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A model for a lattice of coupled cat maps has been recently introduced. This new and specific choice of the coupling makes the description especially easy and nontrivial quantities as Lyapunov exponents determined exactly. We studied the ergodic property of the dynamics along such a chain for a local perturbation. While the perturbation spreads across a front growing ballistically, the position and momentum profiles show large fluctuations due to chaos leading to diffusive transport in the phase space. It provides an example where the diffusion can be directly inferred from the microscopic chaos., Comment: 28 pages, 7 figures

Quantum tunneling in a two-dimensional integrable map is studied. The orbits of the map are all confined to the curves specified by the one-dimensional Hamiltonian. It is found that the behavior of tunneling splitting for the integrable map and the associated Hamiltonian system is qualitatively the same, with only a slight difference in magnitude. However, the tunneling tails of the wave functions, obtained by superposing the eigenfunctions that form the doublet, exhibit significant difference. To explore the origin of the difference, we observe the classical dynamics in the complex plane and find that the existence of branch points appearing in the potential function of the integrable map could play the role for yielding non-trivial behavior in the tunneling tail. The result highlights the subtlety of quantum tunneling, which cannot be captured in nature only by the dynamics in the real plane., Comment: 11 pages, 10 figures

The strong enhancement of tunneling couplings typically observed in tunneling splittings in the quantum map is investigated. We show that the transition from instanton to noninstanton tunneling, which is known to occur in tunneling splittings in the space of the inverse Planck constant, takes place in a parameter space as well. By applying the absorbing perturbation technique, we find that the enhancement invoked as a result of local avoided crossings and that originating from globally spread interactions over many states should be distinguished and that the latter is responsible for the strong and persistent enhancement. We also provide evidence showing that the coupling across the separatrix in phase-space is crucial in explaining the behavior of tunneling splittings by performing the wave-function-based observation. In the light of these findings, we examine the validity of the resonance-assisted tunneling theory.

In recent years fractal Weyl laws and related quantum eigenfunction hypothesis have been studied in a plethora of numerical model systems, called quantum maps. In some models studied there one can easily prove uniform hyperbolicity. Yet, a numerically sound method for computing quantum resonance states, did not exist. To address this challenge, we recently introduced a new class of quantum maps. For these quantum maps, we showed that, quantum resonance states can numerically be computed using theoretically grounded methods such as complex scaling or weak absorbing potentials. However, proving uniform hyperbolicty for this class of quantum maps was not straight forward. Going beyond that work this article generalizes the class of scattering maps and provides mathematical proofs for their uniform hyperbolicity. In particular, we show that the suggested class of two-dimensional symplectic scattering maps satisfies the topological horseshoe condition and uniform hyperbolicity. In order to prove these properties, we follow the procedure developed in the paper by Devaney and Nitecki. Specifically, uniform hyperbolicity is shown by identifying a proper region in which the non-wandering set satisfies a sufficient condition to have the so-called sector bundle or cone field. Since no quantum map is known where both a proof of uniform hyperbolicity and a methodologically sound method for numerically computing quantum resonance states exist simultaneously, the present result should be valuable to further test fractal Weyl laws and related topics such as chaotic eigenfunction hypothesis., Comment: 50 pages, 23 figures

As a continuation of a previous paper (arXiv:2303.05769 [nlin.CD]), we introduce examples of H\'enon-type mappings that exhibit new horseshoe topologies in three and four dimensional spaces that are otherwise impossible in two dimensions., Comment: 39 pages, 19 figures

We derive a sufficient condition for topological horseshoe and uniform hyperbolicity of a 4-dimensional symplectic map, which is introduced by coupling the two 2-dimensional H\'enon maps via linear terms. The coupled H\'enon map thus constructed can be viewed as a simple map modeling the horseshoe in higher dimensions. We show that there are two different types of horseshoes, each of which is realized around different anti-integrable limits in the parameter regime., Comment: 26 pages, 12 figures

We study the effect of homogeneous noise on the escape rate of strongly chaotic area-preserving maps with a small opening. While in the noiseless dynamics the escape rate analytically depends on the instability of the shortest periodic orbit inside the hole, adding noise overall enhances escape, which, however, exhibits a non-trivial response to the noise amplitude, featuring an initial plateau and a successive rapid growth up to a saturation value. Numerical analysis is performed on cat maps with a hole, and the salient traits of the response to noise of the escape rate are reproduced analytically by an approximate model., Comment: 25 pages, 20 figures

We outline formal and physical similarities between the quantum dynamics of open systems, and the mesoscopic description of classical systems affected by weak noise. The main tool of our interest is the dissipative Wigner equation, that, for suitable timescales, becomes analogous to the Fokker-Planck equation describing classical advection and diffusion. This correspondence allows in principle to surmise a finite resolution, other than the Planck scale, for the quantized state space of the open system, particularly meaningful when the latter underlies chaotic classical dynamics. We provide representative examples of the quantum-stochastic parallel with noisy Hopf cycles and Van der Pol type oscillators., Comment: 14 pages, 6 figures

Viruses constantly undergo mutations with genomic changes. The propagation of variants of viruses is an interesting problem. We perform numerical simulations of the microscopic epidemic model based on network theory for the spread of variants. Assume that a small number of individuals infected with the variant are added to widespread infection with the original virus. When a highly infectious variant that is more transmissible than the original lineage is added, the variant spreads quickly to the wide space. On the other hand, if the infectivity is about the same as that of the original virus, the infection will not spread. The rate of spread is not linear as a function of the infection strength but increases non-linearly. This cannot be explained by the compartmental model of epidemiology but can be understood in terms of the dynamic absorbing state known from the contact process., Comment: accepted for publication in Scientific Reports

The subleading eigenvalues and associated eigenfunctions of the Perron-Frobenius operator for 2-dimensional area-preserving maps are numerically investigated. We closely examine the validity of the so-called Ulam method, a numerical scheme believed to provide eigenvalues and eigenfunctions of the Perron-Frobenius operator, both for linear and nonlinear maps on the torus. For the nonlinear case, the second-largest eigenvalues and the associated eigenfunctions of the Perron-Frobenius operator are investigated by calculating the Fokker-Planck operator with sufficiently small diffusivity. On the basis of numerical schemes thus established, we find that eigenfunctions for the subleading eigenvalues exhibit spatially inhomogeneous patterns, especially showing localization around the region where unstable manifolds are sparsely running. Finally, such spatial patterns of the eigenfunction are shown to be very close to the distribution of the maximal finite-time Lyapunov exponents., Comment: 38 pages, 22 figures

We report the numerical observation of scarring, that is enhancement of probability density around unstable periodic orbits of a chaotic system, in the eigenfunctions of the classical Perron-Frobenius operator of noisy Anosov ("cat") maps, as well as in the noisy Bunimovich stadium. A parallel is drawn between classical and quantum scars, based on the unitarity or non-unitarity of the respective propagators. For uniformly hyperbolic systems such as the cat map, we provide a mechanistic explanation for the classical phase-space localization detected, based on the distribution of finite-time Lyapunov exponents, and the interplay of noise with deterministic dynamics. Classical scarring can be measured by studying autocorrelation functions and their power spectra., Comment: 6 pages, 6 figures

The topology of complex classical paths is investigated to discuss quantum tunnelling splittings in one-dimensional systems. Here the Hamiltonian is assumed to be given as polynomial functions, so the fundamental group for the Riemann surface provides complete information on the topology of complex paths, which allows us to enumerate all the possible candidates contributing to the semiclassical sum formula for tunnelling splittings. This naturally leads to action relations among classically disjoined regions, revealing entirely non-local nature in the quantization condition. The importance of the proper treatment of Stokes phenomena is also discussed in Hamiltonians in the normal form., Comment: 28 pages,18 figures

We consider quantum maps induced by periodically-kicked scattering systems and discuss the computation of their resonance spectra in terms of complex scaling and sufficiently weak absorbing potentials. We also show that strong absorptive and projective openings, as commonly used for open quantum maps, fail to produce the resonance spectra of kicked scattering systems, even if the opening does not affect the classical trapped set. The results are illustrated for a concrete model system whose dynamics resembles key features of ionization and exhibits a trapped set which is organized by a topological horseshoe at large kick strength. Our findings should be useful for future tests of fractal Weyl conjectures and investigations of dynamical tunneling., Comment: 24 pages, 12 figures

The instanton-noninstanton (I-NI) transition in the tunneling process, which has been numerically observed in classically nonintegrable quantum maps, can be described by a perturbation theory based on an integrable Hamiltonian renormalized so as to incorporate the integrable part of the map. The renormalized perturbation theory is successfully applied to the two quantum maps, the H\'enon and standard maps. In spite of different nature of tunneling in the two systems, the I-NI transition exhibits very common characteristics. In particular, the manifestation of I-NI transition is obviously explained by a remarkable quenching of the renormalized transition matrix element. The enhancement of tunneling probability after the transition can be understood as a sudden change of the tunneling mechanism from the instanton to quite a different mechanism supported by classical flows just outside of the stable-unstable manifolds of the saddle on the top of the potential barrier., Comment: 6 pages, 4 figures

The enhancement of tunneling probability in the nearly integrable system is closely examined, focusing on tunneling splittings plotted as a function of the inverse of the Planck's constant. On the basis of the analysis using the absorber which efficiently suppresses the coupling creating spikes in the plot, we found that the splitting curve should be viewed as the staircase-shaped skeleton accompanied by spikes. We further introduce renormalized integrable Hamiltonians, and explore the origin of such a staircase structure by investigating the nature of eigenfunctions closely. It is found that the origin of the staircase structure could trace back to the anomalous structure in tunneling tail which manifests itself in the representation using renormalized action bases. This also explains the reason why the staircase does not appear in the completely integrable system.

We unveil chaotic behavior hidden in the energy spectrum of a Jahn-Teller ion vibrating in a cubic anharmonic potential as a typical model for rattling in cage-structure materials. When we evaluate the nearest-neighbor level-spacing distribution $P(s)$ of eigenenergies of the present oscillator system, we observe the transition of $P(s)$ from the Poisson to the Wigner distribution with the increase of cubic anharmonicity, showing the occurrence of chaos in the anharmonic Jahn-Teller vibration. The energy scale of the chaotic region is specified from the analysis of $P(s)$ and we discuss a possible way to observe chaotic behavior in the experiment of specific heat. It is an intriguing possibility that chaos in nonlinear physics could be detected by a standard experiment in condensed matter physics., Comment: 5 pages, 4 figures

In generic Hamiltonian systems tori of regular motion are dynamically separated from regions of chaotic motion in phase space. Quantum mechanically these phase-space regions are coupled by dynamical tunneling. We introduce a semiclassical approach based on complex paths for the prediction of dynamical tunneling rates from regular tori to the chaotic region. This approach is demonstrated for the standard map giving excellent agreement with numerically determined tunneling rates., Comment: 5 pages, 4 figures

A generalization of the Weyl law to systems with a sharply divided mixed phase space is proposed. The ansatz is composed of the usual Weyl term which counts the number of states in regular islands and a term associated with sticky regions in phase space. For a piecewise linear map, we numerically check the validity of our hypothesis, and find good agreement not only for the case with a sharply divided phase space, but also for the case where tiny island chains surround the main regular island. For the latter case, a non-trivial power law exponent appears in the survival probability of classical escaping orbits, which may provide a clue to develop the Weyl law for more generic mixed systems., Comment: 8 pages, 14 figures

We investigate dynamical tunneling in many dimensional systems using a quasi-periodically modulated kicked rotor, and find that the tunneling rate from the torus to the chaotic region is drastically enhanced when the chaotic states become delocalized as a result of the Anderson transition. This result strongly suggests that amphibious states, which were discovered for a one-dimensional kicked rotor with transporting islands [L. Hufnagel et al., Phys. Rev. Lett. 89, 154101 (2002)], quite commonly appear in many dimensional systems., Comment: 4 pages, 5 figures

Quantum tunneling in the presence of chaos is analyzed, focusing especially on the interplay between quantum tunneling and dynamical localization. We observed flooding of potentially existing tunneling amplitude by adding noise to the chaotic sea to attenuate the destructive interference generating dynamical localization. This phenomenon is related to the nature of complex orbits describing tunneling between torus and chaotic regions. The tunneling rate is found to obey a perturbative scaling with noise intensity when the noise intensity is sufficiently small and then saturate in a large noise intensity regime. A relation between the tunneling rate and the localization length of the chaotic states is also demonstrated. It is shown that due to the competition between dynamical tunneling and dynamical localization, the tunneling rate is not a monotonically increasing function of Planck's constant. The above results are obtained for a system with a sharp border between torus and chaotic regions. The validity of the results for a system with a smoothed border is also explained., Comment: 14 pages, 15 figures

The interplay between chaotic tunneling and dynamical localization in mixed phase space is investigated. Semiclassical analysis using complex classical orbits reveals that tunneling through torus regions and transport in chaotic regions are not independent processes, rather they are strongly correlated and described by complex orbits with both properties. This predicts a phenomenon analogous to the quantum suppression of classical diffusion: chaotic tunneling is suppressed as a result of dynamical localization in chaotic regions. This hypothesis is confirmed by numerical experiments where the effect of destructive interference is attenuated., Comment: 10 pages, 4 figures

For a higher order linear ordinary differential operator P, its Stokes curve bifurcates in general when it hits another turning point of P. This phenomenon is most neatly understandable by taking into account Stokes curves emanating from virtual turning points, together with those from ordinary turning points. This understanding of the bifurcation of a Stokes curve plays an important role in resolving a paradox recently found in the Noumi-Yamada system, a system of linear differential equations associated with the fourth Painleve equation., Comment: 7 pages, 4 figures

For a differentiable map $(x_1,x_2,..., x_n)\to (X_1,X_2,..., X_n)$ that has an inverse, we show that there exists a Nambu-Hamiltonian flow in which one of the initial value, say $x_n$, of the map plays the role of time variable while the others remain fixed. We present various examples which exhibit the map-flow correspondence., Comment: 19 pages

For a given differentiable map $(x,y)\to (X(x,y),Y(x,y))$, which has an inverse, we show that there exists a Hamiltonian flow in which x plays the role of the time variable while y is fixed., Comment: 4 pages

Isospectrality of planar domains which are obtained by successive unfolding of a fundamental building block is studied in relation to iso-length spectrality of the corresponding domains. Although an explicit and exact trace formula such as Poisson's summation formula or Selberg's trace formula is not known to exist for planar domains, equivalence between isospectrality and iso-length spectrality in a certain setting can be proved by employing the matrix representation of "transplantation of eigenfunctions". As an application of the equivalence, transplantable pairs of domains, which are all isospectral pairs of planar domains and therefore counter examples of Kac's question "can one hear the shape of a drum?", are numerically enumerated and it is found at least up to the domain composed of 13 building blocks transplantable pairs coincide with those constructed by the method due to Sunada., Comment: 12 pages, 5 figures, submitted to J.Phys.A