Can People Predict Chaotic Sequences?

Previous studies suggesting that people predict chaotic sequences better than chance have not discriminated between sensitivity to nonlinear determinism and facilitation using autocorrelation. Since prediction accuracy declines with increases in the look-ahead window in both cases, a decline in prediction accuracy does not imply chaos sensitivity. To overcome this problem, phase-randomized surrogate time series are used as a control. Such series have the same linear properties as the original chaotic sequence but contain no nonlinear determinism, i.e. chaos. In the experimental task, using a chaotic Hénon attractor, participants viewed the previous eight days temperatures and then predicted temperatures for the next four days, over 120 trials. The control group experienced a sample from a corresponding phase-randomized surrogate series. Both time series were linearly transformed to provide a realistic temperature range. A transformation of the correlation between observed and predicted values decreased over days for the chaotic time series, but remained constant and high for the surrogate series. The interaction between the days and series factors was statistically significant, suggesting that people are sensitive to chaos, even when the autocorrelation functions and power spectra of the control and experimental series are identical. Implications for the psychological assessment of individual differences in human prediction are discussed.     

Spatial and Dynamical Chaos Generated by Reaction–Diffusion Systems in Unbounded Domains

We consider in this article a nonlinear reaction–diffusion system with a transport term (L,∇ _x )u, where L is a given vector field, in an unbounded domain Ω. We prove that, under natural assumptions, this system possesses a locally compact attractor in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov’s ɛ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584–637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on . As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of .     

A centrosymmetric chaos

Symmetry plays an important part in the research of the dynamical behavior of nonlinear system. It is proved in this paper that, for a class of centrosymmetric systems with parametric excitation, chaos behaves in centrosymmetric manner, which implies that chaos need not to be an unsymmetric dynamical state. The project supported by National Natural Science Foundation of China     

垂直冲击消振系统简谐激励响应及稳定性分析

运用迭代映射及其稳定性分析原理,研究了垂直冲击消振系统的简谐激励响应及其周期响应的稳定性．首先建立了稳定周期响应的参数区域边界方程,分析了稳定周期运动向混沌转变的一般规律．然后以典型的二阶主振系为例,得到了几个对消振效果影响较大的稳态周期响应区域的详细数值结果,讨论了稳态周期响应区域及附近的消振效果．     

The dynamics of coupled planar rigid bodies. II. Bifurcations,periodic solutions,and chaos

We give a complete bifurcation and stability analysis for the relative equilibria of the dynamics of three coupled planar rigid bodies. We also use the equivariant Weinstein-Moser theorem to show the existence of two periodic orbits distinguished by symmetry type near the stable equilibrium. Finally we prove that the dynamics is chaotic in the sense of Poincaré-Birkhoff-Smale horseshoes using the version of Melnikov's method suitable for systems with symmetry due to Holmes and Marsden.     

耦合时空混沌的模糊混沌神经网络鲁棒自适应控制

将模糊逻辑系统和混沌神经网络结合起来,利用模糊逻辑系统的逼近能力和混沌神经网络的时空混沌行为,对模型未知的耦合时空混沌系统提出了一种模糊混沌神经网络自适应控制方案;同时考虑系统扰动、未建模动态特性和建模误差的影响,设计自适应补偿器,增强时空混沌系统控制的鲁棒性;并用Laypunov方法证明了该方案的稳定性;仿真验证了方案的有效性和鲁棒性。     

Chaotic Analysis of Nonlinear Viscoelastic Panel Flutter in Supersonic Flow

In this paper chaotic behavior of nonlinear viscoelastic panels in asupersonic flow is investigated. The governing equations, based on vonKàarmàn's large deflection theory of isotropic flat plates, areconsidered with viscoelastic structural damping of Kelvin's modelincluded. Quasi-steady aerodynamic panel loadings are determined usingpiston theory. The effect of constant axial loading in the panel middlesurface and static pressure differential have also been included in thegoverning equation. The panel nonlinear partial differential equation istransformed into a set of nonlinear ordinary differential equationsthrough a Galerkin approach. The resulting system of equations is solvedthrough the fourth and fifth-order Runge–Kutta–Fehlberg (RKF-45)integration method. Static (divergence) and Hopf (flutter) bifurcationboundaries are presented for various levels of viscoelastic structuraldamping. Despite the deterministic nature of the system of equations,the dynamic panel response can become random-like. Chaotic analysis isperformed using several conventional criteria. Results are indicative ofthe important influence of structural damping on the domain of chaoticregion.     

Stabilization of Unstable Limit Cycles in Systems with Limited Controllability: Expanding the Basin of Convergence of OGY-Type Controllers

The underlying geometric structure of the standard OGY control scheme is analyzed. Some of the main mechanisms that under certain conditions lead to failure of the control algorithm are revealed. The limited controllability available in a system is investigated and it is shown that it may lead to serious problems that will significantly enlarge the state space region of failure of the standard OGY controller. A minimal distance algorithm is analyzed and shown to be, for some problems, more advantageous than the standard OGY technique. Nevertheless, for a broad category of problems, the minimal distance scheme is also shown to fail. As a solution for these problems, two new techniques are proposed: the penalized minimal distance and the multi-step OGY-type scheme. The standard OGY and minimal distance algorithms are particular cases of the new techniques proposed. Finally, we give a necessary condition that estimates the region of controllability under the multi-step OGY-type control. We demonstrate a significantly improved basin of convergence for the new multi-step OGY-type algorithm.     

裂纹转子分岔、混沌行为研究

分析非线性涡动中裂纹转子在裂纹存在和裂纹扩展两种情形下的典型非线性动力学行为－－混沌和分岔现象。在2／3倍临界转速区，裂纹深度浅于R／2的裂纹转子有分岔和拟周期响应出现；深度超过R／2的裂纹转子会出现分岔及混沌现象。裂纹角β对这些非线性动力学行为有很大影响。     

Chaotic Oscillation of Saddle Form Cable-Suspended Roofs under Vertical Excitation Action

The purpose of this paper is to investigate the dynamic behaviour of saddle form cable-suspended roofs under vertical excitation action. The governing equations of this problem are system of nonlinear partial differential and integral equations. We first establish a spectral equation, and then consider a model with one coefficient, i.e., a perturbed Duffing equation. The analytical solution is derived for the Duffing equation. Successive approximation solutions can be obtained in likely way for each time to only one new unknown function of time. Numerical results are given for our analytical solution. By using the Melnikov method, it is shown that the spectral system has chaotic solutions and subharmonic solutions under determined parametric conditions.