Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space

We investigate the dynamics of chaotic trajectories in simple yet physically important Hamiltonian systems with non-hierarchical borders between regular and chaotic regions with positive measures. We show that the stickiness to the border of the regular regions in systems with such a sharply divided phase space occurs through one-parameter families of marginally unstable periodic orbits and is characterized by an exponent \gamma= 2 for the asymptotic power-law decay of the distribution of recurrence times. Generic perturbations lead to systems with hierarchical phase space, where the stickiness is apparently enhanced due to the presence of infinitely many regular islands and Cantori. In this case, we show that the distribution of recurrence times can be composed of a sum of exponentials or a sum of power-laws, depending on the relative contribution of the primary and secondary structures of the hierarchy. Numerical verification of our main results are provided for area-preserving maps, mushroom billiards, and the newly defined magnetic mushroom billiards.Comment: To appear in Phys. Rev. E. A PDF version with higher resolution figures is available at http://www.pks.mpg.de/~edugal

A New Type of Irregular Motion in a Class of Game Dynamics Systems

A new type of asymptotic behavior in a game dynamics system is discovered. The system exhibits behavior which combines chaotic motion and attraction to heteroclinic cycles; the trajectory visits several unstable stationary states repeatedly with an irregular order, and the typical length of the stay near the steady states grows exponentially with the number of visits. The dynamics underlying this irregular motion is analyzed by introducing a dynamically rescaled time variable, and its relation to the low-dimensional chaotic dynamics is thus uncovered. The relation of this irregular motion with a strange type of instability of heteroclinic cycles is also examined.Comment: 7 pages (Revtex) + 4 figures (postscript

Cycle expansions for intermittent maps

In a generic dynamical system chaos and regular motion coexist side by side, in different parts of the phase space. The border between these, where trajectories are neither unstable nor stable but of marginal stability, manifests itself through intermittency, dynamics where long periods of nearly regular motions are interrupted by irregular chaotic bursts. We discuss the Perron-Frobenius operator formalism for such systems, and show by means of a 1-dimensional intermittent map that intermittency induces branch cuts in dynamical zeta functions. Marginality leads to long-time dynamical correlations, in contrast to the exponentially fast decorrelations of purely chaotic dynamics. We apply the periodic orbit theory to quantitative characterization of the associated power-law decays.Comment: 22 pages, 5 figure

Delay time modulation induced oscillating synchronization and intermittent anticipatory/lag and complete synchronizations in time-delay nonlinear dynamical systems

Existence of a new type of oscillating synchronization that oscillates between three different types of synchronizations (anticipatory, complete and lag synchronizations) is identified in unidirectionally coupled nonlinear time-delay systems having two different time-delays, that is feedback delay with a periodic delay time modulation and a constant coupling delay. Intermittent anticipatory, intermittent lag and complete synchronizations are shown to exist in the same system with identical delay time modulations in both the delays. The transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay with suitable stability condition is discussed. The intermittent anticipatory and lag synchronizations are characterized by the minimum of similarity functions and the intermittent behavior is characterized by a universal asymptotic $-{3/2}$ power law distribution. It is also shown that the delay time carved out of the trajectories of the time-delay system with periodic delay time modulation cannot be estimated using conventional methods, thereby reducing the possibility of decoding the message by phase space reconstruction.Comment: accepted for publication in CHAOS, revised in response to referees comment

Chaos in credit–constrained emerging economies with Leontief technology

This work provides a framework to analyze the role of financial development as a source of endogenous instability in emerging economies subject to moral hazard problems. We study a piecewise linear dynamic model describing a small open economy with a tradable good produced by internationally mobile capital and a country specific production factor, using Leontief technology. We demonstrate that emerging markets could be endogenously unstable when large capital in–flows increase risk and exacerbate asymmetric information problems, according to empirical evidence. Using bifurcation and stability analysis we describe the properties of the system attractors, we assess the plausibility for complex dynamics and we find out that border collision bifurcations can emerge.border collision bifurcations,,complex dynamics,,emerging economies,,CEECs,,Endogenous instability,,moral hazard,,piecewise linear map.