Volume-preserving Maps Research Articles

Diffusion of magnetic field lines in a confined RFP plasma

A volume-preserving symplectic map is proposed to describe the magnetic field lines when the Taylor equilibriumis perturbed in a generic way. The standard scenario is observed by varying the perturbation strength, but the statistical properties in the chaotic regions are not simple due to the presence of boundaries and remnants of invariant structures. Simpler models of volume-preserving maps are proposed. The slowly modulated standard map captures the basic topological and statistical features. The diffusion is analytically described for large perturbations (above the break-up of the last KAM torus) in terms of correlation functions and for small perturbations using the adiabatic theory, provided that the modulation is sufficiently slow.

Synchronizing hyperchaotic volume-preserving maps and circuits

We show that it is possible to synthesize chaotic systems that have improved characteristics for use in communications. These include whiter spectrum, low-time correlation, hyperchaotic behavior, and little or no phase space structure. These systems are based on locally volume-preserving (or expanding) maps. We show how to construct a circuit that produces such characteristics.

Quadratic volume-preserving maps

We study quadratic, volume-preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the Hénon area-preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family of quadratic volume-preserving maps in three-space for which we find a normal form and study invariant sets. We also give an alternative proof of a theorem by Moser classifying quadratic symplectic maps.

Invariant tori and chaotic streamlines in the ABC flow

We study the dynamical system associated with fluid particle motions of the Arnold-Beltrami-Childress (ABC) flow, defined by x ̇ = A sin z + C cos y , y ̇ = B sin x + A cos z , z ̇ = C sin y + B cos x , where A, B, C are real parameters and | C| ⪡ 1. First, we reduce this system to action-angle-angle coordinates. Then, by using the new-KAM-like theorems for perturbations of a three-dimensional, volume-preserving map, we obtain the conditions of existence of invariant tori in the ABC flow. In addition, by using a high-dimensional generalization of the Melnikov method, we obtain the analytical criterion for the existence of chaotic streamlines in the ABC flow.

Chaotic motion of nonspherical particles settling in a cellular flow field

Small, spheroidal particles undergo a tumbling motion as they settle out under gravity through a fluid flow. This motion is in response to the fluid forces that act on the particle causing it to rotate due to the local fluid vorticity and rate of strain. In this paper the nonlinear system of differential equations governing the particle motion in a spatially periodic, cellular flow field is considered. The motion of spherical particles is completely regular, and in some regions of the flow they may be permanently suspended. Using recent results on volume-preserving maps, it is shown that this behavior can persist for nonspherical particles. More usually the tumbling motion is chaotic. This is characterized by results on Poincar\'e sections of the motion and determination of Lyapunov exponents. The response to flow vorticity and rate of strain is analyzed for both full three-dimensional motion and restricted planar motion to determine the main factors governing the chaotic motion.

Volume-preserving and volume-expanding synchronized chaotic systems

Many schemes for synchronizing chaotic systems have emerged in recent years. Several of them have been suggested for use in communications. In this paper we pay special attention to communications issues. We show that one can use dynamical systems techniques to produce synchronized, chaotic systems that have features that are desirable in private and secure communications systems. These features are (i) less structure or pattern to the attractor, (ii) very low time correlations, (iii) more than one positive Lyapunov exponent (hyperchaotic), (iv) synchronizable with one transmitted (scalar) signal, (v) rapid synchronization in few time steps, (vi) the ability to add information to the chaotic carrier in a nonlinear fashion, and (vii) easy to make and analyze. We accomplish this by introducing a simple expanding-folding view of $n$-dimensional maps which leads to volume-preserving and volume-expanding maps in many cases whose trajectories cover a large nonzero volume of phase space and which have the desired properties.

Average exit time for volume-preserving maps.

For a volume-preserving map, we show that the exit time averaged over the entry set of a region is given by the ratio of the measure of the accessible subset of the region to that of the entry set. This result is primarily of interest to show two things: First, it gives a simple bound on the algebraic decay exponent of the survival probability. Second, it gives a tool for computing the measure of the accessible set. We use this to compute the measure of the bounded orbits for the Henon quadratic map. (c) 1997 American Institute of Physics.

A study of the ANNNI model using a dissipative map

It is well known that extrema of classical, one-dimensional systems can be viewed as trajectories of a nonlinear, volume-preserving map. However, in general, thermodynamically stable extrema (i.e., local minima) of the free energy are numerically unstable trajectories of the map and so difficult to study by this technique. Here we explore a recent idea involving the use of a dissipative map to study the same kind of problem. Such a map may be designed so that its attractors are equal to, or close to, local minima of the energy. We apply such a dissipative map to the mean-field ANNNI model. We find that the technique reliably locates many kinds of metastable state: ferromagnetic, paramagnetic, commensurate, and incommensurate. However, the map also has two notable failings: it (apparently) has no chaotic attractors; and it fails to find correctly metastable states in a region of the phase diagram which is dominated by incommensurate states.

Numerical Study of Invariant Sets of a Quasiperiodic Perturbation of a Symplectic Map

We study the behavior of invariant setsof a volume-preserving map that is a quasiperiodic perturbation of a symplectic map, using approximation by periodic orbits. We present numerical results for analyticity domains of invariant surfaces, behavior after breakdown, and a critical function describing breakdown of invariant surfacesas a function of their rotation vectors. We discuss implications of our results to the existence of a renormalization group operator describing breakdown of invariant surfaces.

Nonergodicity, accelerator modes, and asymptotic quadratic-in-time diffusion in a class of volume-preserving maps.

Using an elementary application of Birkhoff's ergodic theorem, we give necessary and sufficient conditions for the existence of asymptotically ${\mathit{n}}^{2}$ diffusion (where n is an integer representing discrete time in the angle variables in a class of volume-preserving twist maps. We show that nonergodicity is the dynamical mechanism giving rise to this behavior. The influence of accelerator modes on diffusion is described. We discuss how additive noise changes the diffusive behavior and we investigate the effective-diffusivity dependence on bare diffusivity and accelerator modes. In particular, we find that the dependence of the effective-diffusivity coefficient on bare diffusivity is universal for small values of bare-diffusivity coefficient \ensuremath{\sigma} if asymptotic ${\mathit{n}}_{2}$ diffusion is present in the \ensuremath{\sigma}=0 case.

A topological invariant for volume preserving diffeomorphisms

AbstractFor a smooth diffeomorphism f in ℝn+2, which possesses an invariant n-torus , such that the restriction f is topologically conjugate to an irrational rotation, we define a number which represents the way the normal bundle to the torus asymptotically wraps around . We prove that this number is a topological invariant among volume-preserving maps. This result can be seen as a generalization of a theorem by Naishul, for which we give a simple proof.

On the integrability and perturbation of three-dimensional fluid flows with symmetry

The purpose of this paper is to develop analytical methods for studyingparticle paths in a class of three-dimensional incompressible fluid flows. In this paper we study three-dimensionalvolume preserving vector fields that are invariant under the action of a one-parameter symmetry group whose infinitesimal generator is autonomous and volume-preserving. We show that there exists a coordinate system in which the vector field assumes a simple form. In particular, the evolution of two of the coordinates is governed by a time-dependent, one-degree-of-freedom Hamiltonian system with the evolution of the remaining coordinate being governed by a first-order differential equation that depends only on the other two coordinates and time. The new coordinates depend only on the symmetry group of the vector field. Therefore they arefield-independent. The coordinate transformation is constructive. If the vector field is time-independent, then it possesses an integral of motion. Moreover, we show that the system can be further reduced toaction-angle-angle coordinates. These are analogous to the familiar action-angle variables from Hamiltonian mechanics and are quite useful for perturbative studies of the class of systems we consider. In fact, we show how our coordinate transformation puts us in a position to apply recent extensions of the Kolmogorov-Arnold-Moser (KAM) theorem for three-dimensional, volume-preserving maps as well as three-dimensional versions of Melnikov's method. We discuss the integrability of the class of flows considered, and draw an analogy with Clebsch variables in fluid mechanics.

Extension of ‘‘Renormalization of period doubling in symmetric four-dimensional volume-preserving maps’’

We numerically reexamine the scaling behavior of period doublings in four-dimensional volume-preserving maps in order to resolve a discrepancy between numerical results on scaling of the coupling parameter and the approximate renormalization results reported by Mao and Greene [Phys. Rev. A {\bf 35}, 3911 (1987)]. In order to see the fine structure of period doublings, we extend the simple one-term scaling law to a two-term scaling law. Thus we find a new scaling factor associated with coupling and confirm the approximate renormalization results.

Quasiperiodic structure of the stochastic web map.

The q-fold web map is extended to a volume-preserving map in q dimensions. The qth iterate of this map is invariant under translations by 2\ensuremath{\pi} in any of the coordinate directions. As a consequence, well-behaved invariant sets that are periodic in ${\mathit{openR}}^{\mathit{q}}$ produce quasiperiodic sets when restricted to a two-dimensional, irrationally placed subspace identified with the original phase space. A quasilattice of fixed points is constructed in this manner. It is conjectured that the stochastic web, considered as the restriction of a q-dimensional periodic set, is itself quasiperiodic.

Normal form theory for volume preserving maps

We discuss the theory of normal forms for volume preserving maps in the non-resonant case. The concept of normal forms is introduced by using the commutative properties with respect to some symmetry groups related to the resonances of the linear eigenvalues. The existence of formal solutions to the conjugation equation, which reduces a map to normal form, is discussed. We also analyze the asymptotic character of the normal forms series and we prove a general theorem which shows that the error between the normal form dynamics and the true dynamics can be exponentially small as a function of the radius of the chosen polydisk centered at the fixed point. As a consequence it is possible to construct analytic manifolds which are approximately invariant under the action of the initial map up to an error which is exponentially small. The connection with a possible KAM theory for volume-preserving maps is suggested. We also show that Noether's theorem can be generalized to volume preserving maps.

Break-up of the spiral mean torus in a volume-preserving map

We consider a modulated standard map, and study the break-up of the spiral mean torus: though the critical line seems smooth, the behaviour is not universal, some discussion on scaling features of the model is included.

Beyond all orders: Singular perturbations in a mapping

We consider a family ofq-dimensional (q>1), volume-preserving maps depending on a small parametere. Ase → 0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for smalle the heteroclinic connection breaks up and that the splitting between its components scales withe likeeγexp[-β/e]. We estimateβ using the singularities of thee → 0+ heteroclinic orbit in the complex plane. We then estimateγ using linearization about orbits in the complex plane. These estimates, as well as the assertions regarding the behavior of the functions in the complex plane, are supported by our numerical calculations.

Scaling pattern of period doubling in four dimensions.

The period-{ital M} ({ital M}=1 and 2) scaling pattern of period doubling in a symmetric four-dimensional volume-preserving map is studied. The period-doubling sequence repeats itself asymptotically from one bifurcation to the next in a period-1 bifurcation route, and to every other one in a period-2 bifurcation route. The parameter-scaling factors {gamma}{sub 1} and {gamma}{sub 2} in a bifurcation route depend on the bifurcation path. They are some combination of {delta}{sub 1} and {delta}{sub 2} (divergence rates from the period-{ital M} map of the renormalization transformation) and {delta}{sub 1}{sup {prime}} and {delta}{sub 2}{sup {prime}} (convergence rates to the period-{ital M} map). Therefore each bifurcation route is characterized by these four scaling factors. The values of {delta}{sub 1}, {delta}{sub 2}, {delta}{sub 1}{sup {prime}}, and {delta}{sub 2}{sup {prime}} are obtained by a numerical calculation and a renormalization analysis. We find that there are three kinds of period-1 scaling patterns and one kind of period-2 scaling pattern. {delta}{sub 1} and {delta}{sub 1}{sup {prime}} in any bifurcation route are the same as those in area-preserving maps; however, {delta}{sub 2} and {delta}{sub 2}{sup {prime}} depend on the bifurcation route.

A combinatorial algorithm for the euler equations of incompressible flows

A Lagrangian time discretization is introduced for the Euler equations of inviscid incompressible flows, using the least square projection onto the manifold of volume preserving diffeomorphisms. A further discretization is performed first by splitting the physical domain into N blocks of equal volume, then by approximating each volume-preserving mapping involved in the time discretization by a permutation of the N blocks. Finally, a combinatorial algorithm is obtained, where the classical assignment problem has to be solved at each time step.

Existence of KAM tori for four-dimensional volume-preserving maps

We study a four-dimensional volume-preserving map with two parameters. In the parameter plane, we compute the regions of linear stability for periodic orbits with winding numberspn/rn andqn/rn, which are rational approximants of a cubic irrational «KAM», pair (α1, α2). Our numerical results suggest that these stable regions converge to a limit region when the order,n, of the rational approximation increases. For parameter values in this limit region, there would exist a KAM torus with winding numbers (α1, α2). The rate of convergence of the stable regions or their boundaries does not appear to be very predictable,i.e. there seems to be no scaling.