Standard-like Maps Research Articles

Orbit determination for standard-like maps: asymptotic expansion of the confidence region in regular zones

We deal with the orbit determination problem for a class of maps of the cylinder generalizing the Chirikov standard map. The problem consists of determining the initial conditions and other parameters of an orbit from some observations. A solution to this problem goes back to Gauss and leads to the least squares method. Since the observations admit errors, the solution comes with a confidence region describing the uncertainty of the solution itself. We study the behavior of the confidence region in the case of a simultaneous increase of the number of observations and the time span over which they are performed. More precisely, we describe the geometry of the confidence region for solutions in regular zones. We prove an estimate of the trend of the uncertainties in a set of positive measure of the phase space, made of invariant curve. Our result gives an analytical proof of some known numerical evidences.

Experiments looking for theoretical predictions

Some models have global properties, discovered by means of massive simulations, which occur for parameter values which seem hard to predict. We consider three simple cases: (a) non-integrable 2D symplectic maps with separatrix splitting such that the related standard-like maps contain a periodic function far from a sinusoidal-like shape; (b) standard-like maps driven in a quasiperiodic way. In both cases one wants to study up to which values of the parameters there exist invariant curves/tori implying the existence of confined dynamics. The case (c) deals with 2D dissipative systems and looks for the existence of “thick” attractors, with dimension close to 2.

Self-Consistent Chaotic Transport in a High-Dimensional Mean-Field Hamiltonian Map Model

Self-consistent chaotic transport is studied in a Hamiltonian mean-field model. The model provides a simplified description of transport in marginally stable systems including vorticity mixing in strong shear flows and electron dynamics in plasmas. Self-consistency is incorporated through a mean-field that couples all the degrees-of-freedom. The model is formulated as a large set of $N$ coupled standard-like area-preserving twist maps in which the amplitude and phase of the perturbation, rather than being constant like in the standard map, are dynamical variables. Of particular interest is the study of the impact of periodic orbits on the chaotic transport and coherent structures. Numerical simulations show that self-consistency leads to the formation of a coherent macro-particle trapped around the elliptic fixed point of the system that appears together with an asymptotic periodic behavior of the mean field. To model this asymptotic state, we introduced a non-autonomous map that allows a detailed study of the onset of global transport. A turnstile-type transport mechanism that allows transport across instantaneous KAM invariant circles in non-autonomous systems is discussed. As a first step to understand transport, we study a special type of orbits referred to as sequential periodic orbits. Using symmetry properties we show that, through replication, high-dimensional sequential periodic orbits can be generated starting from low-dimensional periodic orbits. We show that sequential periodic orbits in the self-consistent map can be continued from trivial (uncoupled) periodic orbits of standard-like maps using numerical and asymptotic methods. Normal forms are used to describe these orbits and to find the values of the map parameters that guarantee their existence. Numerical simulations are used to verify the prediction from the asymptotic methods.

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.

Oseledets' splitting of standard-like maps.

For the class of differentiable maps of the plane and, in particular, for standard-like maps (McMillan form), a simple relation is shown between the directions of the local invariant manifolds of a generic point and its contribution to the finite-time Lyapunov exponents (FTLE) of the associated orbit. By computing also the point-wise curvature of the manifolds, we produce a comparative study between local Lyapunov exponent, manifold's curvature and splitting angle between stable/unstable manifolds. Interestingly, the analysis of the Chirikov-Taylor standard map suggests that the positive contributions to the FTLE average mostly come from points of the orbit where the structure of the manifolds is locally hyperbolic: where the manifolds are flat and transversal, the one-step exponent is predominantly positive and large; this behaviour is intended in a purely statistical sense, since it exhibits large deviations. Such phenomenon can be understood by analytic arguments which, as a by-product, also suggest an explicit way to point-wise approximate the splitting.

Invariant manifolds at infinity of the RTBP and the boundaries of bounded motion

Invariant manifolds of a periodic orbit at infinity in the planar circular RTBP are studied. To this end we consider the intersection of the manifolds with the passage through the barycentric pericenter. The intersections of the stable and unstable manifolds have a common even part, which can be seen as a displaced version of the two-body problem, and an odd part which gives rise to a splitting. The theoretical formulas obtained for a Jacobi constant C large enough are compared to direct numerical computations showing improved agreement when C increases. A return map to the pericenter passage is derived, and using an approximation by standard-like maps, one can make a prediction of the location of the boundaries of bounded motion. This result is compared to numerical estimates, again improving for increasing C. Several anomalous phenomena are described.

Twist number and order properties of periodic orbits

A less studied numerical characteristic of periodic orbits of area preserving twist maps of the annulus is the twist or torsion number, called initially the amount of rotation Mather (1984) [2]. It measures the average rotation of tangent vectors under the action of the derivative of the map along that orbit, and characterizes the degree of complexity of the dynamics.The aim of this paper is to give new insights into the definition and properties of the twist number and to relate its range to the order properties of periodic orbits. We derive an algorithm to deduce the exact value or a demi-unit interval containing the exact value of the twist number.We prove that at a period-doubling bifurcation threshold of a mini-maximizing periodic orbit, the new born doubly periodic orbit has the absolute twist number larger than the absolute twist of the original orbit after bifurcation. We give examples of periodic orbits having large absolute twist number, that are badly ordered, and illustrate how characterization of these orbits only by their residue can lead to incorrect results.In connection to the study of the twist number of periodic orbits of standard-like maps we introduce a new tool, called 1-cone function. We prove that the location of minima of this function with respect to the vertical symmetry lines of a standard-like map encodes a valuable information on the symmetric periodic orbits and their twist number.

Birkhoff Periodic Orbits in the Standard-Like Maps

The structure of Birkhoff periodic orbits in the standard-like maps is studied. It is proved that there exists a parameter value such that the Birkhoff elliptic periodic orbit of rotation number p/q has its orbital point in the dominant axis.

Non-Symmetric Periodic Points in Reversible Maps: Examples from the Standard Map

The existence of non-symmetric non-Birkhoff periodic points and non-symmetric periodic points of the accelerator mode in standard-like maps is proved. Positions of these points are approximately determined for large parameter values. The number of such points is shown to diverge as the parameter value goes to infinity. We have found two routes for the appearance of non-symmetric periodic points. One is the equi-period bifurcation of a symmetric periodic point and the other is simultaneous saddle-node bifurcations. These two bifurcations seem to disprove the necessity of ‘hidden symmetry’ introduced by Murakami et al.(2001). We do not know whether or not these are the only routes for the appearance of these points. Numerical examples are considered for the standard map.

Local scaling of the flux for standardlike maps.

Results of systematic numerical explorations of local scaling of the flux near a critical invariant circle for a class of standardlike maps are reported. The scaling law is universal and the universality class is determined apparently only by the tail in the rotation number of the critical invariant circle. The flux near just created cantori is also studied, and another type of scaling is indicated, which is a combination of the scaling laws in the critical and the strongly supercritical regimes.

Singularities of periodic orbits near invariant curves

We show numerically, for standard-like maps, how the singularities (in the complex parameter) of the function which conjugates the map to a rotation of rational period behave when the period goes to an irrational number. Furthermore, we propose a numerical method to extrapolate the radius of convergence of the series parametrizing the solution of periodic orbits. The results are compared with analyses performed by Padé approximants, Greene’s method, root criterion and the prediction by renormalization theory.

NONTWIST AREA PRESERVING MAPS WITH REVERSING SYMMETRY GROUP

The aim of this paper is to give a theoretical explanation of the rich phenomenology exhibited by nontwist mappings of the cylinder, in numerical experiments reported in [del-Castillo et al., 1996; Howard & Humpherys, 1995], and to give new insights on the dynamics of nontwist standard-like maps. Our approach is based on the reversing symmetric properties of the nontwist standard-like systems. We relate the bifurcation of periodic orbits of such maps to their position with respect to a group-invariant circle, and to a critical circle, at whose points the twist property is violated. Moreover, we show that appearance of the meanders, i.e. homotopically nontrivial invariant circles of the cylinder, that are not graphs of real functions of angular variable, is a global bifurcation of a curve on the cylinder, with nodal-type singularities, and invariant with respect to the action of a reversing symmetry group.

Poincaré - Melnikov - Arnold method for analytic planar maps

The Poincaré - Melnikov - Arnold method for planar maps gives rise to a Melnikov function defined by an infinite and ( a priori) analytically uncomputable sum. Under an assumption of meromorphicity, residues theory can be applied to provide an equivalent finite sum. Moreover, the Melnikov function turns out to be an elliptic function and a general criterion about non-integrability is provided. Several examples are presented with explicit estimates of the splitting angle. In particular, the non-integrability of non-trivial symmetric entire perturbations of elliptic billiards is proved, as well as the non-integrability of standard-like maps. 33E05

On the singularity structure of invariant curves of symplectic mappings.

We study invariant curves in standard-like maps conjugate to rigid rotation with complex frequencies. The main goal is to study the analyticity domain of the functions defined perturbatively by Lindstedt perturbation expansions. We argue, based on infinite-dimensional bifurcation theory that the boundary of analyticity should typically consist of branch points of order two and we verify it in some examples using nonperturbative numerical methods. We show that this nature of the singularities of the analyticity domain can explain previously reported numerical results and also suggests other numerical methods. (c) 1995 American Institute of Physics.

A family of integrable symplectic standard-like maps related to symmetric spaces

We propose a family of symplectic standard-like maps, providing finite-difference approximations for all integrable nonlinear oscillators related to the classical series of symmetric spaces, introduced by Fordy et al. The Lax pair representations are found and the complete integrability is proved for all but the BDI maps. For the latter a large but incomplete set of integrals is found. For the AIII system also the r-matrix structure is presented.

On the complex separatrices of some standard-like maps

We propose a new method for studying asymptotic behaviour of complex separatrices of some symplectic maps. Our approach gives an alternative method (as opposed to iterating the maps) for computing some constants to measure the splitting of separatrices. Details are presented for semistandard and cubic standard-like maps.

Splitting of separatrices and symbolic dynamics for some degenerate standard-like maps

We consider standard-like maps of class C 1 with parabolic fixed points and such that the second derivative has a jump discontinuity at the fixed points. An asymptotic formula for the splitting of the separatrices is obtained. The structure of certain chaotic set in the stochastic layer around the separatrices is described in terms of symbolic dynamics. It turns out, in particular, that swinging and rotating regimes of motion may alternate in time arbitrarily. Analogous results are stated for the so-called separatrix mapping and for the more smooth standard-like maps

New fixed points of the renormalization operator for invariant circles

The scaling behavior of a family of piecewise-linear standard-like maps is investigated. We find a line of period-two fixed points in the space of non-analytic functions for the renormalization operator for invariant circles with golden mean winding number. These fixed points influence analytic standard-like maps in that they make KAM tori reappear after they have disaapeared when the nonlinearity is increased.