic field line trajectories in natural canonical coordinates. The unperturbed axisymmetric Hamiltonian for magnetic field lines is constructed from the experimental data in the DIII-D. The equilibrium Hamiltonian is a highly accurate, analytic, and realistic representation of the magnetic geometry of the DIII-D. These symplectic mathematical maps are used to calculate the trajectories of magnetic field lines in the DIII-D. Internal statistical topological noise and field errors are irreducible and ubiquitous in magnetic confinement schemes for fusion. It is important to know the stochasticity and magnetic footprint from noise and error fields. The estimates of the spectrum and mode amplitudes of the spatial topological noise and magnetic errors in the DIII-D are used as magnetic perturbation. The discrete and continuous symplectic maps are used to calculate the magnetic footprint on the inboard and outboard collector plates of the DIII-D by inverting the natural coordinates to physical coordinates. Radial variation of magnetic perturbation and the response of plasma to perturbation are not included. The footprints are in the form of toroidally winding helical strips. The area of footprint scales linearly with amplitude. The outboard footprint has higher transverse extent, higher area, and higher fractal dimension. The field diffusion near the X-point for backward lines is higher than that for forward lines. Diffusion for lines that strike the plate is about three to four orders of magnitude higher than for lines that do not strike. Line loss and flux loss show a drop when the amplitude of perturbation is about 18–19 × 10−6, indicating the possible presence of a barrier. The physical parameters such as toroidal angle, length, and poloidal angle covered before striking and the safety factor all have a fractal structure.

Strong and Weak Chaos in Weakly Nonintegrable Many-Body Hamiltonian Systems

We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology.

A comparison of different indicators of chaos based on the deviation vectors: application to symplectic mappings

The aim of this research work is to compare the reliability of several variational indicators of chaos on mappings. The Lyapunov Indicator (LI); the Mean Exponential Growth factor of Nearby Orbits (MEGNO); the Smaller Alignment Index (SALI); the Fast Lyapunov Indicator (FLI); the Dynamical Spectra of stretching numbers (SSN) and the corresponding Spectral Distance (D); and the Relative Lyapunov Indicator (RLI), which is based on the evolution of the difference between two close orbits, have been included. The experiments presented herein allow us to reliably suggest a group of chaos indicators to analyze a general mapping. We show that a package composed of the FLI and the RLI (to analyze the phase portrait globally) and the MEGNO and the SALI (to analyze orbits individually) is good enough to make a description of the systems' dynamics.

Chaotic transport in Hamiltonian systems perturbed by a weak turbulent wave field

Chaotic transport in a hamiltonian system perturbed by a weak turbulent wave field is studied. It is assumed that a turbulent wave field has a wide spectrum containing up to thousands of modes whose phases are fluctuating in time with a finite correlation time. To integrate the hamiltonian equations a fast symplectic mapping is derived. It has a large time-step equal to one full turn in angle variable. It is found that the chaotic transport across tori caused by the interactions of small-scale resonances have a fractal-like structure with the reduced or zero values of diffusion coefficients near low-order rational tori thereby forming transport barriers there. The density of rational tori is numerically calculated and its properties are investigated. It is shown that the transport barriers are formed in the gaps of the density of rational tori near the low-order rational tori. The dependencies of the depth and width of transport barriers on the wave field spectrum and the correlation time of fluctuating turbulent field (or the Kubo number) are studied. These numerical findings may have importance in understanding the mechanisms of transport barrier formation in fusion plasmas.

Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds

We assign to each nondegenerate Hamiltonian on a closed symplectic manifold a Floer-theoretic quantity called its “boundary depth,” and establish basic results about how the boundary depths of different Hamiltonians are related. As applications, we prove that certain Hamiltonian symplectomorphisms supported in displaceable subsets have infinitely many nontrivial geometrically distinct periodic points, and we also significantly expand the class of coisotropic submanifolds which are known to have positive displacement energy. For instance, any coisotropic submanifold of contact type (in the sense of Bolle) in any closed symplectic manifold has positive displacement energy, as does any stable coisotropic submanifold of a Stein manifold. We also show that any stable coisotropic submanifold admits a Riemannian metric that makes its characteristic foliation totally geodesic, and that this latter, weaker, condition is enough to imply positive displacement energy under certain topological hypotheses.

Kolmogorov-Sinai entropy of many-body Hamiltonian systems

The Kolmogorov-Sinai (KS) entropy is a central measure of complexity and chaos. Its calculation for many-body systems is an interesting and important challenge. In this paper, the evaluation is formulated by considering N-dimensional symplectic maps and deriving a transfer matrix formalism for the stability problem. This approach makes explicit a duality relation that is exactly analogous to one found in a generalized Anderson tight-binding model and leads to a formally exact expression for the finite-time KS entropy. Within this formalism there is a hierarchy of approximations, the final one being a diagonal approximation that only makes use of instantaneous Hessians of the potential to find the KS entropy. By way of a nontrivial illustration, the KS entropy of N identically coupled kicked rotors (standard maps) is investigated. The validity of the various approximations with kicking strength, particle number, and time are elucidated. An analytic formula for the KS entropy within the diagonal approximation is derived and its range of validity is also explored.

A numerical study of the stabilization effect of steepness

We present the results of numerical experiments about the influence of steepness on the resonant structure, stability and diffusion in a 4-dimensional symplectic map. The map is designed so that by changing a parameter, we smoothly switch steepness on and off by the change of the so called steepness coefficients. In both cases we measure the diffusion coefficients of the actions within a resonance. According to Nekhoroshev theorem we find that, in the steep case, the diffusion coefficients are definitely smaller than in the non steep one, thus confirming the threshold effect of the steepness coefficients which comes from the proof of Nekhoroshev theorem.

Symplectic calculation of the outboard magnetic footprint from noise and error fields in the DIII-D

AbstractThe backward symplectic DIII-D map and continuous symplectic analog of the map for magnetic field line trajectories in the DIII-D [10] (Luxon, J. L. and Davis, L. E. 1985 Fusion Technol.8, 441) in natural canonical coordinates are used to calculate the magnetic footprint on the outboard collector plate of the DIII-D tokamak from the field errors and internal topological noise. The equilibrium generating function for the DIII-D used in the map very accurately represents the magnetic geometry of the DIII-D. The step-size of the map is kept considerably small so that the magnetic perturbation added from symplectic discretization of the Hamiltonian equations of the magnetic field line trajectories is very small. The natural canonical coordinates allow inverting to the real physical space. The combination of highly accurate equilibrium generating function, natural canonical coordinates, symplecticity, and small step-size then together gives a very accurate calculation of magnetic footprint. Radial variation of magnetic perturbation and the response of plasma to perturbation are not included. The footprint is in the form of toroidally winding helical strips. The area of footprint scales as 1st power of amplitude. The physical parameters as toroidal angle, length, and poloidal angle covered before striking, and the safety factor all have fractal structure. The average field diffusion near X-point for lines that strike and that do not strike differs by about four orders of magnitude. The flux loss decreases for high values of amplitude of perturbation.

Nontwist symplectic maps in tokamaks

We review symplectic nontwist maps that we have introduced to describe Lagrangian transport properties in magnetically confined plasmas in tokamaks. These nontwist maps are suitable to describe the formation and destruction of transport barriers in the shearless region (i.e., near the curve where the twist condition does not hold). The maps can be used to investigate two kinds of problems in plasmas with non-monotonic field profiles: the first is the chaotic magnetic field line transport in plasmas with external resonant perturbations. The second problem is the chaotic particle drift motion caused by electrostatic drift waves. The presented analytical maps, derived from plasma models with equilibrium field profiles and control parameters that are commonly measured in plasma discharges, can be used to investigate long-term transport properties.

Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew's triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange-Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange-d'Alembert-Pontryagin and Hamilton-d'Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.

The destruction of tori in volume-preserving maps

Invariant tori are prominent features of symplectic and volume-preserving maps. From the point of view of chaotic transport the most relevant tori are those that are barriers, and thus have codimension one. For an n-dimensional volume-preserving map, such tori are prevalent when the map is nearly “integrable,” in the sense of having one action and n − 1 angle variables. As the map is perturbed, numerical studies show that the originally connected image of the frequency map acquires gaps due to resonances and domains of nonconvergence due to chaos. We present examples of a three-dimensional, generalized standard map for which there is a critical perturbation size, ε c , above which there are no tori. Numerical investigations to find the “last invariant torus” reveal some similarities to the behavior found by Greene near a critical invariant circle for area preserving maps: the crossing time through the newly destroyed torus appears to have a power law singularity at ε c , and the local phase space near the critical torus contains many high-order resonances.

Symplectic tracking and compensation of dynamic field integrals in complex undulator structures

In first approximation storage ring multipole magnets are described as simple two-dimensional magnet structures and many linear and nonlinear beam optic features of a magnet lattice can already be derived from this model. In contrast, undulators, and in particular variably polarizing devices, employ complicated three-dimensional magnetic fields which may have a severe impact on the electron beam, in particular, in low energy third generation storage rings. A Taylor expanded generating function method is presented to generate a fast, flexible, and symplectic mapping routine for particle tracking in magnetic fields. This method is quite general and is based on the solution of the Hamilton-Jacobi equation. It requires an analytical representation of the fields, which can be differentiated and integrated. For undulators of the APPLE II type, an accurate analytic field model is derived which is suitable for the tracking routine. This field model is fully parametrized representing all operation modes for the production of elliptical or linear polarized light with an arbitrary inclination angle or even arbitrary polarization. Based on this field model, analytic expressions for 2nd order kicks are derived. They are used to estimate the influence of APPLE II undulators on the electron beam dynamic. Furthermore, an analytic model for the description of shims is given. The shims are needed for field and performance optimization. Passive and active shimming concepts for the compensation of linear and nonlinear effects of variably polarizing undulators are discussed.

Scarring for quantum maps with simple spectrum

AbstractIn [Kelmer, Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms, Comm. Math. Phys. 276 (2007), 381–395] we introduced a family of symplectic maps of the torus whose quantization exhibits scarring on invariant co-isotropic submanifolds. The purpose of this note is to show that in contrast to other examples, where failure of quantum unique ergodicity is attributed to high multiplicities in the spectrum, for these examples the spectrum is (generically) simple.