Nonintegrability Parameter Research Articles

Generic superweak chaos induced by Hall effect.

We introduce and study the "kicked Hall system" (KHS), i.e., charged particles periodically kicked in the presence of uniform magnetic (B) and electric (E) fields that are perpendicular to each other and to the kicking direction. We show that for resonant values of B and E and in the weak-chaos regime of sufficiently small nonintegrability parameter κ (the kicking strength), there exists a generic family of periodic kicking potentials for which the Hall effect from B and E significantly suppresses the weak chaos, replacing it by "superweak" chaos (SWC). This means that the system behaves as if the kicking strength were κ^{2} rather than κ. For E=0, SWC is known to be a classical fingerprint of quantum antiresonance, but it occurs under much less generic conditions, in particular only for very special kicking potentials. Manifestations of SWC are a decrease in the instability of periodic orbits and a narrowing of the chaotic layers, relative to the ordinary weak-chaos case. Also, for global SWC, taking place on an infinite "stochastic web" in phase space, the chaotic diffusion on the web is much slower than the weak-chaos one. Thus, the Hall effect can be relatively stabilizing for small κ. In some special cases, the effect is shown to cause ballistic motion for almost all parameter values. The generic global SWC on stochastic webs in the KHS appears to be the two-dimensional closest analog to the Arnol'd web in higher dimensional systems.

Rotating accelerator-mode islands

The existence of rotating accelerator-mode islands (RAIs), performing quasiregular motion in rotational resonances of order m > 1 of the standard map, is firmly established by an accurate numerical analysis of all the known data. It is found that many accelerator-mode islands for relatively small nonintegrability parameter K are RAIs visiting resonances of different orders m < or = 3. For sufficiently large K, one finds also "pure" RAIs visiting only resonances of the same order, m = 2 or m = 3. RAIs, even quite small ones, are shown to exhibit sufficient stickiness to produce an anomalous chaotic transport. The RAIs are basically different in nature from accelerator-mode islands in resonances of the "forced" standard map which was extensively studied recently in the context of quantum accelerator modes.

Avoided crossings in driven systems

We characterize the avoided crossings in a two-parameter, time-periodic system which has been the basis for a wide variety of experiments. By studying these avoided crossings in the near-integrable regime, we are able to determine scaling laws for the dependence of their characteristic features on the non-integrability parameter. As an application of these results, the influence of avoided crossings on dynamical tunneling is described and applied to the recent realization of multiple-state tunneling in an experimental system.

A discrete variational approach for investigation of stationary localized states in a discrete nonlinear Schrödinger equation, named IN‐DNLS

IN‐DNLS considered here is a countable infinite set of coupled one‐dimensional nonlinear ordinary differential difference equations with a tunable nonintegrability parameter. When this parameter vanishes, IN‐DNLS reduces to the famous integrable Ablowitz‐Ladik (AL) equation. The formation of unstaggered and staggered stationary localized states (SLSs) in IN‐DNLS is studied here using a discrete variational method. The functional form of stationary soliton of AL equation is used as the ansatz for SLSs. Derivation of the appropriate functional and its equivalence to the effective Lagrangian are presented. Formation of on‐site peaked and intersite peaked unstaggered SLSs and their dependence on the nonintegrability parameter are investigated. On‐site peaked states are found to be energetically stable. Results are explained using the effective mass picture. Also, the properties of staggered SLSs of Sievers‐Takeno‐ (ST‐) like mode and Page‐ (P‐) like mode are investigated and explained using the same effective mass picture. It is further shown here that an unstable SLS which is found in the truncated analysis of the problem does not survive in the exact calculation. For large‐width and small‐amplitude SLSs, the known asymptotic result for the amplitude is obtained. Further scope and possible extensions of this work are discussed.

The Universal Plateau Structure of Lyapunov Exponents for Period Doubling Cascade Attractors**The project supported by the National Basic Research Climbing-Up Project “Nonlinear Science” and National Natural Science Foundation of China (No. 1880132 and No. 49474216)

Using algebraic. analysis method for periodic orbits of Hénon map, we derive the boundary equations of stable window and Lyapunov exponent plateau region on the space of nonintegrability parameter and dissipation parameter . Ekom the real root of these equations, we obtain the plateau width of Lyapunov exponent and the stable window width for high periodic orbits. The calculated result of plateau structure ratio for period-4 orbit agrees with the conjectural analytical formula: . Hence our result presents further evidence of universal dependence of Lyapunov exponent plateau structure on the dissipation parameter for period doubling cascade attractors of nonlinear system in transition from order to chaos.

Band Husimi distributions and the classical-quantum correspondence on the torus

Band Husimi distributions (BHDs) are introduced in the quantum-chaos problem on a toral phase space. In the framework of this phase space, a quantum state must satisfy Bloch boundary conditions (BCs) on a torus and the spectrum consists of a finite number of levels for given BCs. As the BCs are varied, a level broadens into a band. The BHD for a band is defined as the uniform average of the Husimi distributions for all the eigenstates in the band. The generalized BHD for a set of adjacent bands is the average of the BHDs associated with these bands. BHDs are shown to be closer, in several aspects, to classical distributions than Husimi distributions for individual eigenstates. The generalized BHD for two adjacent bands is shown to be approximately conserved in the passage through a degeneracy between the bands as a nonintegrability parameter is varied. Finally, it is shown how generalized BHDs can be defined so as to achieve physical continuity under small variations of the scaled Planck constant. A generalization of the topological (Chern-index) characterization of the classical-quantum correspondence is then obtained.

Limited sensitivity to analyticity: a manifestation of quantum chaos

Classical Hamiltonian systems generally exhibit an intricate mixture of regular and chaotic motions on all scales of phase space. As a nonintegrability parameter K (the “strength of chaos”) is gradually increased, the analyticity domains of functions describing regular-motion components [e.g., Kolmogorov-Arnol’d-Moser (KAM) tori] usually shrink and vanish at the onset of global chaos (breakup of all KAM tori). It is shown that these phenomena have quantum-dynamical analogs in simple but representative classes of model systems, the kicked rotors and the two-sided kicked rotors. Namely, as K is gradually increased, the analyticity domain ℛQE of the quantum-dynamical eigenstates decreases monotonically, and the width of ℛQE in the global-chaos regime vanishes in the semiclassical limit. These phenomena are presented as particular aspects of a more general scenario: As K is increased, ℛQE gradually becomes less sensitive to an increase in the analyticity domain of the system.

Solitonlike solutions of the generalized discrete nonlinear Schrödinger equation.

We investigate the solution properties of a generalized discrete nonlinear Schr\"odinger equation describing a nonlinear lattice chain. The generalized equation interpolates between the integrable discrete Ablowitz-Ladik equation and the nonintegrable discrete Schr\"odinger equation. Special interest is paid to the creation of stationary localized solutions called breathers. To tackle this problem we apply a map approach and illuminate the linkage of homoclinic and heteroclinic map orbits with localized lattice solutions. The homoclinic and heteroclinic orbits correspond to exact nonlinear solitonlike eigenstates of the lattice. Normal forms and the Melnikov method are used for analytical determinations of homoclinic orbits. Nonintegrability of the map leads to soliton pinning on the lattice. The soliton pinning energy is calculated and it is shown that it can be tuned by varying the ratio of the nonintegrability parameter versus the integrability parameter. The heteroclinic map orbit is derived on the basis of a variational principle. Finally, we use homoclinic and heteroclinic orbits as initial conditions to excite designed stationary localized solutions of desired width in the dynamics of the discrete nonlinear Schr\"odinger equation. In this way we are able to construct coherent solitonlike structures of profile determined by the map parameters. \textcopyright{} 1996 The American Physical Society.

Chaotic streamlines in convective cells

Streamlines in Rayleigh-Benard cells are investigated. The emergence of tori and a robust chaotic region surrounding these tori in the Arter flow are studied. One may construct a Hamiltonian of the flow, which oscillates on a fast time scale, and whose average is conserved along the invariant tori. The chaotic region arises due to the intersection of unstable manifolds emanating from stagnation points, with constant average Hamiltonian surfaces. The scaling of the size of the chaotic region with the nonintegrability parameter is defined by the location of the stagnation points.

Quantum dynamics in a chaotic separatrix layer.

The chaotic dynamics in a separatrix layer are investigated using the ``separatrix map.'' Quantizing this map we show that localization effects inhibit the quantum transport across the separatrix layer. Depending on the size of the nonintegrability parameter, the mechanism of localization is connected with either the remnants of the barely broken separatrix, or with barely broken Kolmogorov-Arnold-Moser tori (``cantori'') inside the separatrix layer, or with dynamical localization due to destructive interference of random transition amplitudes.

Diffusion in Hamiltonian dynamical systems with many degrees of freedom.

Hamiltonian dynamical systems with many degrees of freedom are investigated using symplectic map lattices. It is shown that anomalous diffusion exists only up to some crossover time beyond which the diffusion is normal. A diffusion constant, which is inversely proportional to the crossover time, exhibits a faster than any power-law dependence on the nonintegrability parameter, strongly suggesting the relevance of a bound by Nekhoroshev for Arnold diffusion. The motion in the standard mapping is also reexamined to show that the flicker noise is seen only down to some crossover frequency.

Quantum chaos of periodically pulsed systems: Underlying complete integrability.

The integrability underlying the quantum mechanics of noninte- grable pulsed systems is examined along the line exploited by Nakamura and Lakshmanan [Phys. Rev. Lett. 57, 1661 (1986)]. Coupled dynamical equations for both quasienergies and quasieigenfunctions (rather than matrix elements) with a nonintegrability parameter \ensuremath{\lambda} taken as ``time'' are shown to be reduced to a classical Sutherland's system with internal complex-vector space. Their complete integrability together with constants of motion are also exhibited.

Complete integrability in a quantum description of chaotic systems.

For quantum bound systems whose classical versions are nonintegrable, coupled dynamical equations for both energy levels and eigenfunctions with a nonintegrability parameter $t$ taken as time are shown to be a completely integrable Calogero-Moser system in 1 + 1 dimensions with internal complex vector space. Lax forms and their complete algebraic solutions are given which, in place of statistical mechanics procedures, determine possible energy spectrum and wave-function patterns at an arbitrary value of $t$.

Semi-classical mechanics in phase space: A study of Wigner’s function

We explore the semi-classical structure of the Wigner functions Ψ( q,p ) representing bound energy eigenstates | Ψ 〉 for systems with f degrees of freedom. If the classical motion is integrable, the classical limit of Ψ is a delta function on the f -dimensional torus to which classical trajectories corresponding to |Ψ〉 are confined in the 2 f -dimensional phase space. In the semi-classical limit of Ψ ( ℏ small but not zero) the delta function softens to a peak of order Ψ−  f and the torus develops fringes of a characteristic ‘Airy’ form. Away from the torus,Ψ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When Ψ the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ⋲ ,the system passes through three semi-classical régimes as ℏ diminishes. (b) For states |Ψ〉 associated with regions in phase space filled with irregular trajectories, Ψ will be a random function confined near that region of the ‘energy shell’ explored by these trajectories (this region has more thanks dimensions). (c) For ⋲ ≠ 0, ℏ blurs the infinitely fine classical path structure, in contrast to the integrable case ⋲ = 0, where ℏ imposes oscillatory quantum detail on a smooth classical path structure.