Delay-Coordinate Maps and the Spectra of Koopman Operators

The Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting. Koopman eigenfunctions represent the non-mixing component of the dynamics. They factor the dynamics, which can be chaotic, into quasiperiodic rotations on tori. Here, we describe a method through which these eigenfunctions can be obtained from a kernel integral operator, which also annihilates the continuous spectrum. We show that incorporating a large number of delay coordinates in constructing the kernel of that operator results, in the limit of infinitely many delays, in the creation of a map into the point spectrum subspace of the Koopman operator. This enables efficient approximation of Koopman eigenfunctions in systems with pure point or mixed spectra. We illustrate our results with applications to product dynamical systems with mixed spectra.

     

Dynamical learning of non-Markovian quantum dynamics

We study the non-Markovian dynamics of an open quantum system with machine learning.The observable physical quantities and their evolutions are generated by using the neural network.After the pre-training is completed,we fix the weights in the subsequent processes thus do not need the further gradient feedback.We find that the dynamical properties of physical quantities obtained by the dynamical learning are better than those obtained by the learning of Hamiltonian and time evolution operator.The dynamical learning can be applied to other quantum many-body systems,non-equilibrium statistics and random processes.     

Dynamics of Transiently Chaotic Neural Network and Its Application to Optimization

Through adding a nonlinear self-feedback term in the evolution equations of nerual network,we introduced a transiently chaotic neural network model.In order to utilize the transiently chaotic dynamics mechanism in optimization problem efficiently,we have analyzed the dynamical pocedure of the transiently chaotic neural network model and studied the function of the crucial bifurcation parameter which governs the chaotic behavior of the system.Based on the dynamical analysis of the transiently chaotic neural network model,Chaotic annealing algorithm is also examined and improved.As an example,we applied chaotic annealing method to the traveling salesman problem and obtained good results.     

Discontinuity and Complex Dynamics of Neural Networks

We focus on the discontinuity of a neural network model with diluted and clipped synaptic connections (±l only). The exact evolution rule of the average firing rate becomes a discontinuous piece-wise nonlinear map when very simple functions of dynamical threshold are introduced into the network. Complex dynamics is observed.     

A variational approach to connecting orbits in nonlinear dynamical systems

We propose a variational method for determining homoclinic and heteroclinic orbits including spiral-shaped ones in nonlinear dynamical systems. Starting from a suitable initial curve, a homotopy evolution equation is used to approach a true connecting orbit. The procedure is an extension of a variational method that has been used previously for locating cycles, and avoids the need for linearization in search of simple connecting orbits. Examples of homoclinic and heteroclinic orbits for typical dynamical systems are presented. In particular, several heteroclinic orbits of the steady-state Kuramoto–Sivashinsky equation are found, which display interesting topological structures, closely related to those of the corresponding periodic orbits.     

Information flow between stock markets: A Koopman decomposition approach

Stock markets in the world are linked by complicated and dynamical relationships into a temporal network.Extensive works have provided us with rich findings from the topological properties and their evolutionary trajectories,but the underlying dynamical mechanism is still not in order.In the present work,we proposed a technical scheme to reveal the dynamical law from the temporal network.The index records for the global stock markets form a multivariate time series.One separates the series into segments and calculates the information flows between the markets,resulting in a temporal market network representing the state and its evolution.Then the technique of the Koopman decomposition operator is adopted to find the law stored in the information flows.The results show that the stock market system has a high flexibility,i.e.,it jumps easily between different states.The information flows mainly from high to low volatility stock markets.And the dynamical process of information flow is composed of many dynamic modes distribute homogenously in a wide range of periods from one month to several ten years,but there exist only nine modes dominating the macroscopic patterns.     

Chaos and Geometrical Optics

Radiophysics and Quantum Electronics - Chaotic evolution of dynamical systems is caused by the divergence of nearby orbits, i. e., by the intrinsic instability of the dynamics. The best way to see...     

Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps

The numerical approximation of Perron-Frobenius operators allows efficient determination of the physical invariant measure of chaotic dynamical systems as a fixed point of the operator. Eigenfunctions of the Perron-Frobenius operator corresponding to large subunit eigenvalues have been shown to describe “almost-invariant” dynamics in one-dimensional expanding maps. We extend these ideas to hyperbolic maps in higher dimensions. While the eigendistributions of the operator are relatively uninformative, applying a new procedure called “unwrapping” to regularised versions of the eigendistributions clearly reveals the geometric structures associated with almost-invariant dynamics. This unwrapping procedure is applied to a uniformly hyperbolic map of the unit square to discover this map’s dominant underlying dynamical structure, and to the standard map to pinpoint clusters of period 6 orbits.     

Statistics of Wave Functions for a Point Scatterer on the Torus

Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of eigenfunctions of the Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions. We show that almost all of the perturbed eigenfunctions are uniformly distributed in configuration space.     

网络交通流动态演化的混沌现象及其控制

本文以含2条平行路径的交通网络为例, 探讨了网络交通流逐日动态演化问题. 首先, 建立了动态系统模型来刻画网络交通流的演化过程, 动态系统模型的不动点就是随机用户平衡解, 证明了平衡解存在且唯一. 然后, 根据非线性动力学理论, 推导出了网络交通流演化的稳定性条件. 其次, 通过数值实验, 分析了网络交通流的演化特征, 发现了在一定条件下流量的周期振荡和混沌现象. 最后, 以OD需求为控制变量推导出了网络交通流混沌控制的方法.     

Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime

The study of optomechanical systems has attracted much attention, most of which are concentrated in the physics in the smallamplitude regime. While in this article, we focus on optomechanics in the extremely-large-amplitude regime and consider both classical and quantum dynamics. Firstly, we study classical dynamics in a membrane-in-the-middle optomechanical system in which a partially reflecting and flexible membrane is suspended inside an optical cavity. We show that the membrane can present self-sustained oscillations with limit cycles in the shape of sawtooth-edged ellipses and exhibit dynamical multistability. Then, we study the dynamics of the quantum fluctuations around the classical orbits. By using the logarithmic negativity, we calculate the evolution of the quantum entanglement between the optical cavity mode and the membrane during the mechanical oscillation. We show that there is some synchronism between the classical dynamical process and the evolution of the quantum entanglement.     

Topological chaos and periodic braiding of almost-cyclic sets

In certain (2+1)-dimensional dynamical systems, the braiding of periodic orbits provides a framework for analyzing chaos in the system through application of the Thurston-Nielsen classification theorem. Periodic orbits generated by the dynamics can behave as physical obstructions that "stir" the surrounding domain and serve as the basis for this topological analysis. We provide evidence that, even in the absence of periodic orbits, almost-cyclic regions identified using a transfer operator approach can reveal an underlying structure that enables topological analysis of chaos in the domain.     

Stability transformation: a tool to solve nonlinear problems

We present an analysis of the properties as well as the diverse applications and extensions of the method of stabilisation transformation. This method was originally invented to detect unstable periodic orbits in chaotic dynamical systems. Its working principle is to change the stability characteristics of the periodic orbits by applying an appropriate global transformation of the dynamical system. The theoretical foundations and the associated algorithms for the numerical implementation of the method are discussed. This includes a geometrical classification of the periodic orbits according to their behaviour when the stabilisation transformations are applied. Several refinements concerning the implementation of the method in order to increase the numerical efficiency allow the detection of complete sets of unstable periodic orbits in a large class of dynamical systems. The selective detection of unstable periodic orbits according to certain stability properties and the extension of the method to time series are discussed. Unstable periodic orbits in continuous-time dynamical systems are detected via introduction of appropriate Poincaré surfaces of section. Applications are given for a number of examples including the classical Hamiltonian systems of the hydrogen and helium atom, respectively, in electromagnetic fields. The universal potential of the method is demonstrated by extensions to several other nonlinear problems that can be traced back to the detection of fixed points. Examples include the integration of nonlinear partial differential equations and the numerical determination of Markov-partitions of one-parametric maps.     

Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear ordinary differential equations

The problem of preserving fidelity in numerical computation of nonlinear ordinary differential equations is studied in terms of preserving local differential structure and approximating global integration structure of the dynamical system. The ordinary differential equations are lifted to the corresponding partial differential equations in the framework of algebraic dynamics, and a new algorithm—algebraic dynamics algorithm is proposed based on the exact analytical solutions of the ordinary differential equations by the algebraic dynamics method. In the new algorithm, the time evolution of the ordinary differential system is described locally by the time translation operator and globally by the time evolution operator. The exact analytical piece-like solution of the ordinary differential equations is expressed in terms of Taylor series with a local convergent radius, and its finite order truncation leads to the new numerical algorithm with a controllable precision better than Runge Kutta Algorithm and Symplectic Geometric Algorithm.     

Dynamics-driven evolution to structural heterogeneity in complex networks

The mutual influence of dynamics and structure is a central issue in complex systems. In this paper we study by simulation slow evolution of network under the feedback of a local-majority-rule opinion process. If performance-enhancing local mutations have higher chances of getting integrated into its structure, the system can evolve into a highly heterogeneous small-world with a global hub (whose connectivity is proportional to the network size), strong local connection correlations and power-law-like degree distribution. Networks with better dynamical performance are achieved if structural evolution occurs much slower than the network dynamics. Structural heterogeneity of many biological and social dynamical systems may also be driven by various dynamics-structure coupling mechanisms.     

Soliton,breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints

The nonlinear Schro¨dinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schro¨dinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schro¨dinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.     

Dynamical correlations near instabilities in nonlinear chemical reaction systems

Projection operator techniques are applied to stochastic master equations for homogeneous chemical reaction systems to derive a continued fraction representation for dynamical correlations of the particle numbers. The formalism is applied to two simple nonlinear chemical reactions which exhibit first and second order phase transition analogies, respectively. Numerical results are obtained and various approximations are investigated to describe memory effects arising at the instability points. The method presented here provides a systematic way of investigating the dynamics of nonlinear chemically reacting systems showing unstable behaviour and enhanced fluctuations far from thermal equilibrium.     

Synchronization in complex delayed dynamical networks with impulsive effects

The present paper is mainly concerned with the issues of synchronization dynamics of complex delayed dynamical networks with impulsive effects. A general model of complex delayed dynamical networks with impulsive effects is formulated, which can well describe practical architectures of more realistic complex networks related to impulsive effects. Based on impulsive stability theory on delayed dynamical systems, some simple but less conservative criterion are derived for global synchronization of such dynamical network. It is shown that synchronization of the networks is heavily dependent on impulsive effects of connecting configuration in the networks. Furthermore, the theoretical results are applied to a typical SF network composing of impulsive coupled chaotic delayed Hopfield neural network nodes, and are also illustrated by numerical simulations.     

Identifying almost invariant sets in stochastic dynamical systems

We consider the approximation of fluctuation induced almost invariant sets arising from stochastic dynamical systems. The dynamical evolution of densities is derived from the stochastic Frobenius-Perron operator. Given a stochastic kernel with a known distribution, approximate almost invariant sets are found by translating the problem into an eigenvalue problem derived from reversible Markov processes. Analytic and computational examples of the methods are used to illustrate the technique, and are shown to reveal the probability transport between almost invariant sets in nonlinear stochastic systems. Both small and large noise cases are considered.