Statistics of Poincaré recurrences and the structure of the stochastic layer of a nonlinear resonance

Abstract

<dm:abstracts xmlns:dm="http://www.elsevier.com/xml/dm/dtd"><ce:abstract xmlns:ce="http://www.elsevier.com/xml/common/dtd" view="all" class="author" id="aep-abstract-id5"><ce:section-title>Publisher Summary</ce:section-title><ce:abstract-sec view="all" id="aep-abstract-sec-id6"><ce:simple-para id="fsabs019" view="all">It is shown that a resonance is a mimic of the pendulum motion for which there are two dynamical realms separated by its separatrix and two associated stable and unstable fixed points semi-classically, a quantum level (state) can be viewed as a subspace in the dynamical phase space. Therefore, levels can be classified as those lying below and above the separatrix (corresponding to a resonance in the energy scale). For a pendulum, its oscillating frequency is expected to be smaller as the motion is close to the separatrix because of the nonlinear effect. Quantum mechanically, this will be reflected in that the nearest level energy spacing for those levels around the separatrix will reach a minimum. This is the so-called Dixon dip. This concept is integrated by the quantal and classical analogs. We note that the Dixon dip appears not only at the levels sharing a well-defined polyad number as stated in the literature, but also when the resonance is very seriously perturbed by other complicated interactions, like in the systems of Henon–Heiles and quartic potentials for which resonance is hardly well-defined. However, for the molecular systems like H<ce:inf loc="post">2</ce:inf>O and DCN with multiple resonances, the Dixon dip will be destroyed by the overlapping of resonances which, as conjectured by Chirikov, will lead to chaos. This will be further analyzed by an independent Lyapunov exponent analysis as shown in this chapter below. The destruction of the Dixon dip under multiple resonances in the H<ce:inf loc="post">2</ce:inf>O system where the emerging of multiple resonances definitely will complicate the level structure and may cause the dip phenomenon to be less obvious. For the system of DCN, the dips are only apparent for the low and high levels while not for those levels in between. In terms of Chirikov's conjecture, this is because of the overlapping of resonances which will lead to chaos. Independently, this issue is analyzed in terms of the degree of chaoticity by the averaged Lyapunov exponent. These two results show consistent interpretations. This algorithm can be of potential value in the elucidation of the dynamical properties of the highly excited vibrational system, including the transitional state, of which multiple resonances, and therefore chaos, is prevailing.</ce:simple-para></ce:abstract-sec></ce:abstract></dm:abstracts>