Amplitude death phenomena in delay--coupled Hamiltonian systems

Hamiltonian systems, when coupled {\it via} time--delayed interactions, do not remain conservative. In the uncoupled system, the motion can typically be periodic, quasiperiodic or chaotic. This changes drastically when delay coupling is introduced since now attractors can be created in the phase space. In particular for sufficiently strong coupling there can be amplitude death (AD), namely the stabilization of point attractors and the cessation of oscillatory motion. The approach to the state of AD or oscillation death is also accompanied by a phase--flip in the transient dynamics. A discussion and analysis of the phenomenology is made through an application to the specific cases of harmonic as well as anharmoniccoupled oscillators, in particular the H\'enon-Heiles system.Comment: To be appeared in Phys. Rev. E (2013

Revisiting linear augmentation for stabilizing stationary solutions: potential pitfalls and their application

Linear augmentation has recently been shown to be effective in targeting desired stationary solutions, suppressing bistablity, in regulating the dynamics of drive response systems and in controlling the dynamics of hidden attractors. The simplicity of the procedure is the highlight of this scheme but at the same time questions related to its general applicability still need to be addressed. Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios. Appropriate examples from conservative as well as dissipative systems are presented in this regard and potential applications for relevant observations in dissipative predator--prey systems are also discussed.Comment: updated version with title change, additional figures, text and explanation

Bifurcation Theory of Dynamical Chaos

The purpose of the present chapter is once again to show on concrete new examples that chaos in one-dimensional unimodal mappings, dynamical chaos in systems of ordinary differential equations, diffusion chaos in systems of the equations with partial derivatives and chaos in Hamiltonian and conservative systems are generated by cascades of bifurcations under universal bifurcation Feigenbaum-Sharkovsky-Magnitskii (FShM) scenario. And all irregular attractors of all such dissipative systems born during realization of such scenario are exclusively singular attractors that are the nonperiodic limited trajectories in finite dimensional or infinitely dimensional phase space any neighborhood of which contains the infinite number of unstable periodic trajectories

Dissipative chaotic scattering

We show that weak dissipation, typical in realistic situations, can have a metamorphic consequence on nonhyperbolic chaotic scattering in the sense that the physically important particle-decay law is altered, no matter how small the amount of dissipation. As a result, the previous conclusion about the unity of the fractal dimension of the set of singularities in scattering functions, a major claim about nonhyperbolic chaotic scattering, may not be observable.Comment: 4 pages, 2 figures, revte

Roundoff-induced attractors and reversibility in conservative two-dimensional maps

We numerically study two conservative two-dimensional maps, namely the baker map (whose Lyapunov exponent is known to be positive), and a typical one (exhibiting a vanishing Lyapunov exponent) chosen from the generalized shift family of maps introduced by C. Moore [Phys Rev Lett {\bf 64}, 2354 (1990)] in the context of undecidability. We calculated the time evolution of the entropy $S_q \equiv \frac{1-\sum_{i=1}^Wp_i^q}{q-1}$ ($S_1=S_{BG}\equiv -\sum_{i=1}^Wp_i \ln p_i$), and exhibited the dramatic effect introduced by numerical precision. Indeed, in spite of being area-preserving maps, they present, {\it well after} the initially concentrated ensemble has spread virtually all over the phase space, unexpected {\it pseudo-attractors} (fixed-point like for the baker map, and more complex structures for the Moore map). These pseudo-attractors, and the apparent time (partial) reversibility they provoke, gradually disappear for increasingly large precision. In the case of the Moore map, they are related to zero Lebesgue-measure effects associated with the frontiers existing in the definition of the map. In addition to the above, and consistently with the results by V. Latora and M. Baranger [Phys. Rev. Lett. {\bf 82}, 520 (1999)], we find that the rate of the far-from-equilibrium entropy production of baker map, numerically coincides with the standard Kolmogorov-Sinai entropy of this strongly chaotic system.Comment: Invited paper to appear in Physica A (PASI Meeting, Mar del Plata, December 2006); 12 pages including 7 figures. Version 2 has an improved Figure