The nature of bursting noises,stochastic resonance and deterministic chaos in excitable neurons

We study the Hindmarsh-Rose model of excitable neurons and show that in the asymptotic limit this monostable model can possess some kind of dynamical bistability: small-amplitude quasiharmonic and large-amplitude relaxational oscillations can be simultaneously excited and their formation is accompanied by a narrow hysteresis. We show that bursting noises, stochastic resonance and deterministic chaos are determined by random transitions between these two dynamical states under slow and small changes of one of the model variables (z). We find that these effects take place even for such model parameters when hysteresis transforms into a step and they disappear when this step is smoothed out enough. We analyze some characteristics and conditions of formation of the deterministic chaos. We emphasize that such dynamical bistability and the effects related to it are universal phenomena and occur in a wide class of dynamical systems of different nature including brusselator.     

Calculation of resonance Raman excitation profiles

The present work describes a novel mathematical scheme for the evaluation of multi-dimensional time domain integrals that arise in the calculation of resonance Raman excitation profiles and the optical absorption spectra when the electronic transitions take place between displaced, distorted, and rotated harmonic potential energy surfaces. A recipe is provided for formulating the time domain integrals in terms of a tractable summation of products involving hermite polynomials. Expressions for the special cases (displaced and/or distorted oscillators) are written in terms of mathematical special functions which are computationally easier to deal with. Relations with the so-called transform theory are discussed. Resonance Raman excitation profiles and the optical absorption spectrum for a 7-mode system are calculated as an illustration of our expressions.     

‘Phase transitions’ in bacteria – From structural transitions in free living bacteria to phenotypic transitions in bacteria within biofilms

Phase transitions are common in inanimate systems and have been studied extensively in natural sciences. Less explored are the rich transitions that take place at the micro- and nano-scales in biological systems. In conventional phase transitions, large-scale properties of the media change discontinuously in response to continuous changes in external conditions. Such changes play a significant role in the dynamic behaviours of organisms. In this review, we focus on some transitions in both free-living and biofilms of bacteria. Particular attention is paid to the transitions in the flagellar motors and filaments of free-living bacteria, in cellular gene expression during the biofilm growth, in the biofilm morphology transitions during biofilm expansion, and in the cell motion pattern transitions during the biofilm formation. We analyse the dynamic characteristics and biophysical mechanisms of these phase transition phenomena and point out the parallels between these transitions and conventional phase transitions. We also discuss the applications of some theoretical and numerical methods, established for conventional phase transitions in inanimate systems, in bacterial biofilms.     

荧光动力学的转移函数理论

为描述弱激发条件下存在能量传递的体系中的荧光动力学特性,引进了响应函数和转移函数,并由此建立了新的描述体系动力学过程的系统表述方法。通过一些实例的讨论,表明新方法是系统研究复杂体系荧光动力学的有效手段。 关键词：     

A mathematical framework for critical transitions: Bifurcations, fast-slow systems and stochastic dynamics

Bifurcations can cause dynamical systems with slowly varying parameters to transition to far-away attractors. The terms “critical transition” or “tipping point” have been used to describe this situation. Critical transitions have been observed in an astonishingly diverse set of applications from ecosystems and climate change to medicine and finance. The main goal of this paper is to give an overview which standard mathematical theories can be applied to critical transitions. We shall focus on early-warning signs that have been suggested to predict critical transitions and point out what mathematical theory can provide in this context. Starting from classical bifurcation theory and incorporating multiple time scale dynamics one can give a detailed analysis of local bifurcations that induce critical transitions. We suggest that the mathematical theory of fast-slow systems provides a natural definition of critical transitions. Since noise often plays a crucial role near critical transitions the next step is to consider stochastic fast-slow systems. The interplay between sample path techniques, partial differential equations and random dynamical systems is highlighted. Each viewpoint provides potential early-warning signs for critical transitions. Since increasing variance has been suggested as an early-warning sign we examine it in the context of normal forms analytically, numerically and geometrically; we also consider autocorrelation numerically. Hence we demonstrate the applicability of early-warning signs for generic models. We end with suggestions for future directions of the theory.     

Loschmidt Echo and Fidelity Decay Near an Exceptional Point

Non‐Hermitian classical and open quantum systems near an exceptional point (EP) are known to undergo strong deviations in their dynamical behavior under small perturbations or slow cycling of parameters as compared to Hermitian systems. Such a strong sensitivity is at the heart of many interesting phenomena and applications, such as the asymmetric breakdown of the adiabatic theorem, enhanced sensing, non‐Hermitian dynamical quantum phase transitions, and photonic catastrophe. Like for Hermitian systems, the sensitivity to perturbations on the dynamical evolution can be captured by Loschmidt echo and fidelity after imperfect time reversal or quench dynamics. Here, a rather counterintuitive phenomenon in certain non‐Hermitian systems near an EP is disclosed, namely the deceleration (rather than acceleration) of the fidelity decay and improved Loschmidt echo as compared to their Hermitian counterparts, despite large (non‐perturbative) deformation of the energy spectrum introduced by the perturbations. This behavior is illustrated by considering the fidelity decay and Loschmidt echo for the single‐particle hopping dynamics on a tight‐binding lattice under an imaginary gauge field.     

Bidirectional Partial Generalized Synchronization in Chaotic and Hyperchaotic Systems via a New Scheme

In this paper, a bidirectional partial generalized (lag, complete, and anticipated) synchronization of a class of continuous-time systems is defined. Then based on the active control idea, a new systematic and concrete scheme is developed to achieve bidirectional partial generalized (lag, complete, and anticipated) synchronization between two chaotic systems or between chaotic and hyperchaotic systems. With the help of symbolic-numerical computation, we choose the modified Chua system, Lorenz system, and the hyperchaotic Tamasevicius-Namajunas-Cenys system to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme. It is interesting that partial chaos synchronization not only can take place between two chaotic systems, but also can take place between chaotic and hyperchaotic systems. The proposed scheme can also be extended to research bidirectional partial generalized (lag, complete, and anticipated) synchronization between other dynamical systems.     

Equilibrium Phase Transitions in Coupled Map Lattices: A Pedestrian Approach

A class of piecewise linear coupled map lattices with simple symbolic dynamics is constructed. It can be solved analytically in terms of the statistical mechanics of spin lattices. The corresponding Hamiltonian is written down explicitly in terms of the parameters of the map. The approach follows the line of recent mathematical investigations. But the presentation is kept elementary so that phase transitions in the dynamical model can be studied in detail. Although the method works only for map lattices with repelling invariant sets some of the conclusions, i.e., the role of local curvature of the single site map and properties of the nearest neighbour coupling might play an important role for phase transitions in general dynamical systems.     

动力学变换法探测连续系统不稳定周期轨道

本文利用动力学变换方法和庞加莱截面方法对两种连续混沌动力学系统进行不稳定周期轨道探测研究, 并对Lorenz系统进行了替代数据法检验.结果表明:基于庞加莱截面的动力学变换改进算法 可有效探测连续混沌动力学系统中的不稳定周期轨道.     

Traffic states and jamming transitions induced by a bus in two-lane traffic flow

We study the traffic states and jamming transitions induced by a bus (slow car) in a two-lane traffic of cars. We use the dynamic model which is an extended one of the optimal velocity model to take into account the lane changing. The fundamental (flow-density) diagram is presented. The fundamental diagram changes highly by introducing a bus on a two-lane roadway. It is found that there are the six distinct states for the two-lane traffic flow including a bus. The spatio-temporal patterns are presented for the distinct traffic states. The dynamical state of traffic changes with density of cars. It is shown that the dynamical transitions among the distinct traffic states occur at some values of density. The phase diagram (region map) is shown for the two-lane traffic flow including a bus.     

Relaxation in establishing a steady-state stochastic process with 1/<Emphasis Type="Italic">f</Emphasis> spectrum and low-frequency spike statistics

According to theory, fluctuations with a power spectrum inversely proportional to frequency (1/f processes) may arise when dissimilar phase transitions simultaneously take place in physical systems with intense white noise. In this work, relaxation effects in establishing a steady-state stochastic process with non-equilibrium phase transitions are described in terms of two nonlinear stochastic differential equations. The results thus obtained carry information on the statistics of large-scale low-frequency spikes. Step “forgetting” of initial conditions is noted. It is numerically shown that the distributions of the durations and maximal values of extreme low-frequency spikes have a power-type form.     

Metallization of the melts of Se,S, I2 under high pressure

Abstract

In the melts of Se S and I₂ the nonmetal-metal transitions were found under pressure. For sulphur and selenium the transitions are accompanied by a decrease of the volume. For iodine two transitions take place. During the first one the volume changes very slightly, the second transition being accompanied by an essential decrease of the volume.

The existence of transitions of this kind gives an explanation for anomalies in the melting curves of some substances.     

Coherent regimes of globally coupled dynamical systems

This Letter presents a method by which the mean field dynamics of a population of dynamical systems with parameter diversity and global coupling can be described in terms of a few macroscopic degrees of freedom. The method applies to populations of any size and functional form in the region of coherence. It requires linear variation or a narrow distribution for the dispersed parameter. Although an approximation, the method allows us to quantitatively study the transitions among the collective regimes as bifurcations of the effective macroscopic degrees of freedom. To illustrate, the phenomenon of oscillator death and the route to full locking are examined for chaotic oscillators with time scale mismatch.     

Thermodynamics via Inducing

We consider a Turing-type reaction-diffusion system involving quadratic and cubic nonlinearities and numerically investigate the role of nonlinear terms in producing spots, stripes, labyrinths, hexagonal arrangement of spots, blotches and transitions among them. From our numerical experiments performed on a square domain with zero-flux boundary conditions, we observe that the system displays a form of multistability for which different stable spatial distribution of concentrations appear for a same set of control parameters depending upon the initial conditions. For varying values of model parameters both in the first- and the second-stage of simulations, we obtain a number of transition states that are found to be sensitive on the relative strength of the quadratic and cubic coupling terms. We obtain a graphical relationship among such model parameters at which the transitions take place.

     

Synchronization in networks of spatially extended systems

Synchronization processes in networks of spatially extended dynamical systems are analytically and numerically studied. We focus on the relevant case of networks whose elements (or nodes) are spatially extended dynamical systems, with the nodes being connected with each other by scalar signals. The stability of the synchronous spatio-temporal state for a generic network is analytically assessed by means of an extension of the master stability function approach. We find an excellent agreement between the theoretical predictions and the data obtained by means of numerical calculations. The efficiency and reliability of this method is illustrated numerically with networks of beam-plasma chaotic systems (Pierce diodes). We discuss also how the revealed regularities are expected to take place in other relevant physical and biological circumstances.     

Heterogeneous recurrence representation and quantification of dynamic transitions in continuous nonlinear processes

Many real-world systems are evolving over time and exhibit dynamical behaviors. In order to cope with system complexity, sensing devices are commonly deployed to monitor system dynamics. Online sensing brings the proliferation of big data that are nonlinear and nonstationary. Although there is rich information on nonlinear dynamics, significant challenges remain in realizing the full potential of sensing data for system control. This paper presents a new approach of heterogeneous recurrence analysis for online monitoring and anomaly detection in nonlinear dynamic processes. A partition scheme, named as Q-tree indexing, is firstly introduced to delineate local recurrence regions in the multi-dimensional continuous state space. Further, we design a new fractal representation of state transitions among recurrence regions, and then develop new measures to quantify heterogeneous recurrence patterns. Finally, we develop a multivariate detection method for on-line monitoring and predictive control of process recurrences. Case studies show that the proposed approach not only captures heterogeneous recurrence patterns in the transformed space, but also provides effective online control charts to monitor and detect dynamical transitions in the underlying nonlinear processes.     

Spin freezing transition and non-Fermi-liquid self-energy in a three-orbital model

A single-site dynamical mean-field study of a three band model with the rotationally invariant interactions appropriate to the t_(2g) levels of a transition metal oxide reveals a quantum phase transition between a paramagnetic metallic phase and an incoherent metallic phase with frozen moments. The Mott transitions occurring at electron densities n=2, 3 per site take place inside the frozen moment phase. The critical line separating the two phases is characterized by a self-energy with the frequency dependence Sigma(omega) approximately sqrt[omega] and a broad quantum critical regime. The findings are discussed in the context of the power law observed in the optical conductivity of SrRuO3.     

Instantaneous frequencies of a chaotic system

The structure and geometry of high-dimensional, complex dynamical systems is usually hidden under a profusion of numerical data. We show that time-frequency analysis allows one to analyze these data regardless of the number of degrees of freedom. Our method takes snapshots of the system in terms of its instantaneous frequencies defined as ridges of the time-frequency landscape. Using the wavelet transform of a single trajectory, it can characterize key dynamical properties like the extent of chaos, resonance transitions and trappings. Unité Mixte de Recherche (UMR 6207) du CNRS, et des universités Aix-Marseille I, Aix-Marseille II et du Sud Toulon-Var. Laboratoire affilié à la FRUMAM (FR 2291).     

Using invariants to change detection in dynamical system with chaos

Change detection is the crucial subject in dynamical systems. There are suitable methods for detecting changes for linear systems and some methods for nonlinear systems, but there is a lack of methods concerning chaotic systems. This paper presents change detection techniques for dynamical systems with chaos. We consider the dynamical system described by the time series which originated from ordinary differential equation and real-world phenomena. We assume that the change parameters are unknown and the change could be either slight or drastic. The process of change detection is based on characteristic dynamical system invariants. Changes in the invariants’ values of the dynamical systems are the indicators of change. We propose a method of change detection based on the fractal dimension and recurrence plot. The automatic detection is provided by control charts. Methods were checked by using small data sets and stream data.