Structure, size, and statistical properties of chaotic components in a mixed-type Hamiltonian system

An investigation of escape and scaling properties of a billiard system

We investigate some statistical properties of escaping particles in a billiard system whose boundary is described by two control parameters with a hole on its boundary. Initially, we analyze the survival probability for different hole positions and sizes. We notice that the survival probability follows an exponential decay with a characteristic power-law tail when the hole is positioned partially or entirely over large stability islands in phase space. We find that the survival probability exhibits scaling invariance with respect to the hole size. In contrast, the survival probability for holes placed in predominantly chaotic regions deviates from the exponential decay. We introduce two holes simultaneously and investigate the complexity of the escape basins for different hole sizes and control parameters by means of the basin entropy and the basin boundary entropy. We find a non-trivial relation between these entropies and the system's parameters and show that the basin entropy exhibits scaling invariance for a specific control parameter interval.

Power-law decay of the fraction of the mixed eigenstates in kicked top model with mixed-type classical phase space

The properties of mixed eigenstates in a generic quantum system with a classical counterpart that has mixed-type phase space, although important to understand several fundamental questions that arise in both theoretical and experimental studies, are still not clear. Here, following a recent work [Č. Lozej, D. Lukman, and M. Robnik, Phys. Rev. E 106, 054203 (2022)2470-004510.1103/PhysRevE.106.054203], we perform an analysis of the features of mixed eigenstates in a time-dependent Hamiltonian system, the celebrated kicked top model. As a paradigmatic model for studying quantum chaos, the kicked top model is known to exhibit both classical and quantum chaos. The types of eigenstates are identified by means of the phase-space overlap index, which is defined as the overlap of the Husimi function with regular and chaotic regions in classical phase space. We show that the mixed eigenstates appear due to various tunneling precesses between different phase-space structures, while the regular and chaotic eigenstates are, respectively, associated with invariant tori and chaotic components in phase space. We examine how the probability distribution of the phase-space overlap index evolves with increasing system size for different kicking strengths. In particular, we find that the relative fraction of mixed states exhibits a power-law decay as the system size increases, indicating that only purely regular and chaotic eigenstates are left in the strict semiclassical limit. We thus provide further verification of the principle of uniform semiclassical condensation of Husimi functions and confirm the correctness of the Berry-Robnik picture.

Spectral Form Factor and Dynamical Localization

Quantum dynamical localization occurs when quantum interference stops the diffusion of wave packets in momentum space. The expectation is that dynamical localization will occur when the typical transport time of the momentum diffusion is greater than the Heisenberg time. The transport time is typically computed from the corresponding classical dynamics. In this paper, we present an alternative approach based purely on the study of spectral fluctuations of the quantum system. The information about the transport times is encoded in the spectral form factor, which is the Fourier transform of the two-point spectral autocorrelation function. We compute large samples of the energy spectra (of the order of 106 levels) and spectral form factors of 22 stadium billiards with parameter values across the transition between the localized and extended eigenstate regimes. The transport time is obtained from the point when the spectral form factor transitions from the non-universal to the universal regime predicted by random matrix theory. We study the dependence of the transport time on the parameter value and show the level repulsion exponents, which are known to be a good measure of dynamical localization, depend linearly on the transport times obtained in this way.

Stickiness and recurrence plots: An entropy-based approach

The stickiness effect is a fundamental feature of quasi-integrable Hamiltonian systems. We propose the use of an entropy-based measure of the recurrence plots (RPs), namely, the entropy of the distribution of the recurrence times (estimated from the RP), to characterize the dynamics of a typical quasi-integrable Hamiltonian system with coexisting regular and chaotic regions. We show that the recurrence time entropy (RTE) is positively correlated to the largest Lyapunov exponent, with a high correlation coefficient. We obtain a multi-modal distribution of the finite-time RTE and find that each mode corresponds to the motion around islands of different hierarchical levels.

Phenomenology of quantum eigenstates in mixed-type systems: Lemon billiards with complex phase space structure

The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between their centers, as introduced by Heller and Tomsovic [E. J. Heller and S. Tomsovic, Phys. Today 46, 38 (1993)0031-922810.1063/1.881358]. This paper is a continuation of our recent papers on a classical and quantum ergodic lemon billiard (B=0.5) with strong stickiness effects [Č. Lozej et al., Phys. Rev. E 103, 012204 (2021)2470-004510.1103/PhysRevE.103.012204], as well as on the three billiards with a simple mixed-type phase space and no stickiness [Č. Lozej et al., Nonlin. Phenom. Complex Syst. 24, 1 (2021)1817-245810.33581/1561-4085-2021-24-1-1-18]. Here we study two classical and quantum lemon billiards, for the cases B=0.1953,0.083, which are mixed-type billiards with a complex structure of phase space, without significant stickiness regions. A preliminary study of their spectra was published recently [ Č. Lozej, D. Lukman, and M. Robnik, Physics 3, 888 (2021)10.3390/physics3040055]. We calculate a very large number (10^{6}) of consecutive eigenstates and their Poincaré-Husimi (PH) functions, and analyze their localization properties by studying the entropy localization measure and the normalized inverse participation ratio. We introduce an overlap index, which measures the degree of the overlap of PH functions with classically regular and chaotic regions. We observe the existence of regular states associated with invariant tori and chaotic states associated with the classically chaotic regions, and also the mixed-type states. We show that in accordance with the Berry-Robnik picture and the principle of uniform semiclassical condensation of PH functions, the relative fraction of mixed-type states decreases as a power law with increasing energy, thus, in the strict semiclassical limit, leaving only purely regular and chaotic states. Our approach offers a general phenomenological overview of the structural and localization properties of PH functions in quantum mixed-type Hamiltonian systems.

The Iris billiard: Critical geometries for global chaos

We introduce the Iris billiard that consists of a point particle enclosed by a unit circle around a central scattering ellipse of fixed elongation (defined as the ratio of the semi-major to the semi-minor axes). When the ellipse degenerates to a circle, the system is integrable; otherwise, it displays mixed dynamics. Poincaré sections are presented for different elongations. Recurrence plots are then applied to the long-term chaotic dynamics of trajectories launched from the unstable period-2 orbit along the semi-major axis, i.e., one that initially alternately collides with the ellipse and the circle. We obtain numerical evidence of a set of critical elongations at which the system undergoes a transition to global chaos. The transition is characterized by an endogenous escape event, E, which is the first time a trajectory launched from the unstable period-2 orbit misses the ellipse. The angle of escape, θ_esc, and the distance of the closest approach, d_min, of the escape event are studied and are shown to be exquisitely sensitive to the elongation. The survival probability that E has not occurred after n collisions is shown to follow an exponential distribution.

Stickiness in generic low-dimensional Hamiltonian systems: A recurrence-time statistics approach

We analyze the structure and stickiness in the chaotic components of generic Hamiltonian systems with divided phase space. Following the method proposed recently in Lozej and Robnik [Phys. Rev. E 98, 022220 (2018)2470-004510.1103/PhysRevE.98.022220], the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid. We perform extensive numerical studies of three example systems: the Chirikov standard map, the family of Robnik billiards, and the family of lemon billiards. The filling of the cells is compared to the random model of chaotic diffusion, introduced in Robnik et al. [J. Phys. A: Math. Gen. 30, L803 (1997)JPHAC50305-447010.1088/0305-4470/30/23/003] for the description of transport in the phase spaces of ergodic systems. The model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential. In generic systems the stickiness induces correlations in the cell visits. The distribution of recurrence times exhibits a separation of timescales because of the dynamical trapping. We model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times. We introduce the variable S, which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness. We use S to globally assess the distributions of recurrence times. We find that in the bulk of the chaotic sea S=1, while S>1 in areas of stickiness. We present the results in the form of animated grayscale plots of the variable S in the largest chaotic component for the three example systems, included as supplemental material to this paper.

Statistical properties of the localization measure of chaotic eigenstates and the spectral statistics in a mixed-type billiard

We study the quantum localization in the chaotic eigenstates of a billiard with mixed-type phase space [J. Phys. A: Math. Gen. 16, 3971 (1983)JPHAC50305-447010.1088/0305-4470/16/17/014; J. Phys. A: Math. Gen. 17, 1049 (1984)JPHAC50305-447010.1088/0305-4470/17/5/027], after separating the regular and chaotic eigenstates, in the regime of slightly distorted circle billiard where the classical transport time in the momentum space is still large enough, although the diffusion is not normal. This is a continuation of our recent papers [Phys. Rev. E 88, 052913 (2013)PLEEE81539-375510.1103/PhysRevE.88.052913; Phys. Rev. E 98, 022220 (2018)2470-004510.1103/PhysRevE.98.022220]. In quantum systems with discrete energy spectrum the Heisenberg time t_{H}=2πℏ/ΔE, where ΔE is the mean level spacing (inverse energy level density), is an important timescale. The classical transport timescale t_{T} (transport time) in relation to the Heisenberg timescale t_{H} (their ratio is the parameter α=t_{H}/t_{T}) determines the degree of localization of the chaotic eigenstates, whose measure A is based on the information entropy. We show that A is linearly related to normalized inverse participation ratio. The localization of chaotic eigenstates is reflected also in the fractional power-law repulsion between the nearest energy levels in the sense that the probability density (level spacing distribution) to find successive levels on a distance S goes like ∝S^{β} for small S, where 0≤β≤1, and β=1 corresponds to completely extended states. We show that the level repulsion exponent β is empirically a rational function of α, and the mean 〈A〉 (averaged over more than 1000 eigenstates) as a function of α is also well approximated by a rational function. In both cases there is some scattering of the empirical data around the mean curve, which is due to the fact that A actually has a distribution, typically with quite complex structure, but in the limit α→∞ well described by the beta distribution. The scattering is significantly stronger than (but similar as) in the stadium billiard [Nonlin. Phenom. Complex Syst. (Minsk) 21, 225 (2018)] and the kicked rotator [Phys. Rev. E 91, 042904 (2015)PLEEE81539-375510.1103/PhysRevE.91.042904]. Like in other systems, β goes from 0 to 1 when α goes from 0 to ∞. β is a function of 〈A〉, similar to the quantum kicked rotator and the stadium billiard.

Characteristic times in the standard map

We study and compare three characteristic times of the standard map: the Lyapunov time t_{L}, the Poincaré recurrence time t_{r}, and the stickiness (or escape) time t_{st}. The Lyapunov time is the inverse of the Lyapunov characteristic number (L) and in general is quite small. We find empirical relations for the L as a function of the nonlinearity parameter K and of the chaotic area A. We also find empirical relations for the Poincaré recurrence time t_{r} as a function of the nonlinearity parameter K, of the chaotic area A, and of the size of the box of initial conditions ε. As a consequence, we find relations between t_{r} and L. We compare the distributions of the stickiness time and the Poincaré recurrence time. The stickiness time inside the sticky regions at the boundary of the islands of stability is orders of magnitude smaller than the Poincaré recurrence time t_{r} and this affects the diffusion exponent μ, which converges always to the value μ=1. This is shown in an extreme stickiness case. The diffusion is anomalous (ballistic motion) inside the accelerator mode islands of stability with μ=2 but it is normal everywhere outside the islands with μ=1. In a particular case of extreme stickiness, we find the hierarchy of islands around islands.