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In this paper, we examine a population model with carrying capacity, time delay, and sources of additive and multiplicative environmental noise. We find that time delay, noise sources and their correlation induce regime shifts and transitions between the population survival state and the extinction state. To explore the transition mechanism between these two states, we analyzed the shift time to extinction, or the delayed extinction time, of populations. The main finding is that the extinction transition time as a function of the noise intensity shows a maximum, indicating the existence of an appropriate noise intensity leading to a maximal delayed extinction. This nonmonotonic behavior, with a maximum, is a signature of the noise-enhanced stability phenomenon, observed in many physical and complex metastable systems. In particular, this maximum increases (or decreases) as the cross-correlation intensity or the delay time in the death process increases. Furthermore, the signal-to-noise ratio as a function of noise intensity shows a maximum, which is a signature of the stochastic resonance phenomenon in the population dynamics model investigated in the presence of time delay and environmental noise. Graphic abstract

By studying the time series of logistic maps, dark lines in bifurcation diagrams and cobweb diagrams, characteristics of sequential iterations of the map are found. Before the merging together two chaotic bands, sequential iterations of the map present an ordered state. After that, with the instability of the hyperstable periodic orbit, sequential iterations of the map appear disordered. Therefore, the statistical results of time correlation of direction phase are introduced to describe the transition of the collective behaviour. Based on the two-dimensional coupled map lattice, the relationship between bifurcation parameters and order parameters with the change of coupling strength is studied. We show that when the coupling strength is weak, the critical bifurcation parameters are positively correlated with the coupling strength. When the coupling strength is large, the phase order of the system is not affected by the coupling strength. The transition of collective behaviour in a modular network is also studied. By fixing modularity and bifurcation parameters, with the change of coupling strength, the collective behaviour presents a transition process from spatiotemporal chaos to phase-ordered state. There are periodic orbits in the spatiotemporal chaotic region. A phase synchronisation region can be present in the antiphase synchronisation region. Furthermore, there exist multistable solutions in the region.

Dynamical patterns in complex networks of coupled oscillators are of both theoretical and practical interest, yet to fully reveal and understand the interplay between pattern emergence and network structure remains to be an open problem. Among the many outstanding issues, a fundamental one is how the network structure affects the stability of dynamical patterns. To address this issue, we focus on the spiral wave patterns and investigate the effects of systematically added random links on their stability and dynamical evolutions. We find that, as the network structure deviates more from the regular topology and thus becomes increasingly more complex, an originally stable spiral wave pattern can disappear but different types of patterns can emerge. In addition, short-distance links added to a small region containing the spiral tip can have a more significant effect on the wave pattern than long-distance connections. As more random links are introduced into the network, distinct pattern transitions can occur, such as the transition of the spiral wave pattern to a global synchronization state, to a chimera-like state, or to a pinned spiral wave. About the transition points, the network dynamics are highly sensitive to small structural perturbations in that the addition of even a single link can change the pattern from one type to another. These findings provide additional insights into the pattern dynamics in complex networks, a problem that is relevant to many physical, chemical, and biological systems.

The disappearance and reappearance of chaos by adjusting the internal parameters of dynamics in Lorenz system are studied. We observe monotonous and periodic time-dependent changes of Rayleigh number. There exists relaxation time for the disappearance of chaos, when we use the snapshot attractors to observe the change of the system attractors. We show that the rate of disappearance and reappearance of chaos is positively correlated with the control parameters. To reflect the relaxation phenomenon of chaotic disappearance and the sensitivity of trajectory, the concept of finite-time Lyapunov exponent is used. Then the statistical characteristics of the system can be presented by standard deviation. The chaotic disappearance and reappearance are manifested in the decrease and increase of the standard deviation. The standard deviation decreases continuously during chaotic disappearance, but increases discontinuously during chaotic reappearance. A distinctive scenario is that no matter which parameter changes, when we use the same rate of change in the process of chaotic disappearance and reappearance, their paths are different.

We uncover a route from low-dimensional to high-dimensional chaos in nonsmooth dynamical systems as a bifurcation parameter is continuously varied. The striking feature is the existence of a finite parameter interval of periodic attractors in between the regimes of low- and high-dimensional chaos. That is, the emergence of high-dimensional chaos is preceded by the system's settling into a totally nonchaotic regime. This is characteristically distinct from the situation in smooth dynamical systems where high-dimensional chaos emerges directly and smoothly from low-dimensional chaos. We carry out an analysis to elucidate the underlying mechanism for the abrupt emergence and disappearance of the periodic attractors and provide strong numerical support for the typicality of the transition route in the pertinent two-dimensional parameter space. The finding has implications to applications where high-dimensional and robust chaos is desired., 9 pages, 8 figures

The speed and paths of synchronization play a key role in the function of a system, which has not received enough attention up to now. In this work, we study the synchronization process of coupled logistic maps that reveals the common features of low-dimensional dissipative systems. A slowing down of synchronization process is observed, which is a novel phenomenon. The result shows that there are two typical kinds of transient process before the system reaches complete synchronization, which is demonstrated by both the coupled multiple-period maps and the coupled multiple-band chaotic maps. When the coupling is weak, the evolution of the system is governed mainly by the local dynamic, i.e., the node states are attracted by the stable orbits or chaotic attractors of the single map and evolve toward the synchronized orbit in a less coherent way. When the coupling is strong, the node states evolve in a high coherent way toward the stable orbit on the synchronized manifold, where the collective dynamics dominates the evolution. In a mediate coupling strength, the interplay between the two paths is responsible for the slowing down. The existence of different synchronization paths is also proven by the finite-time Lyapunov exponent and its distribution.

The most basic local conversion is local operations and classical communications (LOCC), which is also the most natural restriction in quantum information processing. We investigate the conversions between the ground states in quantum critical systems via LOCC and propose an novel method to reveal the different convertibility via majorization relation when a quantum phase transition occurs. The ground-state local convertibility in the one-dimensional transverse field Ising model is studied. It is shown that the LOCC convertibility changes nearly at the phase transition point. The relation between the order of quantum phase transitions and the LOCC convertibility is discussed. Our results are compared with the corresponding results using the Renyi entropy and the LOCC convertibility with assisted entanglement., Comment: 4 pages, 4 figures

The multiple devil's staircase, that was observed in 1997 in a one-dimensional map with two discontinuous regions, showed a kind of consonant tower-like structures. All the branches that connected top and bottom phase-locked steps of the towers could be divided into two kinds. Our recent study found that actually 16 different kinds of tower branches existed in most parts of parameter space. 14 of them belong to the so-called dissonant structures. The number of types of corresponding dissonant branches is employed to describe the dissonance of the staircase. When the number of discontinuous regions, n, in the system function develops, the dissonance of the staircase increases with 2n3−n rule.

A multifractal measure for the relaxation time distribution in Davidson–Cole dielectric relaxation is analytically obtained. A general first-order multifractal phase transition is reported. It is characterized by the existence of three different phases and the corresponding triple point.

This letter reports the specific phase-locking property of a 1-d map that includes two discontinuous regions. The devil's staircase observed in the system loses monotonicity, self-similarity, and the “Farey tree rule” for description of the plateau length distribution. However this staircase is composed of many conventional devil's staircases that have all three characteristics.

The leakage of iterations from chaotic attractor near crisis due to dynamical interaction between discontinuity and noninvertibility has been studied. The scaling law of the mean life time is derived analytically. The numerical result shows very good agreement.

An interior crisis with striking new characteristics has been observed in a one-dimensional piece-wise linear map. Induced by the collision between a chaotic attractor and a “hole”, the crisis exhibits a scaling law of the characteristic time as 〈 τ 〉 ∼ ϵ −1 . The analytical calculation shows that this scaling behavior sensitively depends on the function form of the map and suggests that a different and more complicated scaling behavior may be observed in some non-linear maps.

The influence of the surface structure and vibration mode on the resistivity of Cu films and the corresponding size effect are investigated. The temperature dependent conductivities of the films with different surface morphologies are calculated by the algorithm based upon the tight-binding linear muffin-tin orbital method and the Green's function technique. The thermal effect is introduced by setting the atomic displacements according to the Gaussian distribution with the mean-square amplitude estimated by the Debye model. The result shows that the surface atomic vibration contributes significantly to the resistivity of the systems. Comparing the conductivities for three different vibration modes, it is suggested that freezing the surface vibration is necessary for practical applications to reduce the resistivity induced by the surface electron-phonon scattering.

In this letter the scaling properties of the period-adding sequences in a so-called "multiple Devil's staircase" are reported. It is certified both analytically and numerically that the width of the i-th phase-locked plateau in a sequence scales as ln|Δe(i)| ∝ i, and the position of the plateau scales as ln|e∞ - ei| ∝ i. These properties are qualitatively different from those of the period-adding sequences in conventional Devil's staircases.

A simple bulk model of electron-phonon coupling in metals has been partially successful in explaining experiments on metal films that actually involve surface- or other low-dimensional phonons. However, by an exact application of this standard model to a semi-infinite substrate with a free surface, making use of the actual vibrational modes of the substrate, we show that such agreement is fortuitous, and that the model actually predicts a low-temperature crossover from the familiar ${T}^{5}$ temperature dependence to a stronger ${T}^{6}\phantom{\rule{0.2em}{0ex}}\mathrm{log}\phantom{\rule{0.2em}{0ex}}T$ scaling. Comparison with existing experiments suggests a widespread breakdown of the standard model of electron-phonon coupling in metals.

There is great interest in the development of novel nanomachines that use charge, spin, or energy transport, to enable new sensors with unprecedented measurement capabilities. Electrical and thermal transport in these mesoscopic systems typically involves wave propagation through a nanoscale geometry such as a quantum wire. In this paper we present a general theoretical technique to describe wave propagation through a curved wire of uniform cross-section and lying in a plane, but of otherwise arbitrary shape. The method consists of (i) introducing a local orthogonal coordinate system, the arclength and two locally perpendicular coordinate axes, dictated by the shape of the wire; (ii) rewriting the wave equation of interest in this system; (iii) identifying an effective scattering potential caused by the local curvature; and (iv), solving the associated Lippmann-Schwinger equation for the scattering matrix. We carry out this procedure in detail for the scalar Helmholtz equation with both hard-wall and stress-free boundary conditions, appropriate for the mesoscopic transport of electrons and (scalar) phonons. A novel aspect of the phonon case is that the reflection probability always vanishes in the long-wavelength limit, allowing a simple perturbative (Born approximation) treatment at low energies. Our results show that, in contrast to charge transport, curvature only barely suppresses thermal transport, even for sharply bent wires, at least within the two-dimensional scalar phonon model considered. Applications to experiments are also discussed., 9 pages, 11 figures, RevTex

Stochastic resonance in an excitable neuron based on the Rulkov map with noise, delay feedback, low-frequency signal and high-frequency signal is investigated numerically. The results show that there exist an optimal noise intensity, optimal time delay and optimal amplitude of the high-frequency signal at which the phase synchronisation between the low-frequency input signal and the output signal is the best. The Fourier coefficient is calculated to measure the stochastic resonance. It is found that the existence of a maximum in the , and plots is the identifying characteristic of the stochastic resonance phenomenon.

Coherence resonance in a discrete excitable neuronal model with noises and time delays is investigated. The effects of the time delays on coherence resonance are revealed in two cases: Gaussian white and Gaussian color noises, respectively. The coefficient of variation of interspike intervals is calculated by numerical simulation. The results show that the coherence resonance is enhanced with the time delay increasing in the weak noises intensity cases, while there is no effect in the large noise intensity. Moreover, the coherence resonance can be held back by the self-correlation time, when the system is driven by a color noise.

The partial and complete periodic synchronization in coupled discontinuous map lattices consisting of both discontinuous and non-invertible maps are discussed. We classify three typical types of periodic synchronization states, which give rise to different spatiotemporal patterns including static partial periodic synchronization, dynamically periodic synchronization, and complete periodic synchronization patterns. A special prelude dynamics of partial and complete periodic synchronization motion, which is shown by five separated concave curves in the time series plots of the order parameters, is observed. The detailed analysis shows that the special prelude dynamics is induced by the competition between two synchronized clusters, and the analytical expression for the corresponding order parameter is obtained.

Neurons in the brain utilize various firing trains to encode the input signals they have received. Firing behavior of one single neuron is thoroughly explained by using a bifurcation diagram from polarized resting to firing, and then to depolarized resting. This explanation provides an important theoretical principle for understanding neuronal biophysical behaviors. This paper reports the novel experimental and modeling results of the modification of such a bifurcation diagram by adjusting small conductance potassium (SK) channel. In experiments, changes in excitability and depolarization block in nucleus accumbens shell and medium-spiny projection neurons are explored by increasing the intensity of injected current and blocking the SK channels by apamin. A shift of bifurcation points is observed. Then, a Hodgkin—Huxley type model including the main electrophysiological processes of such neurons is developed to reproduce the experimental results. The reduction of SK channel conductance also shifts the bifurcations, which is in consistence with experiment. A global bifurcation paradigm of this shift is obtained by adjusting two parameters, intensity of injected current and SK channel conductance. This work reveals the dynamics underpinning modulation of neuronal firing behaviors by biologically important ionic conductance. The results indicate that small ionic conductance other than that responsible for spike generation can modify bifurcation points and shift the bifurcation diagram and, thus, change neuronal excitability and adaptation.

The multifractal measures of the relaxation-time distributions are analytically obtained for some typical non-Debye dielectric relaxation processes. The characteristics of the corresponding multifractal thermodynamics are discussed. It is shown that the probability of the relaxation times near the poles of their distribution function is fractal scaling. The corresponding Lipschitz-H\"old singular exponent is, or can be, determined by the so-called shape parameters in the empirical dielectric relaxation formulas. The relationship to some analytical proofs of the empirical dielectric formulas based upon the fractal models is also analyzed. Some generalized multifractal phase transitions with interesting features are reported in this paper. The recent experiment results on the molten-crystal transition in organic glass systems are also discussed to support our conclusions.