Two-dimensional Area-preserving Map Research Articles

Application of the Slater criteria to localize invariant tori in Hamiltonian mappings.

We investigate the localization of invariant spanning curves for a family of two-dimensional area-preserving mappings described by the dynamical variables I and θ by using Slater's criterion. The Slater theorem says there are three different return times for an irrational translation over a circle in a given interval. The returning time, which measures the number of iterations a map needs to return to a given periodic or quasi periodic region, has three responses along an invariant spanning curve. They are related to a continued fraction expansion used in the translation and obey the Fibonacci sequence. The rotation numbers for such curves are related to a noble number, leading to a devil's staircase structure. The behavior of the rotation number as a function of invariant spanning curves located by Slater's criterion resulted in an expression of a power law in which the absolute value of the exponent is equal to the control parameter γ that controls the speed of the divergence of θ in the limit the action I is sufficiently small.

A mechanism explaining the metamorphoses of KAM islands in nonhyperbolic chaotic scattering

In the context of nonhyperbolic chaotic scattering, it has been shown that the evolution of the KAM islands exhibits four abrupt metamorphoses that strongly affect the predictability of Hamiltonian systems. It has been suggested that these metamorphoses are related to significant changes in the structure of the KAM islands. However, previous research has not provided an explanation of the mechanisms underlying the metamorphoses. Here, we show that they occur due to the formation of a homoclinic or heteroclinic tangle that breaks the internal structure of the main KAM island. We obtain similar qualitative results in a two-dimensional Hamiltonian system and a two-dimensional area-preserving map. The equivalence of the results obtained in both systems suggests that the same four metamorphoses play an important role in conservative systems.

Eigenfunctions of the Perron–Frobenius operator and the finite-time Lyapunov exponents in uniformly hyperbolic area-preserving maps

The subleading eigenvalues and associated eigenfunctions of the Perron–Frobenius operator for two-dimensional area-preserving maps are numerically investigated. We closely examine the validity of the so-called Ulam method, a numerical scheme believed to provide eigenvalues and eigenfunctions of the Perron–Frobenius operator, both for linear and nonlinear maps on the torus. For the nonlinear case, the second-largest eigenvalues and the associated eigenfunctions of the Perron–Frobenius operator are investigated by calculating the Fokker–Planck operator with sufficiently small diffusivity. On the basis of numerical schemes thus established, we find that eigenfunctions for the subleading eigenvalues exhibit spatially inhomogeneous patterns, especially showing localization around the region where unstable manifolds are sparsely running. Finally, such spatial patterns of the eigenfunction are shown to be very close to the distribution of the maximal finite-time Lyapunov exponents.

Transport Barriers in Symplectic Maps

Chaotic transport is a subject of paramount importance in a variety of problems in plasma physics, specially those related to anomalous transport and turbulence. On the other hand, a great deal of information on chaotic transport can be obtained from simple dynamical systems like two-dimensional area-preserving (symplectic) maps, where powerful mathematical results like KAM theory are available. In this work we review recent works on transport barriers in area-preserving maps, focusing on systems which do not obey the so-called twist property. For such systems KAM theory no longer holds everywhere and novel dynamical features show up as non-resistive reconnection, shearless curves and shearless bifurcations. After presenting some general features using a standard nontwist mapping, we consider magnetic field line maps for magnetically confined plasmas in tokamaks.

Uniform Hyperbolicity of a Scattering Map with Lorentzian Potential

We show that a two-dimensional area-preserving map with Lorentzian potential is a topological horseshoe and uniformly hyperbolic in a certain parameter region. In particular, we closely examine the so-called sector condition, which is known to be a sufficient condition leading to the uniformly hyperbolicity of the system. The map will be suitable for testing the fractal Weyl law as it is ideally chaotic yet free from any discontinuities which necessarily invokes a serious effect in quantum mechanics such as diffraction or nonclassical effects. In addition, the map satisfies a reasonable physical boundary condition at infinity, thus it can be a good model describing the ionization process of atoms and molecules.

On the dynamics of a kicked harmonic oscillator

In this paper the dynamics of a kicked harmonic oscillator is studied. This is done by constructing a stroboscopic map for the kicked harmonic oscillator by following the time t-flow of the differential equation and taking phase portrait for some integrals multiple of the period of the kicked. The resulting map is a two-dimensional area-preserving (or symplectic) map. We provide a proof for the continuity of the position, and also a local stability analysis for the trivial equilibrium. Complete analysis for a general kick function for the 1:1, and for the 1:2 resonances are presented. For the 1:4 resonance, we describe the dynamics numerically for some specific kick functions.

On the Existence of Chaos in a Discontinuous Area-Preserving Map Arising in Financial Markets

In this paper, we further study a discontinuous piecewise-linear financial market model established in our previous paper. The model dynamics is driven by a two-dimensional discontinuous area-preserving map. We exploit its complex dynamics from the viewpoint of open conservative systems. We first give the classification of fixed points, and then theoretically present the existence of periodic saddle orbit and a homoclinic chaos for some parameter setting. We finally apply the Conley–Moser conditions to verify the existence of Smale horseshoe-like dynamics and chaos for another parameter setting. The result is helpful for understanding the internal change rule of the finance market.

Chaotic magnetic field lines and fractal structures in a tokamak with magnetic limiter

Open hamiltonian systems have typically fractal structures underlying chaotic dynamics with a number of physical consequences on transport. We consider such fractal structures related to the formation of a chaotic magnetic field line region near the tokamak wall and the corresponding field line dynamics, described by a two-dimensional area-preserving map. We focus on the exit basins, which are sets of points which originate orbits escaping through some exit and are typically fractal, also exhibiting the so-called Wada property. We show qualitative as well as quantitative evidences of these properties.

Scaling invariance of the diffusion coefficient in a family of two-dimensional Hamiltonian mappings

We consider a family of two-dimensional nonlinear area-preserving mappings that generalize the Chirikov standard map and model a variety of periodically forced systems. The action variable diffuses in increments whose phase is controlled by a negative power of the action and hence effectively uncorrelated for small actions, leading to a chaotic sea in phase space. For larger values of the action the phase space is mixed and contains a family of elliptic islands centered on periodic orbits and invariant Kolmogorov-Arnold-Moser (KAM) curves. The transport of particles along the phase space is considered by starting an ensemble of particles with a very low action and letting them evolve in the phase until they reach a certain height h. For chaotic orbits below the periodic islands, the survival probability for the particles to reach h is characterized by an exponential function, well modeled by the solution of the diffusion equation. On the other hand, when h reaches the position of periodic islands, the diffusion slows markedly. We show that the diffusion coefficient is scaling invariant with respect to the control parameter of the mapping when h reaches the position of the lowest KAM island.

Diffraction and tunneling in systems with mixed phase space

The role of diffraction is investigated for two-dimensional area-preserving maps with sharply or almost sharply divided phase space, in relation to the issue of dynamical tunneling. The diffraction effect is known to appear in general when the system contains indifferentiable or discontinuous points. We find that it controls the quantum transition between regular and chaotic regions in mixed phase space in the case where the border between these regions is set to be sharp. However, its manifestation is rather subtle: it would be possible to identify the diffraction effect under suitable coordinates if the support of the wave function contains indifferentiable or discontinuous points, whereas it is mixed with the tunneling effect and the whole process becomes hybrid if the support does not contain the sources of diffraction. We make detailed analyses, including the semiclassical treatment of edge contributions of the one-step propagator, to clarify the nature of diffraction in mixed phase space. Our result implies that chaos does not play any roles in the regular-to-chaotic transition process when the phase space is sharply divided.

Scaling dynamics for a particle in a time-dependent potential well

Some dynamical properties for a classical particle confined in an infinitely deep box of potential containing a periodically oscillating square well are studied. The dynamics of the system is described by using a two-dimensional non-linear area-preserving map for the variables energy and time. The phase space is mixed and the chaotic sea is described using scaling arguments. Scaling exponents are obtained as a function of all the control parameters, extending the previous results obtained in the literature.

A family of stadium-like billiards with parabolic boundaries under scaling analysis

Some chaotic properties of a family of stadium-like billiards with parabolic focusing components, which is described by a two-dimensional nonlinear area-preserving map, are studied. Critical values of billiard geometric parameters corresponding to a sudden change of the maximal Lyapunov exponent are found. It is shown that the maximal Lyapunov exponent obtained for chaotic orbits of this family is scaling invariant with respect to the control parameters describing the geometry of the billiard. We also show that this behavior is observed for a generic one-parameter family of mapping with the nonlinearity given by a tangent function.

The Kepler map in the three-body problem

The Kepler map was derived by Petrosky (1986) and Chirikov and Vecheslavov (1986) as a tool for description of the long-term chaotic orbital behaviour of the comets in nearly parabolic motion. It is a two-dimensional area-preserving map, describing the motion of a comet in terms of energy and time. Its second equation is based on Kepler’s third law, hence the title of the map. Since 1980s the Kepler map has become paradigmatic in a number of applications in celestial mechanics and atomic physics. It represents an important kind of general separatrix maps. Petrosky and Broucke (1988) used refined methods of mathematical physics to derive analytical expressions for its single parameter. These methods became available only in the second half of the 20th century, and it may seem that the map is inherently a very modern mathematical tool. With the help of the Jacobi integral I show that the Kepler map, including analytical formulae for its parameter, can be derived by quite elementary methods. The prehistory and applications of the Kepler map are considered and discussed.

Existence of homoclinic orbits of an area-preserving map with a nonhyperbolic structure

Basic phenomena in chaos can be associated with the existence of homoclinic orbits. In this paper we provide a simple mathematical example of an existence of homoclinic orbits embedded on a nonhyperbolic invariant set. To do it, we study a two-dimensional area-preserving piecewise linear map. Exploiting its dynamical behaviors from the point of view of open conservative systems, we analytically show that outside of the existing nonhyperbolic invariant set which is enclosed by heteroclinic saddle connections of an unstable periodic orbit, dynamical behaviors exhibit unbounded, but inside of such invariant set, regular and chaotic motions. Moreover, on such the invariant set, we theoretically present the existence of a homoclinic orbit by proving homoclinic intersections of the stable and unstable manifolds of a saddle point and also provide numerical evidences for geometrical structures of the nonhyperbolic invariant set and dynamical behaviors.

Leading Pollicott-Ruelle Resonances and Transport in Area-Preserving Maps

The leading Pollicott-Ruelle resonance is calculated analytically for a general class of two-dimensional area-preserving maps. Its wave number dependence determines the normal transport coefficients. In particular, a general exact formula for the diffusion coefficient D is derived without any high stochasticity approximation, and a new effect emerges: The angular evolution can induce fast or slow modes of diffusion even in the high stochasticity regime. The behavior of D is examined for three particular cases: (i) the standard map, (ii) a sawtooth map, and (iii) a Harper map as an example of a map with a nonlinear rotation number. Numerical simulations support this formula.

Corrugated Waveguide under Scaling Investigation

Some scaling properties for classical light ray dynamics inside a periodically corrugated waveguide are studied by use of a simplified two-dimensional nonlinear area-preserving map. It is shown that the phase space is mixed. The chaotic sea is characterized using scaling arguments revealing critical exponents connected by an analytic relationship. The formalism is widely applicable to systems with mixed phase space, and especially in studies of the transition from integrability to nonintegrability, including that in classical billiard problems.

Creation of coherent structures in area-preserving maps

We propose a new method for the creation of coherent structures in two-dimensional area-preserving maps. The coherent structures are created by applying a parameter perturbation in the neighbourhood of the periodic points of the map. We demonstrate the success of our method in the case of the standard map and in two cases of maps which represent realistic fluid flows, the blinking vortex map of Aref and the journal bearing map of Chaiken. We also find the signatures of the coherent structures in quantifiers like the local exit time distribution and of the local Lyapunov exponents of the system. Our method may provide useful insights into the ways in which coherent structures are created in natural systems.

Volume-preserving maps, source-free systems and their local structures

In this paper, we study local structures of volume-preserving maps and source-free vector fields, which are defined in the Euclidean n-space Rn with n ≥ 3. First, we prove that any volume-preserving map, defined in some neighbourhood of the origin, can be represented as a composition of n − 1 essentially two-dimensional area-preserving maps. This result can be viewed as an analogue of the following known fact (Feng and Shang 1995 Volume-preserving algorithms for source-free dynamical systems Numer. Math. 71 451–63): any source-free vector field on Rn can be represented as a sum of n − 1 essentially two-dimensional Hamiltonian vector fields. Then, we present a local representation of source-free vector fields under volume-preserving coordinate changes. Finally, we construct a Lie algebra of skew-symmetric tensor potentials of second order associated with source-free vector fields. The Lie algebra turns out to be isomorphic to the Lie algebra of source-free vector fields.

A new type of period-doubling scaling behavior in two-dimensional area-preserving map

A novel type of period-doubling scaling behavior in two-dimensional area-preserving maps is reported, a conservative analog of the critical behavior in period-doubling one-dimensional maps with quartic extremum. We present data of numerical solution of the two-dimensional version of the Feigenbaum–Cvitanović RG equation and accurate estimates for the universal constants. Illustrations are given for self-similarity in the phase space and in the parameter space of the model map.

Scaling properties for a classical particle in a time-dependent potential well

Some scaling properties for a classical particle interacting with a time-dependent square-well potential are studied. The corresponding dynamics is obtained by use of a two-dimensional nonlinear area-preserving map. We describe dynamics within the chaotic sea by use of a scaling function for the variance of the average energy, thereby demonstrating that the critical exponents are connected by an analytic relationship.