The cat maps: quantum mechanics and classical motion

Periodic orbit is one of the most important issues among all kinds of large-scale motion types, and has been regarded as the breakthrough of three-body problem by Poincare. Specifically, to orbital dynamics around small bodies, there are at least two reasons for us to focus on the periodic motion, which is related to orbital evolution of real small body system, and is crucial for mission design of approaching the target body. Differing from equilibrium points, to search periodic orbits in the phase space is still an art to date, especially for the highly dimensional systems like Eq. 2.23. The orbital motion equation near a small body Eq. 2.22 has similar formulation with the equation of CRTBP, which suggests that it might include periodic motion of the same level of abundance as the latter. This chapter starts with a specific asteroid. Section 4.2 proposes an algorithm to search large-scale periodic orbits around irregular bodies, which is then applied to find out the periodic orbital families of the target small body. Section 4.3 surveys the stabilities of these orbits, and Sect. 4.4 further describes the topologies of different orbital families, based on which a classification method is proposed to track the topological evolution. Section 4.5 discusses the general motion forms about the periodic orbits, which discriminates different orbital patterns according to the linearized map on the section.

Eigenlevel correlation diagrams has proven to be a very useful tool to understand eigenstate characteristics of classically chaotic systems. In particular, we showed in a previous publication [Phys. Rev. Lett. 80, 944 (1998)0031-900710.1103/PhysRevLett.80.944] how to unveil the scarring mechanism, a cornerstone in the theory of quantum chaos, using the Planck constant as the correlation parameter. By increasing the Planck constant, we induced a transition from order to chaos, in which scarred wave functions appeared as the interaction of pairs of eigenstates in broad avoided crossings, forming a well-defined frontier in the correlation diagram. In this paper, we demonstrate that this frontier can be obtained by means of the semiclassical quantization of the involved scarring periodic orbits. Additionally, in order to calculate the Maslov index of each scarring periodic orbit, which is necessary for the semiclassical quantization procedure, we introduce a straightforward method based on Lagrangian descriptors. We illustrate the theory using the vibrational eigenstates of the LiCN molecular system.

We study the classical and quantum correspondence for a two-dimensional Sinai billiard system. By using the Stationary state expansion method and Gutzwiller's periodic orbit theory, we analyze the quantum length spectrum obtained through the Fourier transformation of the quantum density of state for the Sinai billiard system, and by comparing the peak position with the length of the classical periodic orbit we find their excellent correspondence. We observe that some quantum states are localized near some short period orbits, forming the quantum scarred states or superscarred states. In this paper we also investigate the nearest-neighbor spacing distribution of levels for both concentric and nonconcentric Sinai billiard systems, and find that the concentric Sinai billiard system is nearintegrable, and for the nonconcentric Sinai billiard system with =3/8 its nearest-neighbor spacing distribution of levels transits from nearintegrable to the Wigner distribution as the distance between the two centers increases.

Innovated by the nonlinear modes concept in the vibrational dynamics, the vertical periodic orbits around the triangular libration points are revisited for the Circular Restricted Three-body Problem. The $\zeta $ -component motion is treated as the dominant motion and the $\xi$ and $\eta $ -component motions are treated as the slave motions. The slave motions are in nature related to the dominant motion through the approximate nonlinear polynomial expansions with respect to the $\zeta $ -position and $\zeta $ -velocity during the one of the periodic orbital motions. By employing the relations among the three directions, the three-dimensional system can be transferred into one-dimensional problem. Then the approximate three-dimensional vertical periodic solution can be analytically obtained by solving the dominant motion only on $\zeta $ -direction. To demonstrate the effectiveness of the proposed method, an accuracy study was carried out to validate the polynomial expansion (PE) method. As one of the applications, the invariant nonlinear relations in polynomial expansion form are used as constraints to obtain numerical solutions by differential correction. The nonlinear relations among the directions provide an alternative point of view to explore the overall dynamics of periodic orbits around libration points with general rules.

An annular billiard is a dynamical system in which a particle moves freely in a disk except for elastic collisions with the boundary and also a circular scatterer in the interior of the disk. We investigate the stability properties of some periodic orbits in annular billiards in which the scatterer is touching or close to the boundary. We analytically show that there exist linearly stable periodic orbits of an arbitrary period for scatterers with decreasing radii that are located near the boundary of the disk. As the position of the scatterer moves away from a symmetry line of a periodic orbit, the stability of periodic orbits changes from elliptic to hyperbolic, corresponding to a saddle-center bifurcation. When the scatterer is tangent to the boundary, the periodic orbit is parabolic. We prove that slightly changing the reflection angle of the orbit in the tangential situation leads to the existence of Kolmogorov-Arnold-Moser islands. Thus, we show that there exists a decreasing to zero sequence of open intervals of scatterer radii, along which the billiard table is not ergodic.

The synchronization transition in coupled piecewise map lattices is studied. The average order parameter is calculated to diagnose synchronization of coupled piecewise maps. As a result, it shows that coexistence of a period cluster and synchronization states in the synchronization transition process. A detailed research shows that the period clusters consist of two coexisting period attractors, and their period orbit is an unstable periodic orbit of the isolate map. The unstable periodic orbit is stabilized again by the mutual interactions of the local nonlinearity and the spatial coupling. The influence of noise on the complete synchronization (CS) in coupled piecewise map systems is also studied in this work. Synchronization probability densities and average order parameters as a function of the coupling strength with various noise intensities are calculated. When the noise intensity is small, the noise can postpone synchronization transition processes. However, if the noise intensity is larger, the noise can destroy the period clusters states, shorten the synchronization transition and enhance the CS in coupled piecewise maps.

Haros graphs have been recently introduced as a set of graphs bijectively related to real numbers in the unit interval. Here we consider the iterated dynamics of a graph operator R over the set of Haros graphs. This operator was previously defined in the realm of graph-theoretical characterization of low-dimensional nonlinear dynamics and has a renormalization group (RG) structure. We find that the dynamics of R over Haros graphs is complex and includes unstable periodic orbits of arbitrary period and nonmixing aperiodic orbits, overall portraiting a chaotic RG flow. We identify a single RG stable fixed point whose basin of attraction is associated with the set of rational numbers, and find periodic RG orbits that relate to (pure) quadratic irrationals and aperiodic RG orbits, related with (nonmixing) families of nonquadratic algebraic irrationals and transcendental numbers. Finally, we show that the graph entropy of Haros graphs is globally decreasing as the RG flows towards its stable fixed point, albeit in a strictly nonmonotonic way, and that such graph entropy remains constant inside the periodic RG orbit associated to a subset of irrationals, the so-called metallic ratios. We discuss the possible physical interpretation of such chaotic RG flow and put results regarding entropy gradients along RG flow in the context of c-theorems.

The semiclassical limit of the quantum few-body problem has not been studied in general terms from the point of view of periodic orbit theory. In a previous paper, we studied noninteracting two-body systems [Phys. Rev. A 62, 042109 (2000)] and discussed the fact that the periodic orbits occur in continuous families. Interactions destroy the periodic orbit families leaving a discrete set of isolated periodic orbits. In this paper, we consider the effect of weak two-body interactions, which can be thought of as symmetry-breaking perturbations and can thus be analyzed using a theory developed by Creagh [Ann. Phys. (N.Y.) 248, 1 (1996)]. The P\"oschl-Teller two-body system confined in a square well is analyzed to illustrate the use of the formalism. It is shown that the effect of the interaction can be evaluated for all two-particle periodic orbits, and that the coarse-grained quantum density of states can be fully reproduced from simply summing the perturbed contributions of each periodic orbit family. Good numerical estimates of the quantum singlet energies can actually be obtained, but it is found that that perturbed trace formulas cannot reproduce the multiplet splittings predicted from quantum mechanics. Several interesting properties are observed depending on the range of the interaction and on whether the interaction is attractive or repulsive.

Classical and quantum mechanics of a strongly chaotic billiard system

Let f : $M\to M$ be a $C^{1+\varepsilon}$-map on a smooth Riemannian manifold $M$ and let $\Lambda\subset M$ be a compact $f$-invariant locally maximal set. In this paper we obtain several results concerning the distribution of theperiodic orbits of $f|_\Lambda$. These results are non-invertible and, in particular, non-uniformly hyperbolic versions of well-known results by Bowen, Ruelle, and others in the case of hyperbolic diffeomorphisms. We show that the topological pressure Ptop$(\varphi)$ can be computed by the values of the potential $\varphi$ on the expanding periodic orbits and also that every hyperbolic ergodic invariant measure is well-approximated by expanding periodic orbits. Moreover, we provethat certain equilibrium states are Bowen measures. Finally, we derive a large deviation resultfor the periodic orbits whose time averages are apart from the space average of a given hyperbolic invariant measure.

Chaotic motion of a harmonically bound charged particle in a magnetic field, in the presence of a half-plane barrier

The quantum form-factor K( tau )-the Fourier transform of the spectral auto-correlation function-may be represented semiclassically in terms of a sum over classical periodic orbits. We consider the problem of how this approximation behaves in the limit of long (scaled) time tau . It is shown that whilst K itself tends to unity, the periodic-orbit sum typically grows exponentially as tau to infinity . This behaviour is related to the fact that leading-order semiclassical quantization methods yield complex eigenvalues with imaginary parts that are of higher order in Planck's constant. Divergence from the quantum limit begins when tau = tau *(h(cross)), which, for typical two-degrees-of-freedom systems and maps, is shown to be independent of h(cross) as h(cross) to 0. In the case of the baker's map, however, quantum diffraction from the classical discontinuity instead causes the analogue of tau * to tend to zero like N-12/, Where N is the integer that corresponds to the inverse of Planck's constant. This is in agreement with recent numerical studies. Finally, we consider the implications of the semiclassical divergence studied here for the method developed by Argaman et al. (1993) of investigating correlations between the periodic orbits of chaotic systems.

Periodic orbits as the skeleton of classical and quantum chaos

We consider a system having decaying states with complex energies ${E}_{j}\ensuremath{-}i{\ensuremath{\Gamma}}_{j}/2$, and we define the ``width-weighted spectrum'' as $D\ensuremath{\Gamma}(E)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{j}^{}{\ensuremath{\Gamma}}_{j}\ensuremath{\delta}({E\ensuremath{-}E}_{j})$. We derive a semiclassical formula for this width-weighted spectrum, a formula which is analogous to the periodic-orbit representation of the density of states. The formula applies if classical motion is regular and if decay occurs by tunneling through a barrier. The semiclassical formula involves not periodic orbits or closed orbits, but action integrals associated with irreducible loops on the ``extremal torus,'' on which the particle climbs up the barrier and hangs at the top. Calculations confirm the validity of the formula.

A nonrelativistic superparticle moving freely on the super-Riemann surface of genus g≥2 is investigated. The classical motion is characterized by the super-Fuchsian group. The periodic orbits are unstable and the system is classically chaotic. Quantization is performed in the path integral method. The kernel function is explicitly given in the semiclassical approximation. The quantized energy spectrum is related to the length spectrum of the classical periodic orbits, which is a superanalog of Selberg’s trace formula.