Existence and Dynamics of Bounded Traveling Wave Solutions to Getmanou Equation

In this paper, we study the existence and dynamics of bounded traveling wave solutions to Getmanou equations by using the qualitative theory of differential equations and the bifurcation method of dynamical systems. We show that the corresponding traveling wave system is a singular planar dynamical system with two singular straight lines, and obtain the bifurcations of phase portraits of the system under different parameters conditions. Through phase portraits, we show the existence and dynamics of several types of bounded traveling wave solutions including solitary wave solutions, periodic wave solutions, compactons, kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions are given. Additionally, we confirm abundant dynamical behaviors of the traveling wave s olutions to the equation, which are summarized as follows: i) We confirm that two types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system. ii) We confirm that two types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, and the homoclinic orbit of associated system, which is tangent to the singular line at the singular point of associated system.     

Measure-theoretical properties of the unstable foliation of two-dimensional differentiable area-preserving systems

This article analyzes in detail the statistical and measure-theoretical properties of the nonuniform stationary measure, referred to as the w-invariant measure, associated with the spatial length distribution of the integral manifolds of the unstable invariant foliation in two-dimensional differentiable area-preserving systems. The analysis is developed starting from a sequence of analytical approximations for the associated density. These approximations are related to the properties of the Jacobian matrix of the nth iteration of a Poincaré map. The w-invariant measure plays a fundamental role in the study of transport phenomena in laminar-chaotic fluid-mixing systems, for which it furnishes the asymptotic invariant distribution of intermaterial contact length between two fluids. The w-invariant measure turns out to be singular and exhibits multifractal features. Its associated density displays local self-similarity in an epsilon neighborhood of hyperbolic periodic points. The cancellation exponent of the signed measure associated with the w measure by attaching at each point the direction of the field of the asymptotic unstable eigenvectors is also analyzed. The only case for which the w-invariant measure is absolutely continuous is given by the conjugation of hyperbolic toral automorphisms with a linear automorphism. The connections with the statistical properties, and in particular with the stretching dynamics, are addressed in detail.     

混沌体系中寻找周期轨迹的方法

一个不可积混沌体系，由于扰动而遭到破坏时，存活的周期轨迹体现了体系的本质特征，是 体系的运动骨架.在一定程度上， 可以由周期轨迹来量子化不可积体系，这充分说明了 周期轨迹的重要性.而寻找周期轨迹，也就成为研究混沌体系动力学特性以及对混沌体系进 行量子化的关键问题.结合具体实例，给出了3种常用的寻找周期轨迹方法，并详细探讨了各 种方法的优缺点和适用范围. 关键词： 周期轨迹 数值方法 混沌     

Deltons, peakons and other traveling-wave solutions of a Camassa-Holm hierarchy

In this letter, we study an integrable Camassa-Holm hierarchy whose high-frequency limit is the Camassa-Holm equation. Phase plane analysis is employed to investigate bounded traveling wave solutions. An important feature is that there exists a singular line on the phase plane. By considering the properties of the equilibrium points and the relative position of the singular line, we find that there are in total three types of phase planes. Those paths in phase planes which represented bounded solutions are discussed one-by-one. Besides solitary, peaked and periodic waves, the equations are shown to admit a new type of traveling waves, which concentrate all their energy in one point, and we name them deltons as they can be expressed as some constant multiplied by a delta function. There also exists a type of traveling waves we name periodic deltons, which concentrate their energy in periodic points. The explicit expressions for them and all the other traveling waves are given.     

Itinerant memory dynamics and global bifurcations in chaotic neural networks

We have considered itinerant memory dynamics in a chaotic neural network composed of four chaotic neurons with synaptic connections determined by two orthogonal stored patterns as a simple example of a chaotic itinerant phenomenon in dynamical associative memory. We have analyzed a mechanism of generating the itinerant memory dynamics with respect to intersection of a pair of alpha branches of periodic points and collapse of a periodic in-phase attracting set. The intersection of invariant sets is numerically verified by a novel method proposed in this paper.     

Characteristics of in-out intermittency in delay-coupled FitzHugh–Nagumo oscillators

We analyze a pair of delay-coupled FitzHugh–Nagumo oscillators exhibiting in-out intermittency as a part of the generating mechanism of extreme events. We study in detail the characteristics of in-out intermittency and identify the invariant subsets involved – a saddle fixed point and a saddle periodic orbit – neither of which are chaotic as in the previously reported cases of in-out intermittency. Based on the analysis of a periodic attractor possessing in-out dynamics, we can characterize the approach to the invariant synchronization manifold and the spiralling out to the saddle periodic orbit with subsequent ejection from the manifold. Due to the striking similarities, this analysis of in-out dynamics also explains in-out intermittency     

Periodic orbit analysis at the onset of the unstable dimension variability and at the blowout bifurcation

Many chaotic dynamical systems of physical interest present a strong form of nonhyperbolicity called unstable dimension variability (UDV), for which the chaotic invariant set contains periodic orbits possessing different numbers of unstable eigendirections. The onset of UDV is usually related to the loss of transversal stability of an unstable fixed point embedded in the chaotic set. In this paper, we present a new mechanism for the onset of UDV, whereby the period of the unstable orbits losing transversal stability tends to infinity as we approach the onset of UDV. This mechanism is unveiled by means of a periodic orbit analysis of the invariant chaotic attractor for two model dynamical systems with phase spaces of low dimensionality, and seems to depend heavily on the chaotic dynamics in the invariant set. We also described, for these systems, the blowout bifurcation (for which the chaotic set as a whole loses transversal stability) and its relation with the situation where the effects of UDV are the most intense. For the latter point, we found that chaotic trajectories off, but very close to, the invariant set exhibit the same scaling characteristic of the so-called on-off intermittency.     

Emergence of chaotic attractor and anti-synchronization for two coupled monostable neurons

The dynamics of two coupled piece-wise linear one-dimensional monostable maps is investigated. The single map is associated with Poincare section of the FitzHugh-Nagumo neuron model. It is found that a diffusive coupling leads to the appearance of chaotic attractor. The attractor exists in an invariant region of phase space bounded by the manifolds of the saddle fixed point and the saddle periodic point. The oscillations from the chaotic attractor have a spike-burst shape with anti-phase synchronized spiking.     

Measures with infinite Lyapunov exponents for the periodic Lorentz gas

We study invariant measures for the periodic Lorentz gas which are supported on the set of points with infinite Lyapunov exponents. We construct examples of such measures which are measures of maximal entropy and ones which are not.     

The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of certain non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.     

Noninvertibility and resonance in discrete-time neural networks for time-series processing

We present a computer-assisted study emphasizing certain elements of the dynamics of artificial neural networks (ANNs) used for discrete time-series processing and nonlinear system identification. The structure of the network gives rise to the possibility of multiple inverses of a phase point backward in time; this is not possible for the continuous-time system from which the time series are obtained. Using a two-dimensional illustrative model in an oscillatory regime, we study here the interaction of attractors predicted by the discrete-time ANN model (invariant circles and periodic points locked on them) with critical curves. These curves constitute a generalization of critical points for maps of the interval (in the sense of Julia-Fatou); their interaction with the model-predicted attractors plays a crucial role in the organization of the bifurcation structure and ultimately in determining the dynamic behavior predicted by the neural network.     

Dynamics of Meandering Spiral Waves Driven by Two-Point Feedback

We numerically study the dynamics of meandering spiral waves in theexcitable system subjected to a feedback signal coming from two measuring points located on a straight line together with the initial spiral core. The core location and size radius of the final attractors are computed, and they change with the position of the moving measuring point in a unique way. By the Fourier Spectral analysis, we find the frequency-locked behaviors different from the
driving scheme of the external periodic force. It is also found that the meandering spiral wave can be eliminated when the moving measuring point approaches closely the boundary and its feedback gain is large enough. This offers an effective and convenient method for eliminating meandering spiral waves.     

基于Riemann解的二维流体力学Lagrange有限点无网格方法

在高维流体力学计算中,对于多介质大变形等一类问题,采用有网格方法常遇到较大的困难.针对二维问题,研究了一种无网格方法——Lagrange有限点方法:在求解区域上设置适当的离散点集,视其中每一点为流体力学Lagrange点;对于点集的任一点,确定邻点集合,并基于该点同邻点集合的联系,应用Godunov方法将流体力学Lagrange方程进行离散;考虑到算法的稳健性,方法中可设置较多邻点并采用最小二乘法.将该方法应用于典型的数值算例,取得了良好效果.     

Intersection properties of invariant manifolds in certain twist maps

We consider the spaceN ofC ² twist maps that satisfy the following requirements. The action is the sum of a purely quadratic term and a periodic potential times a constantk (hereafter called the nonlinearity). The potential restricted to the unit circle is bimodal, i.e. has one local minimum and one local maximum. The following statements are proven for maps inN with nonlinearityk large enough. The intersection of the unstable and stable invariant manifolds to the hyperbolic minimizing periodic points contains minimizing homoclinic points. Consider two finite pieces of these manifolds that connect two adjacent homoclinic minimizing points (hereafter called fundamental domains). We prove that all such fundamental domains have precisely one point in their intersection (the Single Intersection theorem). In addition, we show that limit points of minimizing points are recurrent, which implies that Aubry Mather sets (with irrational rotation number) are contained in diamonds formed by local stable and unstable manifolds of nearby minimizing periodic orbits (the Diamond Configuration theorem). Another corollary concerns the intersection of the minimax orbits with certain symmetry lines of the map.     

On the correlation structure of some random point processes on the line

The correlation structure of some remarkable point processes on the one-dimensional real line is investigated. More specifically, focus is on translation invariant determinantal, permanental and/or renewal point processes. In some cases, anomalous (non-Poissonian) fluctuations for the number of points in a large window can be observed. This may be read from the total correlation function of the point process. We try to understand when and why this occurs and what are the anomalous behaviors to be expected.From examples, it is shown that determinantal (fermion) point processes can be super-homogeneous (the number variance grows slower than the number mean) and even hyper-uniform (when variance growth saturates).Renewal point processes with bounded spacings variance are essentially Poissonian (the number variance grows like the number mean as in Poisson models).Under certain conditions, permanental (boson) point processes can be sub-homogeneous or critical (in the sense that the number variance grows faster than the number mean).We give several detailed examples illustrating these properties of interest together with unexpected behaviors.     

Šilnikov manifolds in coupled nonlinear Schrödinger equations

We consider two coupled nonlinear Schrödinger equations with even, periodic boundary conditions, that are damped and quasiperiodically forced. We prove the existence of invariant manifolds with Šilnikov-type dynamics that are homoclinic to a spatially independent invariant torus. Such manifolds appear to induce complex behavior in numerical experiments.     

Multi-scale energy exchanges between a nonlinear oscillator of Bouc-Wen type and another coupled nonlinear system

The concept of energy exchange between coupled oscillators can be endowed for wide variety of applications such as control and energy harvesting. It has been proved that by coupling an essential nonlinear oscillator (cubic nonlinearity) to a main system (mostly linear), the latter system can be controlled in a one way and almost irreversible manner. The phenomenon is called energy pumping and the coupled nonlinear system is named as nonlinear energy sink (NES). The process of energy transfer from the main system to the nonlinear smooth or non-smooth attachment at different scales of time can present several scenarios: It can be attracted to periodic behaviors which present low or high energy levels for the main system and/or to quasi-periodic responses of two oscillators by persistent bifurcations between their stable zones. In this paper we analyze multi-scale dynamics of two attached oscillators: a Bouc-Wen type in general (in particular: a Dahl type and a modified hysteresis system) and a NES (nonsmooth and cubic). The system behavior at fast and first slow times scales by detecting its invariant manifold, its fixed points and singularities will be analyzed. Analytical developments will be accompanied by some numerical examples for systems that present quasi-periodic responses. The endowed Bouc-Wen models correspond to the hysteretic behavior of materials or structures. This paper is clearly connected with the dynamics of systems with hysteresis and nonlinear dynamics based energy harvesting.     

On Dirac Physical Measures for Transitive Flows

We discuss some examples of smooth transitive flows with physical measures supported at fixed points. We give some conditions under which stopping a flow at a point will create a Dirac physical measure at that indifferent fixed point. Using the Anosov-Katok method, we construct transitive flows on surfaces with the only ergodic invariant probabilities being Dirac measures at hyperbolic fixed points. When there is only one such point, the corresponding Dirac measure is necessarily the only physical measure with full basin of attraction. Using an example due to Hu and Young, we also construct a transitive flow on a three-dimensional compact manifold without boundary, with the only physical measure the average of two Dirac measures at two hyperbolic fixed points.     

Invariant curves,attractors, and phase diagram of a piecewise linear map with chaos

We introduce equations describing the invariant curves associated with periodic points in a wide class of two-dimensional invertible maps, which in the special case of the mapT(x, z)=(1?a¦x¦+bz,x) can be solved by analytical methods. In the dissipative case several branches of the separatrices of the fixed points, as well as, of the period-2 and -4 points, are constructed. The regions of the parameter space where a given type of strange attractor exists are located. We point out that the disappearance of homoclinic intersections between the separatrices of the fixed point and that of heteroclinic intersections between the unstable manifolds of the period-2 points and the stable manifold of the fixed point may occur separately, and the latter leads already to the appearance of a two-piece strange attractor. This phenomenon may happen at weak dissipation in other maps, too. In the conservative caseb=1 separatrices and certain invariant tori are calculated.