Vector field editing and periodic orbit extraction using Morse decomposition

Design and control of vector fields is critical for many visualization and graphics tasks such as vector field visualization, fluid simulation, and texture synthesis. The fundamental qualitative structures associated with vector fields are fixed points, periodic orbits, and separatrices. In this paper, we provide a new technique that allows for the systematic creation and cancellation of fixed points and periodic orbits. This technique enables vector field design and editing on the plane and surfaces with desired qualitative properties. The technique is based on Conley theory, which provides a unified framework that supports the cancellation of fixed points and periodic orbits. We also introduce a novel periodic orbit extraction and visualization algorithm that detects, for the first time, periodic orbits on surfaces. Furthermore, we describe the application of our periodic orbit detection and vector field simplification algorithms to engine simulation data demonstrating the utility of the approach. We apply our design system to vector field visualization by creating data sets containing periodic orbits. This helps us understand the effectiveness of existing visualization techniques. Finally, we propose a new streamline-based technique that allows vector field topology to be easily identified.     

Input-to-state stability for cascade systems with multiple invariant sets

In a recent paper Angeli and Efimov (2015), the notion of Input-to-State Stability (ISS) has been generalized for systems with decomposable invariant sets and evolving on Riemannian manifolds. In this work, we analyze the cascade interconnection of such ISS systems and we characterize the finest possible decomposition of its invariant set for three different scenarios: 1. the driving system exhibits multistability (convergence to fixed points only); 2. the driving system exhibits multi-almost periodicity (convergence to fixed points as well as periodic and almost-periodic orbits) and the driven system is assumed to be incremental ISS; 3. the driving system exhibits multiperiodicity (convergence to fixed points and periodic orbits) whereas the driven system is ISS in the sense of Angeli and Efimov (2015). Furthermore, we provide marginal results on the backward/forward asymptotic behavior of incremental ISS systems and on the response of a contractive system under asymptotically almost-periodic forcing. Three examples illustrate the potentiality of the proposed framework.     

计及Casimir效应时旋转式NEMS致动器的动态稳定性研究

分析了Casimir力对旋转式纳电子机械(NEMS)致动器动态稳定性的影响.计及Casimir效应时,旋转式致动器失稳时的临界倾角和临界电压和结构的几何尺寸有关.外加电压为零时,导出了旋转式致动器在Casimir力作用下发生吸合的临界特征几何尺寸.给出了旋转式致动器的无量纲化运动控制方程,并对它作了定性分析.定性分析表明:该运动方程相应的自治系统在相平面上的平衡点包括中心点、稳定的焦点和不稳定的鞍点;相图呈现周期轨道、同宿轨道以及异宿轨道.另外系统还存在分叉现象.     

Hyperbolic automorphisms of tori and pseudo-random sequences

A method for generating pseudo-random sequences of d-dimensional vectors is considered; it is based on theergodic theory of periodic orbits in the sense of [2] for unstable dynamical systems such as the hyperbolic automorphisms of the d-dimensional Torus. Since these systems enjoy strong chaotic properties, their orbits are both dense andchaotic in some sense, however the ergodic property holds only for orbits having initial points with irrational coordinates, the remaining ones being periodic. Unfortunately, those orbits are the only ones that a computer is able to generate. Since a pseudo-random sequence in [0,1] ^d is a long periodic orbit which has chaotic behaviour similar in some sense to the one of aperiodic orbits, in this note, we shall prove lower and upper bounds for the length of the period of orbits of the hyperbolic automorphisms of the d-dimensional Torus, expressed in terms of the (rational) starting point. The algorithms proposed are free of computational error, since they work in integer arithmetic. Surprisingly the elimination of the round off errors turns out in anincrease of the length of the period. Statistical testing and the problem of estimating the discrepancy of the obtained sequences are also treated.     

用状态变量的一阶微分反馈法实现蔡电路的混沌控制

提出了用系统变量的一阶微分反馈实现Chua电路的分岔和混沌控制的方法.首先根据理论分析,求出使混沌Chua电路中不动点稳定时控制参数k1的取值范围和使系统发生Hopf分岔时控制参数k₁的临界值k₀.由分岔图可以得到系统被控制到各种nP周期轨道时k₁的取值范围.然后根据该控制方法,设计了实现Chua电路混沌控制的电路,并进行了电路仿真.数值计算和仿真结果验证了这种控制分岔和混沌方法的正确性和有效性.     

Computing periodic orbits with high accuracy

This paper introduces a new family of algorithms for computing periodic orbits of vector fields. These global methods achieve high accuracy with relatively coarse discretizations of periodic orbits through the use of automatic differentiation. High degree Taylor series expansions of trajectories are computed at mesh points. On a fixed mesh, we construct closed curves that converge smoothly to periodic orbits as the degree of the Taylor series expansions increase. The algorithms have been implemented in Matlab together with the use of the automatic differentiation code ADOL-C. Numerical tests of our codes are compared with AUTO calculations using the Hodgkin-Huxley equations as a test problem.     

有界闭域上的线性赋值循环终止性分析

对有界闭域上的线性赋值循环程序终止性问题进行研究.利用Jordan 标准型技术将原循环程序的终止性问题约减为终止性等价的具有简单结构的循环程序的终止性问题.证明了当线性迭代映射满足一定条件时,该类循环程序不可终止的充分必要条件是：迭代映射在有界闭域上有不动点或周期轨.     

Nonlinear Analysis and Attitude Control of a Gyrostat Satellite with Chaotic Dynamics Using Discrete‐Time LQR‐OGY

Quasi‐periodic and chaotic behavior, along with the control of chaos for a Gyrostat satellite (GS), is investigated in this work. The quaternion‐based dynamical model of the GS is first derived, and then the influences of the reaction wheels in the GS structure, under the gravity gradient perturbation that causes a route to chaos through quasi‐periodicity mechanism, is investigated. For the suppression of chaos in the system, a chaos control system with the quaternion feedback is designed for the GS based on the extension of the Ott‐Grebogi‐Yorke (OGY) method using the linearization of the Poincaré map. In the extended OGY controller, the Poincaré map is estimated using the Least Square Support Vector Machine (LSSVM) technique. After linearization of the Poincaré map, the Discrete‐time Linear Quadratic Regulator (DLQR) is applied on the linearized Poincaré map, making the DLQR‐OGY controller for chaos. The DLQR‐OGY control system stabilizes the orbits to the fixed points providing a small control input signal, which leads to a decrease in the control effort and energy consumption in the GS system.     

Adaptive competitive self-organizing associative memory

This work presents the design of an adaptive competitive self-organizing associative memory (ACSAM) system for use in classification and recognition of pattern information. Volterra and Lotka's models of interacting species in biology motivated the ACSAM model; a model based on a system of nonlinear ordinary differential equations (ODEs). Self-organizing behavior is modeled for unsupervised neural networks employing the concept of interacting/competing species in biology. In this model, self-organizing properties can be implicitly coded within the systems trajectory structure using only ODEs. Among the features of this continuous-time system are: 1) the dynamic behavior is well-understood and characterized; 2) the desired fixed points are the only asymptotically stable states of the system; 3) the trajectories of ACSAM derived from the weight activities of the gradient system have no periodic or homoclinic orbits; and 4) the heteroclinic orbits that exist between equilibrium states are structurally unstable and can be removed by small perturbations.     

基于多项式展开的三角平动点垂直周期轨道解析构建方法研究

平动点是圆型限制性三体问题中的五个平衡解,其附近存在着大量的周期轨道,研究这些周期轨道的构建方法在深空探测中具有重要的理论及工程意义.本文从模态运动的角度出发,分析三角平动点附近周期轨道,通过多项式展开法构建出主坐标下周期轨道三个运动方向之间的渐近关系,从新的角度分析了系统的动力学特性和三维周期运动三个方向内在关联以及物理规律.同时可以为设计真实力学模型下的飞行器轨道提供借鉴.文中提出的方法可以被拓展至椭圆型限制性三体问题的三维周期轨道构建或共线平动点附近的轨道构建中.     

Floquet-based design and control approach to spacecraft formation flying in libration point orbits

A method for spacecraft formation flying (SFF) design and control near libration point orbits was developed by making use of the Floquet theory for periodic orbits. Firstly, the Floquet theory used in libration point orbits was introduced and the coefficients of four Floquet periodic modes were proved to be nearly constant when the amplitude in z direction of earth-moon L1 halo orbits is less than 20000 km. On this basis, a configuration design approach to SFF in L1 halo orbits was proposed, and several typ...     

Existence of periodic orbits of stable saturated systems

The saturation of linear controllers produces the undesirable existence of equilibrium points or periodic orbits of the closed-loop system. This typical nonlinear behavior has been observed in real systems or by means of simulation of certain examples. However, there are only a few studies in which the properties of saturated systems have been examined rigorously and, a proof of the existence of periodic orbits created by the saturation of the controller is lacking. In this paper we choose an example of an open-loop stable linear control system with an stabilizing saturated linear feedback to prove rigorously the existence of a periodic orbit.     

A new shooting algorithm for the search of periodic orbits

The periodic orbit theory gives the basic framework to study a quantum and classical correspondence. In this paper, we firstly report that we have found the existence of a certain surface, which we call the devil’s staircase surface. Secondly, taking the advantage of some intriguing properties of this surface, we propose a new method to exhaustively search for periodic orbits in the anisotropic Kepler problem. Our method fully takes into account of an intriguing property of the initial value problem of the anisotropic Kepler problem, and it reduces the two-dimensional search into the one-dimensional search. Using this method, all of the periodic orbits up to the length \(2N=20\) (altogether 19284 distinct periodic orbits) have been successfully obtained, which exceeds the world record of 76 periodic orbits up to \(2N=10\) .     

原子力显微镜微悬臂梁的混沌运动控制

通过数值模拟方法分析了原子力显微镜（atomic force microscopy,AFM）微悬臂梁的混沌运动与分岔特性,研究了“时间延迟反馈控制”、“周期信号控制”分岔特性和混沌运动控制参数的取值范围,以及同一周期轨道不同控制参数的值域．研究结果对复杂系统非线性动力学行为分析和混沌运动控制提供了有意义的理论参考,同时对控制原子力显微镜主要构件的运动和改善其测量精度,具有工程实用价值．     

一类磁性刚体航天器的混沌控制研究

采用基于误差线性系统稳定性准则的混沌控制方法,控制具有结构内阻尼的磁性刚体航天器在重力场与磁场共同作用下在圆形轨道的混沌姿态运动.讨论了航天器姿态运动方程中部分参数的取值对于运动姿态的影响,给出了这些参数通过倍周期分岔或逆倍周期分岔通往混沌的途径.当参数使系统做混沌姿态运动时,采用上述方法将混沌运动控制至周期-4轨道,并实现周期-1、2、4轨道之间转换的灵活控制.此外,分析了控制参数的变化对于控制效果的影响,并分别给出了控制至不同轨道时的输入扰动范围及控制参数范围.仿真结果表明,该方法能够实现混沌姿态运动在预定周期轨道间的灵活控制,且输入扰动量小、控制速度快、具有高精度,从而验证了该方法在航天器混沌姿态运动控制方面的有效性.     

Application of efficient composite methods for computing with certainty periodic orbits in molecular systems

Recently, we have proposed a technique for the computation of periodic orbits in molecular systems, based on the characteristic bisection method [Vrahatis et al., Comput. Phys. Commun. 138 (2001) 53]. The main advantage of the characteristic bisection method is that it converges with certainty within a given starting rectangular region. In this paper we further improve this technique by applying, on a surface of section of a Poincaré map, an iterative scheme based on the composition of the characteristic bisection method with other more rapid root-finding methods such as Newton's or Broyden's methods. Thus, the composite schemes compute rapidly with certainty periodic orbits of molecular systems. By applying these methods to the LiNC/LiCN molecular system we obtain promising results. We have reproduced previous results using considerable less CPU time. Also, we have located and computed new asymmetric families of periodic orbits.     

Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes

The aim of this paper is to present two methods for the calculation of the nonlinear normal modes of vibration for undamped nonlinear mechanical systems: the time integration periodic orbit method and the modal representation method. In the periodic orbit method, the nonlinear normal mode is obtained by making the continuation of branches of periodic orbits of the equation of motion. The terms “periodic orbits” means a closed trajectory in the phase space, which is obtained by time integration. In the modal representation method, the nonlinear normal mode is constructed in terms of amplitude, phase, mode shape, and frequency, with the distinctive feature that the last two quantities are amplitude and total phase dependent. The methods are compared on two DOF strongly nonlinear systems.     

一种Lyapunov指数算法及其实现

提出了一种利用周期轨道不同权重计算Lyapunov指数的算法。对混沌序列的周期轨道进行统计，并计算不同的周期轨道的Lyapunov指数，依据周期轨道的权重加权求和得到整个混沌吸引子的平均Lyapunov指数。深入讨论了初始值等对平均Lyapunov指数的影响。该算法不用舍去开始迭代点，适用于复杂混沌系统。     

On the existence of periodic solutions in time-invariant fractional order systems

Periodic solutions and their existence are one of the most important subjects in dynamical systems. Fractional order systems like integer ones are no exception to this rule. Tavazoei and Haeri (2009) have shown that a time-invariant fractional order system does not have any periodic solution. In this article, this claim has been investigated and it is shown that although in any finite interval of time the solutions do not show any periodic behavior, when the steady state responses of fractional order systems are considered, periodic orbits can be detected.     

基于低频周期参数扰动统一混沌系统的分岔分析

本文以低频周期参数扰动下的统一混沌系统为研究对象,应用动力学基础知识,讨论了系统的平衡点的分布及其稳定性,得到了周期扰动系统的静态分岔和Hopf分岔的条件。根据Melnikov方法,计算得到了系统的同宿轨道,以及得到了系统发生同宿轨道分岔的条件。为了验证理论研究结果的正确性,本文采用数值模拟的方法进行了验证,结果表明,理论研究结果正确。本文的研究结果可以看作是对周期激励的Lorenz类系统和Chen类系统的总结,可以有助于混沌系统在计算机应用领域的推广和应用。