A. Yilmaz, G. Hacibekiroglu, E. Bolcal, Y. Polatoglu
The authors of this paper studied Schrodinger wave equation to investiagate the chaotic behavior of atomic energy levels in relation with three quantum numbers n, l, m by means of derived inequality. It could give rise to the siplitting of atomic spectral lines. Keywords: Chaos, Schrodinger wave equation, atomic energy levels
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Coupled map lattices are a paradigm of higher-dimensional dynamical systems exhibiting spatio-temporal chaos. A special case of non-hyperbolic maps are one-dimensional map lattices of coupled Chebyshev maps with periodic boundary conditions, called chaotic strings. In this short note we show that the fine structure of the self energy of this chaotic string in the scaling region (i.e. for very small coupling) is retained if we reduce the length of the string to three lattice points.
Stationary wave solutions of the perturbed Korteweg-de Vries equation are considered in the presence of external hamiltonian perturbations. Conditions of their chaotic behaviour are studied with the help of Melnikov theory. For the homoclinic chaos Poincaré sections are constructed to demonstrate the complicated behaviour, and Lyapunov exponents are also numerically calculated.
Strong nonlinear effects combined with diffusive coupling may give rise to unpredictable evolution in spatially extended deterministic dynamical systems even in the presence of a fully negative spectrum of Lyapunov exponents. This regime, denoted as ``stable chaos'', has been so far mainly characterized by numerical studies. In this manuscript we investigate the mechanisms that are at the basis of this form of unpredictable evolution generated by a nonlinear information flow through the boundaries. In order to clarify how linear stability can coexist with nonlinear instability, we construct a …
For classical billiards we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distinguish chaotic from integrable systems. As examples of 2D chaotic billiards we considered the Bunimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral rigidity we found GOE behaviour consistent with predictions from random matrix theory. We studied transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, wh…
For a system of globally coupled chaotic maps with bistable behaviour we relate the rate function for large deviations in the system size at finite time to dynamical properties of the self consistent Perron-Frobenius operator (SCPFO) that describes the system in the infinite size limit.
The separation of the Schrödinger equation into a Markovian and an interference term provides a new insight in the quantum dynamics of classically chaotic systems. The competition between these two terms determines the localized or diffusive character of the dynamics. In the case of the Kicked Rotor, we show how the constrained maximization of the entropy implies exponential localization.
We characterize the synchronization of an array of coupled chaotic elements as a phase transition where order parameters related to the joint probability at two sites obey power laws versus the mutual coupling strength; the phase transition corresponds to a change in the exponent of the power law. Since these studies are motivated by the behaviour of the cortical neurons in cognitive tasks, we account for the short time available to any brain decision by studying how the mutual coupling affects the transient behaviour of a synchronization transition over a fixed time interval. We present a nov…
The concept of environmental interface is defined and analyzed from the point of view of the possible source of non-standard behaviour. The energy balance equation is written for the interface where all kinds of energy transfer occur. It is shown that under certain conditions, the discrete version of the equation for the temperature time rate turns in to the well-known logistic equation and the conditions for chaotic behaviour are studied. They are determined by the Lyapunov exponent. The realistic situation when the coefficients of the equation vary with time, is studied for the Earth-environ…
A three species food chain model is studied analytically as well as numerically. Integrability of the model is studied using Painleve analysis while chaotic behaviour is studied using numerical techniques, such as calculation of Lyapunov exponents, plotting the bifurcation diagram and phase plots. We correct and critically comment on the wrong results reported recently on this ecological model, in a paper by Rai ([1995] ``Modelling ecological systems'', Int. J. Bifurcation and Chaos 5, 537-543).
We predict synchronization of the chaotic dynamics of two atomic ensembles coupled to a heavily damped optical cavity mode. The atoms are dissipated collectively through this mode and pumped incoherently to achieve a macroscopic population of the cavity photons. Even though the dynamics of each ensemble are chaotic, their motions repeat one another. In our system, chaos first emerges via quasiperiodicity and then synchronizes. We identify the signatures of synchronized chaos, chaos, and quasiperiodicity in the experimentally observable power spectra of the light emitted by the cavity.
This paper considers properties of nonlinear waves and solitons of Korteweg-de Vries equation in the presence of external perturbation. For time-periodic hamiltonian perturbation the width of the stochastic layer is calculated. The conclusions about chaotic behaviour in long-period waves and solitons are inferred. Obtained theoretical results find experimental confirmation in experiments with the propagation of ion-acoustic waves in plasma.
The driven non-linear duffing osillator is a very good, and standard, example of a quantum mechanical system from which classical-like orbits can be recovered from unravellings of the master equation. In order to generated such trajectories in the phase space of this oscillator in this paper we use a the quantum jumps unravelling together with a suitable application of the correspondence principle. We analyse the measured readout by considering the power spectra of photon counts produced by the quantum jumps. Here we show that localisation of the wave packet from the measurement of the oscilla…
We study dissipative transport of spontaneously emitting atoms in a 1D standing-wave laser field in the regimes where the underlying deterministic Hamiltonian dynamics is regular and chaotic. A Monte Carlo stochastic wavefunction method is applied to simulate semiclassically the atomic dynamics with coupled internal and translational degrees of freedom. It is shown in numerical experiments and confirmed analytically that chaotic atomic transport can take the form either of ballistic motion or a random walking with specific statistical properties. The character of spatial and momentum diffusion…
A discontinuous area-preserving mapping derived from a sinusoidally-forced impacting system is studied. This system, the elastic impact oscillator, is very closely related to the accelerator models of particle physics such as the Fermi map. The discontinuity in the mapping is due to grazing which can have a surprisingly large effect upon the phase space. In particular, at the boundary of the stochastic sea, the discontinuity set and its images can act as a partial barrier which allows trajectories to move between chaotic and regular regions. The system at higher energies is also analysed and M…
We present evidence for chaotic dynamics within the spin-down rates of 17 pulsars originally presented by Lyne et al. Using techniques that allow us to re-sample the original measurements without losing structural information, we have searched for evidence of a strange attractor in the time series of frequency derivatives for each of the 17 pulsars. We demonstrate the effectiveness of our methods by applying them to a component of the Lorenz and Rössler attractors that were sampled with similar cadence to the pulsar time series. Our measurements of correlation dimension and Lyapunov exponent s…
This thesis focuses on the mechanisms of energy transport in multidimensional heterogeneous lattice models, studying in particular the case of the Klein-Gordon model of coupled anharmonic oscillators in one and two spatial dimensions. We report the effects of the type of the impurity (heterogeneity) parameter on the systems' transport properties and classifies the transport mechanisms of the nonlinear versions of the models into various dynamical regimes. We also identify computationally efficient numerical integration techniques, including the so-called symplectic integrators, and implement t…
Chaotic Behaviour of the Earth System in the Anthropocene
该研究基于Landau-Ginzburg模型，证明人类世中受人类活动影响的地球系统可出现混沌行为，还分析了人类影响遵循逻辑斯蒂映射时的分岔现象及行星边界相互作用对模型可预测性的影响。
We demonstrate a previously unobserved nonlinear phenomenon in an ultra-low loss quartz Bulk Acoustic Wave cavity ($Q>3\times10^9$), which only occurs below 20 milli-Kelvin in temperature and under relatively weak pumping. The phenomenon reveals the emergence of several stable equilibria (at least two foci and two nodes) and jumps between these quasi states at random times. The degree of this randomness as well as separations between levels can be controlled by the frequency of the incident carrier signal. It is demonstrated that the nature of the effect lays beyond the standard Duffing model.
Recovering trajectories of quantum systems whose classical counterparts display chaotic behavior has been a subject that has received a lot of interest over the last decade. However, most of these studies have focused on driven and dissipative systems. The relevance and impact of chaoticlike phenomena to quantum systems has been highlighted in recent studies which have shown that quantum chaos is significant in some aspects of quantum computation and information processing. In this paper we study a three-body system comprising of identical particles arranged so that the system's classical traj…
We consider the dynamics of a single atom submitted to periodic pulses of a far-detuned standing wave generated by a high-finesse optical cavity, which is an atomic version of the well-known ``kicked rotor''. We show that the classical phase-space map can be ``reconstructed'' by monitoring the transmission of the cavity. We also studied the effect of spontaneous emission on the reconstruction, and put limits to the maximum acceptable spontaneous emission rate.
A review of some recent results and ideas about the expected behaviour of large chaotic systems and fluids.
The mode-fluctuation distribution $P(W)$ is studied for chaotic as well as for non-chaotic quantum billiards. This statistic is discussed in the broader framework of the $E(k,L)$ functions being the probability of finding $k$ energy levels in a randomly chosen interval of length $L$, and the distribution of $n(L)$, where $n(L)$ is the number of levels in such an interval, and their cumulants $c_k(L)$. It is demonstrated that the cumulants provide a possible measure for the distinction between chaotic and non-chaotic systems. The vanishing of the normalized cumulants $C_k$, $k\geq 3$, implies a…
The understanding of the relationship between topology and behaviour in interconnected networks would allow to characterise and predict behaviour in many real complex networks since both are usually not simultaneously known. Most previous studies have focused on the relationship between topology and synchronisation. In this work, we provide analytical formulas that shows how topology drives complex behaviour: chaos, information, and weak or strong synchronisation; in multiplex networks with constant Jacobian. We also study this relationship numerically in multiplex networks of Hindmarsh-Rose n…
Modelling the intermittency of wind speed has got significant relevance on many fields, including wind power generation and distribution. Most of the available modelling techniques assume the temporal fluctuations in the wind is stochastic in nature and the underlying dynamics is best described by statistical methods or the probabilistic distribution. The advent of chaos theory have changed the perception about irregular fluctuations of dynamic systems and it has demonstrated that random-like fluctuations can also arise from deterministic chaotic systems. In this paper, we have analysed the de…
We study the quantum behaviour of chaotic billiards which exhibit classically diffusive behaviour. In particular we consider the stadium billiard and discuss how the interplay between quantum localization and the rich structure of the classical phase space influences the quantum dynamics. The analysis of this model leads to new insight in the understanding of quantum properties of classically chaotic systems.
Many dynamical systems of different complexity, e.g. 1D logistic map, the Lorentz equations, or real phenomena, like turbulent convection, show chaotic behaviour. Despite huge differences, the dynamical scenarios for these systems are strikingly similar: chaotic bands are born through the series of period doubling bifurcations and merge through interior crises. Within chaotic bands periodic windows are born through the tangent bifurcations, preceded by the intermittent behaviour. This is a universal behaviour of dynamical systems (Feigenbaum 1983). We demonstrate such behaviour in models of pu…
We consider a simple model of lossless interaction between a two-level single atom and a standing-wave single-mode laser field which creates a one-dimensional optical lattice. Internal dynamics of the atom is governed by the laser field which is treated to be classical with a large number of photons. Center-of-mass classical atomic motion is governed by the optical potential and the internal atomic degree of freedom. The resulting Hamilton-Schrödinger equations of motion are a five-dimensional nonlinear dynamical system with two integrals of motion. The main focus of the paper is chaotic atomi…
We review some recent work on the synchronization of coupled dynamical systems on a variety of networks. When nodes show synchronized behaviour, two interesting phenomena can be observed. First, there are some nodes of the floating type that show intermittent behaviour between getting attached to some clusters and evolving independently. Secondly, two different ways of cluster formation can be identified, namely self-organized clusters which have mostly intra-cluster couplings and driven clusters which have mostly inter-cluster couplings.
High dimensional tori and chaotic and intermittent transients in magnetohydrodynamic Couette flows
本研究通过磁化球形库埃特问题，探究高维不变环面分岔序列引发的混沌瞬态与间歇行为，发现危机事件破坏混沌吸引子并产生瞬态，且不稳定环面参与演化，其中一种序列呈现开-关间歇性。
Chaotic studies in Magnetic Dipoles
本文研究外部旋转磁场下磁针的混沌行为，通过引入阻尼与重力修正运动方程，分析分岔、成对耦合及二维阵列同步，并考察噪声对同步的影响。
A detailed analysis of the century long visual light curve of the long-period Mira star R Cygni is presented and discussed. The data were collected from the publicly available databases of the AFOEV, the BAAVSS and the VSOLJ. The full light curve consists of 26655 individual points obtained between 1901 and 2001. The light curve and its periodicity were analysed with help of the O-C diagram, Fourier analysis and time-frequency analysis. The results demonstrate the limitations of these linear methods. The next step was to investigate the possible presence of low-dimensional chaos in the light c…
We show that diffusion due to chaotic mixing in the Neighbourhood of the Sun may not be as relevant as previously suggested in erasing phase space signatures of past Galactic accretion events. For this purpose, we analyse Solar Neighbourhood-like volumes extracted from cosmological simulations that naturally account for chaotic orbital behaviour induced by the strongly triaxial and cuspy shape of the resulting dark matter haloes, among other factors. In the approximation of an analytical static triaxial model, our results show that a large fraction of stellar halo particles in such local volum…
Individual chaotic behaviour of the S-stars in the Galactic centre
本文通过N体模拟和符号回归研究银河系中心S星的个体混沌行为，发现其位置空间平均分离量随初始偏心率以e的2.3次方迅速增大。
A numerical study of the quantum double pendulum is conducted. A suitable quantum scaling is found which allows to have as the only parameters the ratios of the lengths and masses of the two pendula and a (quantum) gravity parameter containing Planck's constant. Comparison with classical and semiclassical results is used to understand the behaviour of the energy curves of the levels, to define regimes in terms of the gravity parameter, and to classify the (resonant) interactions among levels by connecting them to various classical phase space structures (resonance islands).
Derivatives and Constraints in Chaotic Flows: Asymptotic Behaviour and a Numerical Method
本文通过广义李雅普诺夫指数描述混沌流中轨迹对初始条件扰动的二阶响应，并利用两种矩阵分解方法分析导数的渐近行为与数值计算，揭示了导数满足的微分约束。
We analyze the origin and properties of the chaotic dynamics of two atomic ensembles in a driven-dissipative experimental setup, where they are collectively damped by a bad cavity mode and incoherently pumped by a Raman laser. Starting from the mean-field equations, we explain the emergence of chaos by way of quasiperiodicity -- presence of two or more incommensurate frequencies. This is known as the Ruelle-Takens-Newhouse route to chaos. The equations of motion have a $\mathbb{Z}_{2}$-symmetry with respect to the interchange of the two ensembles. However, some of the attractors of these equat…
Chaotic walking of cold atoms in a tilted optical lattice, created by two counter propagating running waves with an additional external field, is demonstrated theoretically and numerically in the semiclassical and Hamiltonian approximations. The effect consists in random-like changing the direction of atomic motion in a rigid lattice under the influence of a constant force due to a specific behavior of the atomic dipole-moment component that changes abruptly in a random-like manner while atoms cross standing-wave nodes. Chaotic walking generates a fractal-like scattering of atoms that manifest…
We study the quantum behaviour of the stadium billiard. We discuss how the interplay between quantum localization and the rich structure of the classical phase space influences the quantum dynamics. The analysis of this model leads to new insight in the understanding of quantum properties of classically chaotic systems.
Certain deterministic non-linear systems may show chaotic behaviour. Time series derived from such systems seem stochastic when analyzed with linear techniques. However, uncovering the deterministic structure is important because it allows for construction of more realistic and better models and thus improved predictive capabilities. This paper describes key features of chaotic systems including strange attractors and Lyapunov exponents. The emphasis is on state space reconstruction techniques that are used to estimate these properties, given scalar observations. Data generated from equations …
In this work we study the non-Markovian behaviour of a qubit coupled to an environment in which the corresponding classical dynamics change from integrable to chaotic. We show that in the transition region, where the dynamics has both regular islands and chaotic areas, the average non-Markovian behaviour is enhanced to values even larger than in the regular regime. This effect can be related to the non-Markovian behaviour as a function of the the initial state of the environment, where maxima are attained at the regions dividing separate areas in classical phase space, particularly at the bord…
In atomic nuclei, ordered and chaotic states generally coexist. In this paper the transition from ordered to chaotic states will be discussed in the framework of roto-vibrational and shell models. In particular for $^{160}Gd$, in the roto-vibrational model, the Poincarè sections clearly show the transition from order to chaos for different values of rotational frequency. Furthermore, the spectral statistics of low-lying states of several $fp$ shell nuclei are studied with realistic shell-model calculations.
We calculate near-threshold bound states and Feshbach resonance positions for atom + rigid-rotor models of the highly anisotropic systems Li+CaH and Li+CaF. We perform statistical analysis on the resonance positions to compare with the predictions of random matrix theory. For Li+CaH with total angular momentum $J=0$ we find fully chaotic behavior in both the nearest-neighbor spacing distribution and the level number variance. However, for $J>0$ we find different behavior due to the presence of a nearly conserved quantum number. Li+CaF ($J=0$) also shows apparently reduced levels of chaotic beh…
Group-theoretical approach is applied to study behavior of lossless two-level atoms in a standing-wave laser field. Due to the recoil effect, the internal and external atomic degrees of freedom become coupled. The internal dynamics is described quantum mechanically in terms of the SU(2) group parameters. The evolution operator is found in an explicit way after solving a single ODE for one of the group parameters. The translational motion in a standing wave is governed by the classical Hamilton equations which are coupled to the SU(2) group equations. It is shown that the full set of equations …
We consider two particles hopping on a chain with a contact interaction between them. At strong interaction, there is a molecular bound state separated by a direct gap from a continuous band of atomic states. Introducing weak disorder in the interaction, the molecular state becomes Anderson localized. At stronger disorder, part of the molecular band delocalizes and dissociates due to its hybridization to the atomic band. We characterize these different regimes by computing the density of states, the inverse participation ratio, the level-spacing statistics and the survival probability of an in…
We consider a general N-degree-of-freedom dissipative system which admits of chaotic behaviour. Based on a Fokker-Planck description associated with the dynamics we establish that the drift and the diffusion coefficients can be related through a set of stochastic parameters which characterize the steady state of the dynamical system in a way similar to fluctuation-dissipation relation in non-equilibrium statistical mechanics. The proposed relationship is verified by numerical experiments on a driven double well system.
We consider a basic model of the lossless interaction between a moving two-level atom and a standing-wave single-mode laser field. Classical treatment of the translational atomic motion provides the semiclassical Hamilton-Schrodinger equations which are a 5D nonlinear dynamical system with two integrals of motion. The atomic dynamics can be regular or chaotic in dependence on values of the control parameters, the atom-field detuning and recoil frequency. We develop a semiclassical theory of the chaotic atomic transport in terms of a random walk of the atomic electric dipole moment $u$. Based o…
For an attractive trapped Bose-Einstein condensate an imaginary three-body recombination loss term and an imaginary linear source term are usually included in the Gross-Pitaevskii (GP) equation for a proper account of dynamics. Under the action of an impulsive force, generated by suddenly changing the atomic interaction or the trapping potential, the solution of this complex GP equation for attractive interaction is found to lead to a very long term chaotic oscillation.
Confining and escaping magnetic field lines in Tokamaks: Analysis via symplectic map
针对托卡马克磁约束等离子体混合相空间混沌输运难题，提出基于递归分析和瞬态动力学研究的两种数值方法，揭示黏性与不变流形的影响，助力理解等离子体行为与壁相互作用。
Semiclassical multimode laser theory is extended to gas lasers with open two-dimensional resonators of arbitrary shape. The Doppler frequency shift of the linear-gain coefficient leads to an additional linear coupling between the modes, which, however, is shown to be negligible. The nonlinear laser equations simplify in the special case of wave-chaotic resonators. In the single-mode regime, the intensity of a chaotic laser, as a function of the mode frequency, displays a local minimum at the frequency of the atomic transition. The width of the minimum scales with the inhomogeneous linewidth, i…
A condition upon which sporadic bursts (intermittent behaviour) of the relative energy become possible is derived for the motion in the chaotic layer around the separatrix of non-linear resonance. This is a condition for the existence of a marginal resonance, i.e. a resonance located at the border of the layer. A separatrix map in Chirikov's form [Chirikov, B. V., Phys. Reports 52, 263 (1979)] is used to describe the motion. In order to provide a straightforward comparison with numeric integrations, the separatrix map is synchronized to the surface of the section farthest from the saddle point…
Spectral form factor in chaotic, localized, and integrable open quantum many-body systems
本文利用耗散谱形状因子研究开放量子多体系统的谱统计，发现混沌系统普遍呈现 Ginibre 系综的二次斜坡-平台行为，并识别出非厄米多体 Thouless 时间，通过十二个模型验证了随机矩阵理论普适性，同时分析了可积与多体局域系统的不同行为。
We investigate the emergence of chaotic dynamics in a quantum Fermi - Pasta - Ulam problem for anharmonic vibrations in atomic chains applying semi-quantitative analysis of resonant interactions complemented by exact diagonalization numerical studies. The crossover energy separating chaotic high energy phase and localized (integrable) low energy phase is estimated. It decreases inversely proportionally to the number of atoms until approaching the quantum regime where this dependence saturates. The chaotic behavior appears at lower energies in systems with free or fixed ends boundary conditions…
We investigate the chaotic behaviour of multiparticle systems, in particular DNA and graphene models, by applying methods of nonlinear dynamics. Using symplectic integration techniques, we present an extensive analysis of chaos in the Peyrard-Bishop-Dauxois (PBD) model of DNA. The chaoticity is quantified by the maximum Lyapunov exponent (mLE) across a spectrum of temperatures, and the effect of base pair (BP) disorder on the dynamics is studied. In addition to heterogeneity due to the ratio of adenine-thymine (AT) and guanine-cytosine (GC) BPs, the distribution of BPs in the sequence is analy…
We consider a multi-dimensional extension of the Thomas-Rössler (TR) systems, that was inspired by Thomas' earlier work on biological feedback circuits, and we report on our first results that shows its ability to sustain a spatio-temporal behaviour, reminiscent of chimera states. The novelty here being that its underlying mechanism is based on "chaotic walks" discovered by René Thomas during the course of his investigations on what he called Labyrinth Chaos. We briefly review the main properties of TR systems and their chaotic and hyperchaotic dynamics and discuss the simplest way of coupling…
I suggest a solution to a persistent mystery in the physics of turbulent shear flows: cumulus clouds rise to towering heights, practically without entraining the ambient medium, while apparently similar turbulent jets in general lose their identity within a small distance through entrainment and mixing. From dynamical systems computations on a model chaotic vortical flow, I show that entrainment and mixing due to coherent structures depend sensitively on the relative speeds of different portions of the flow. A small change in these speeds, effected for example by heating, drastically alters th…
We analyse the dynamics of the improved discretised version of the well known Izhikevich neuronmodel under the action of external electromagnetic field. It is found that the three-dimensional IZHmap shows rich dynamics. With the variation of the electromagnetic field, period-doubling routeto chaos in a repeating fashion is observed from the bifurcation diagram. Even the forward andbackward continuation bifurcation diagram which do not completely overlap suggests that there is multistability in the system. The phenomenon of bistability (coexistence of periodic and chaotic attractors) is observe…
Asymptotic behaviour of the confidence region in orbit determination for hyperbolic maps with a parameter
本文研究含参数双曲映射轨道确定中置信椭球的渐近行为，发现混沌情形下参数估计导致不确定性衰减速率不同，并给出非快于多项式衰减的条件，应用于分段扩张映射和环面变换。
Intensities of LEED and PED are analyzed from a statistical point of view. The probability distribution is compared with a Porter-Thomas law, characteristic of a chaotic quantum system. The agreement obtained is understood in terms of analogies between simple models and Berry's conjecture for a typical wavefunction of a chaotic system. The consequences of this behaviour on surface structural analysis are qualitatively discussed by looking at the behaviour of standard correlation factors.
The $M$-dimensional unitary matrix $S(E)$, which describes scattering of waves, is a strongly fluctuating function of the energy for complex systems such as ballistic cavities, whose geometry induces chaotic ray dynamics. Its statistical behaviour can be expressed by means of correlation functions of the kind $\left \langle S_{ij}(E+ε)S^†_{pq}(E-ε)\right\rangle$, which have been much studied within the random matrix approach. In this work, we consider correlations involving an arbitrary number of matrix elements and express them as infinite series in $1/M$, whose coefficients are rational func…
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