Regularity of critical invariant circles of non-twist maps (original) (raw)

Regularity of critical invariant circles of the standard nontwist map

2005

We study critical invariant circles of several noble rotation numbers at the edge of break-up for an area-preserving map of the cylinder, which violates the twist condition. These circles admit essentially unique parametrizations by rotational coordinates. We present a high accuracy computation of about 10 7 Fourier coefficients. This allows us to compute the regularity of the conjugating maps and to show that, to the extent of numerical precision, it only depends on the tail of the continued fraction expansion.

Regularity Properties of Critical Invariant Circles of Twist Maps, and Their Universality

Siam Journal on Applied Dynamical Systems, 2008

We compute accurately the golden critical invariant circles of several area-preserving twist maps of the cylinder. We define some functions related to the invariant circle and to the dynamics of the map restricted to the circle (for example, the conjugacy between the circle map giving the dynamics on the invariant circle and a rigid rotation on the circle). The global Hölder regularities of these functions are low (some of them are not even once differentiable). We present several conjectures about the universality of the regularity properties of the critical circles and the related functions. Using a Fourier analysis method developed by R. de la Llave and one of the authors, we compute numerically the Hölder regularities of these functions. Our computations show that -withing their numerical accuracy -these regularities are the same for the different maps studied. We discuss how our findings are related to some previous results: (a) to the constants giving the scaling behavior of the iterates on the critical invariant circle (discovered by Kadanoff and Shenker); (b) to some characteristics of the singular invariant measures connected with the distribution of iterates. Some of the functions studied have pointwise Hölder regularity that is different at different points. Our results give a convincing numerical support to the fact that the points with different Hölder exponents of these functions are interspersed in the same way for different maps, which is a strong indication that the underlying twist maps belong to the same universality class. *

Critical invariant circles in asymmetric and multiharmonic generalized standard maps

Communications in Nonlinear Science and Numerical Simulation, 2014

Invariant circles play an important role as barriers to transport in the dynamics of areapreserving maps. KAM theory guarantees the persistence of some circles for near-integrable maps, but far from the integrable case all circles can be destroyed. A standard method for determining the existence or nonexistence of a circle, Greene's residue criterion, requires the computation of long-period orbits, which can be difficult if the map has no reversing symmetry. We use de la Llave's quasi-Newton, Fourier-based scheme to numerically compute the conjugacy of a Diophantine circle conjugate to rigid rotation, and the singularity of a norm of a derivative of the conjugacy to predict criticality. We study near-critical conjugacies for families of rotational invariant circles in generalizations of Chirikov's standard map. A first goal is to obtain evidence to support the long-standing conjecture that when circles breakup they form cantori, as is known for twist maps by Aubry-Mather theory. The location of the largest gaps is compared to the maxima of the potential when anti-integrable theory applies. A second goal is to support the conjecture that locally most robust circles have noble rotation numbers, even when the map is not reversible. We show that relative robustness varies inversely with the discriminant for rotation numbers in quadratic algebraic fields. Finally, we observe that the rotation number of the globally most robust circle generically appears to be a piecewise-constant function in two-parameter families of maps.

Universal scaling of critical quasiperiodic orbits in a class of twist maps

Journal of Physics A: Mathematical and General, 1998

Recently we have shown that the fractal properties of the critical invariant circles of the standard map, as summarized by the f (α) spectrum and the generalized dimensions D(q), depend only on the tails in the continued fraction expansion of the corresponding rotation numbers in (Burić N, Mudrinić M and Todorović K 1997 J. Phys. A: Math. Gen. 30 L161). In the present paper this result is extended on the whole class of sufficiently smooth area-preserving twist maps of cylinders. We present numerical evidence that the f (α) and D(q) are the same for all critical invariant circles of any such map which have the rotation numbers with the same tail.

Equivalent classes of critical circles

Journal of Physics A: Mathematical and General, 1997

We present numerical evidence that the fractal properties of the critical invariant circles of a typical area-preserving twist map, as summarized by the f (α) spectrum and the generalized dimensions D(q), depend only on the tails in the continued fraction expansion of the corresponding rotation numbers. f (α) and D(q) are numerically the same for all critical invariant circles of the standard and sine maps which have the rotation numbers with the same periodic tail.

Natural boundaries for area-preserving twist maps

Journal of Statistical Physics, 1992

We consider KAM invariant curves for generalizations of the standard map of the form (x 0 ; y 0 ) = (x + y 0 ; y + "f(x)), where f(x) is an odd trigonometric polynomial. We study numerically their analytic properties by Pad e approximant method applied to the function which conjugates the dynamics to a rotation 7 ! + !. In the complex " plane, natural boundaries of di erent shapes are found. In the complex plane the analyticity region appears to be a strip bounded by a natural boundary, whose width tends linearly to 0 as " tends to the critical value.

Efficient and Reliable Algorithms for the Computation of Non-Twist Invariant Circles

Foundations of Computational Mathematics

This paper presents a methodology to study non-twist invariant circles and their bifurcations for area preserving maps. We recall that nontwist invariant circles are characterized not only by being invariant, but also by having some specified normal behavior. The normal behavior may endow them with extra stability properties (e.g. against external noise) and hence, they appear as design goals in some applications. Our methodology leads to efficient algorithms to compute and continue, with respect to parameters, non-twist invariant circles. The algorithms are quadratically convergent, have low storage requirement and low operations count per step. Furthermore, the algorithms are backed up by rigorous aposteriori theorems which give sufficient conditions guaranteeing the existence of a true non-twist invariant circle, provided an approximate invariant circle is known. Hence, one can compute confidently even very close to breakdown. With some extra effort, the calculations could be turned into computer assisted proofs. Our algorithms are also guaranteed to converge up to the breakdown of the invariant circles, then they are suitable to compute regions of parameters where the non-twist invariant circles exist. The calculations involved in the computation of the boundary of these regions are very robust, they do not require symmetries and can run without continuous manual adjustments. This paper contains a detailed description of our algorithms, the corresponding implementation and some numerical results, obtained by running the computer programs. In particular, we include estimates for two-dimensional parameter regions where non-twist invariant circles (with a prescribed frequency) exist. These numerical explorations lead to some new mathematical conjectures.

Asymptotic rigidity of scaling ratios for critical circle mappings

Ergodic Theory and Dynamical Systems, 1999

Let fff be a smooth homeomorphism of the circle having one cubic-exponent critical point and irrational rotation number of bounded combinatorial type. Using certain pull-back and quasiconformal surgical techniques, we prove that the scaling ratios of fff about the critical point are asymptotically independent of fff. This settles in particular the golden mean universality conjecture. We introduce the notion of a holomorphic commuting pair, a complex dynamical system that, in the analytic case, represents an extension of fff to the complex plane and behaves somewhat as a quadratic-like mapping. We define a suitable renormalization operator that acts on such objects. Through careful analysis of the family of entire mappings given by zmapstoz+theta−(1/2pi)sin2pizz\mapsto z+\theta -(1/2\pi)\sin{2\pi z}zmapstoz+theta−(1/2pi)sin2piz, theta\thetatheta real, we construct examples of holomorphic commuting pairs, from which certain necessary limit set pre-rigidity results are extracted. The rigidity problem for fff is thereby reduced to one of renormaliza...

On the mode-locking universality for critical circle maps

Nonlinearity, 1990

The conjectured universality of the Hausdorff dimension of the fractal set formed by the set of the irrational winding parameter values for critical circle maps is shown to follow from the universal scalings for quadratic irrational winding numbers.