Regular and Chaotic Dynamics最新文献

In this short note, we prove that singular Reeb vector fields associated with generic (b)-contact forms on three dimensional manifolds with compact embedded critical surfaces have either (at least) (2N) or an infinite number of escape orbits, where (N) denotes the number of connected components of the critical set. In case where the first Betti number of a connected component of the critical surface is positive, there exist infinitely many escape orbits. A similar result holds in the case of (b)-Beltrami vector fields that are not (b)-Reeb. The proof is based on a more detailed analysis of the main result in [19].

For total collision solutions of the (n)-body problem, Chazy showed that the overall size of the configuration converges to zero with asymptotic rate proportional to (|T-t|^{frac{2}{3}}) where (T) is thecollision time. He also showed that the shape of the configuration converges to the set ofcentral configurations. If the limiting central configuration is nondegenerate, the rate of convergence of the shape is of order (O(|T-t|^{p})) for some (p>0). Here we show by example that in the planar four-bodyproblem there exist total collision solutions whose shape converges to a degenerate central configuration at a rate which is slower that any power of (|T-t|).

A brake orbit for the N-body problem is a solution for which, at some instant,all velocities of all bodies are zero. We reprove two “lost theorems” regarding brake orbits and use them to establish some surprising properties of the completion of theJacobi – Maupertuis metric for the N-body problem at negative energies.

Affine transformations in Euclidean space generate a correspondence between integrable systemson cotangent bundles to a sphere, ellipsoid and hyperboloid embedded in (R^{n}). Using thiscorrespondence and the suitable coupling constant transformations, we can get real integrals of motion in the hyperboloid case starting with real integrals of motion in the sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic and sextic polynomials in momenta.

We discuss the holomorphic properties of the complex continuation of the classical Arnol’d – Liouville action-angle variables for real analytic 1 degree-of-freedom Hamiltonian systems dependingon external parameters in suitable Generic Standard Form, with particular regard to the behaviour near separatrices.In particular, we show that near separatrices the actions, regarded as functions of the energy, have a special universal representation in terms of affine functions of the logarithm with coefficientsanalytic functions.Then, we study the analyticity radii of the action-angle variables in arbitrary neighborhoods of separatrices and describe their behaviour in terms of a (suitably rescaled) distance from separatrices.Finally, we investigatethe convexity of the energy functions (defined as the inverse of the action functions) near separatrices, and prove that, in particular cases (in the outer regions outside the main separatrix, and in the case the potential is close to a cosine), the convexity is strictly defined, while in general it can be shown that inside separatrices there are inflection points.

This paper is a summary of results that prove the abundance ofone-dimensional strange attractors near a Shil’nikov configuration, as wellas the presence of these configurations in generic unfoldings ofsingularities in (mathbb{R}^{3}) of minimal codimension.Finding these singularities in families of vector fields is analytically possible and thus provides a tractable criterion for the existence of chaotic dynamics.Alternative scenarios for the possible abundance of two-dimensional attractors in higherdimension are also presented. The role of Shil’nikov configuration is now played by a certain type of generalised tangency which should occur for families of vector fields (X_{mu})unfolding generically some low codimension singularity in (mathbb{R}^{n})with (ngeqslant 4).

In studying general perturbations of a dissipative twist map depending on two parameters, a frequency (nu) and a dissipation (eta), the existence of a Cantor set (mathcal{C}) of curves in the ((nu,eta)) plane such that the corresponding equation possesses a Diophantine quasi-periodic invariant circle can be deduced, up to small values of the dissipation, as a direct consequence of a normal form theorem in the spirit of Rüssmann and the “elimination of parameters” technique. These circles are normally hyperbolic as soon as (etanot=0), which implies that the equation still possesses a circle of this kind for values of the parameters belonging to a neighborhood (mathcal{V}) of this set of curves. Obviously, the dynamics on such invariant circles is no more controlled and may be generic, but the normal dynamics is controlled in the sense of their basins of attraction.

As expected, by the classical graph-transform method we are able to determine a first rough region where the normal hyperbolicity prevails and a circle persists, for a strong enough dissipation (etasim O(sqrt{varepsilon}),) (varepsilon) being the size of the perturbation. Then, through normal-form techniques, we shall enlarge such regions and determine such a (conic) neighborhood (mathcal{V}), up to values of dissipation of the same order as the perturbation, by using the fact that the proximity of the set (mathcal{C})allows, thanks to Rüssmann’s translated curve theorem, an introduction of local coordinates of the type (dissipation, translation) similar to the ones introduced by Chenciner in [7].

In P. D. McSwiggen’s article, it was proposed Derived from Anosov type construction which leads to a partially hyperbolic diffeomorphism of the 3-torus. The nonwandering set of this diffeomorphism contains a two-dimensional attractor which consists of one-dimensional unstable manifolds of its points. The constructeddiffeomorphism admits an invariant one-dimensional orientable foliation such that it containsunstable manifolds of points of the attractor as its leaves. Moreover, this foliation has aglobal cross section (2-torus) and defines on it a Poincaré map which is a regular Denjoytype homeomorphism. Such homeomorphisms are the most natural generalization of Denjoyhomeomorphisms of the circle and play an important role in the description of the dynamicsof aforementioned partially hyperbolic diffeomorphisms. In particular, the topologicalconjugacy of corresponding Poincaré maps provides necessary conditions for the topologicalconjugacy of the restrictions of such partially hyperbolic diffeomorphisms totheir two-dimensional attractors. The nonwandering set of each regular Denjoy type homeomorphismis a Sierpiński set and each such homeomorphism is, by definition, semiconjugate to theminimal translation of the 2-torus. We introduce a complete invariant of topological conjugacyfor regular Denjoy type homeomorphisms that is characterized by the minimal translation,which is semiconjugation of the given regular Denjoy type homeomorphism, with a distinguished,no more than countable set of orbits.

In this work an exhaustive numerical and analytical investigation of the dynamics of a bi-parametric symplecticmap, the so-called rationalstandard map, at moderate-to-large values of theamplitude parameter is addressed. After reviewing the model, a discussion concerning an analyticaldetermination of the maximum Lyapunov exponent is provided together with thorough numerical experiments.The theoretical results are obtained in the limit of a nearly uniform distribution of the phase values.Correlations among phases lead to departures from the expected estimates.In this direction, a detailed study of the role of stable periodic islands of periods 1, 2 and 4 is included.Finally, an experimental relationship between the Lyapunov and instability times is shown,while an analytical one applies when correlations are irrelevant, which is the case, in general,for large values of the amplitude parameter.

In this study, we analyze a planar mathematical pendulum with a suspension point that oscillates harmonically in the vertical direction. The bob of the pendulum is electrically charged and is located between two wires with a uniform distribution of electric charges, both equidistant from the suspension point. The dynamics of this phenomenon is investigated. The system has three parameters, and we analyze the parametric stability of the equilibrium points, determining surfaces that separate the regions of stability and instability in the parameter space. In the case where the parameter associated with the charges is equal to zero, we obtain boundary curves that separate the regions of stability and instability for the Mathieu equation.