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.     

动力学变换法探测连续系统不稳定周期轨道

本文利用动力学变换方法和庞加莱截面方法对两种连续混沌动力学系统进行不稳定周期轨道探测研究, 并对Lorenz系统进行了替代数据法检验.结果表明:基于庞加莱截面的动力学变换改进算法 可有效探测连续混沌动力学系统中的不稳定周期轨道.     

Crisis-induced unstable dimension variability in a dynamical system

Unstable dimension variability is an extreme form of non-hyperbolic behavior in chaotic systems whose attractors have periodic orbits with a different number of unstable directions. We propose a new mechanism for the onset of unstable dimension variability based on an interior crisis, or a collision between a chaotic attractor and an unstable periodic orbit. We give a physical example by considering a high-dimensional dissipative physical system driven by impulsive periodic forcing.     

Multistability and the control of complexity

We show how multistability arises in nonlinear dynamics and discuss the properties of such a behavior. In particular, we show that most attractors are periodic in multistable systems, meaning that chaotic attractors are rare in such systems. After arguing that multistable systems have the general traits expected from a complex system, we pass to control them. Our controlling complexity ideas allow for both the stabilization and destabilization of any one of the coexisting states. The control of complexity differs from the standard control of chaos approach, an approach that makes use of the unstable periodic orbits embedded in an extended chaotic attractor. (c) 1997 American Institute of Physics.     

Analytical study of chaos and applications

We summarize various cases where chaotic orbits can be described analytically. First we consider the case of a magnetic bottle where we have non-resonant and resonant ordered and chaotic orbits. In the sequence we consider the hyperbolic Hénon map, where chaos appears mainly around the origin, which is an unstable periodic orbit. In this case the chaotic orbits around the origin are represented by analytic series (Moser series). We find the domain of convergence of these Moser series and of similar series around other unstable periodic orbits. The asymptotic manifolds from the various unstable periodic orbits intersect at homoclinic and heteroclinic orbits that are given analytically. Then we consider some Hamiltonian systems and we find their homoclinic orbits by using a new method of analytic prolongation. An application of astronomical interest is the domain of convergence of the analytical series that determine the spiral structure of barred-spiral galaxies.     

Control of chaotic systems using an on-line trained linear neural controller

We propose a control system including an on-line trained linear neural controller to control chaotic systems. The control system stabilizes a chaotic orbit onto an unstable fixed point without using the knowledge of the location of the point and the local linearized dynamics at the point. Furthermore, the control system can track the stabilized orbit to the unstable fixed point whose location and local dynamics vary slowly with a variation of the system parameter. This paper extends a previous paper (Konishi and Kokame, 1995) for more general situations and improves the neural controller proposed in the previous paper both to simplify the training algorithm and to guarantee the convergence of the neural controller. The stability analysis of the control system reveals that some unstable fixed points cannot be stabilized in the control system. Numerical experiments show that the control system works well for controlling high-dimensional chaotic systems.     

Local control of chaotic motion

On the basis of the Ott, Grebogi and Yorke method (OGY) of controlling chaotic motion by stabilizing unstable periodic orbits we propose a control method which allows a nearly continuous adjusting of the control parameter and which therefore is capable also for controlling noisy systems. Any motion which is a solution of the system's equation of motion can be stabilized, unstable periodic orbits as well as chaotic trajectories. We demonstrate the feasibility of the method by stabilizing experimentally arbitrarily chosen chaotic trajectories of a driven damped pendulum affected by noise.     

From generalized synchrony to topological decoherence: emergent sets in coupled chaotic systems

We consider the evolution of the unstable periodic orbit structure of coupled chaotic systems. This involves the creation of a complicated set outside of the synchronization manifold (the emergent set). We quantitatively identify a critical transition point in its development (the decoherence transition). For asymmetric systems we also describe a migration of unstable periodic orbits that is of central importance in understanding these systems. Our framework provides an experimentally measurable transition, even in situations where previously described bifurcation structures are inapplicable.     

Stable quantum computation of unstable classical chaos

We show on the example of the Arnold cat map that classical chaotic systems can be simulated with exponential efficiency on a quantum computer. Although classical computer errors grow exponentially with time, the quantum algorithm with moderate imperfections is able to simulate accurately the unstable chaotic classical nonlinear dynamics for long times. The algorithm can be easily implemented on systems of a few qubits.     

Control of chaos in excitable physiological systems: A geometric analysis

Model-independent chaos control techniques are inherently well-suited for the control of physiological systems for which quantitative system models are unavailable. The proportional perturbation feedback (PPF) control paradigm, which uses electrical stimulation to perturb directly the controlled system variable (e.g., the interbeat or interspike interval), was developed for excitable physiological systems that do not have an easily accessible system parameter. We develop the stable manifold placement (SMP) technique, a PPF-type technique which is simpler and more robust than the original PPF control algorithm. We use the SMP technique to control a simple geometric model of a chaotic system in the neighborhood of an unstable periodic orbit (UPO). We show that while the SMP technique can control a chaotic system that has UPO dynamics which are characterized by one stable manifold and one unstable manifold, the success of the SMP technique is sensitive to UPO parameter estimation errors. (c) 1997 American Institute of Physics.     

Controlling chaos in spatially extended beam-plasma system by the continuous delayed feedback

In this paper we discuss the control of complex spatio-temporal dynamics in a spatially extended nonlinear system (fluid model of Pierce diode) based on the concepts of controlling chaos in the systems with few degrees of freedom. A presented method is connected with stabilization of unstable homogeneous equilibrium state and the unstable spatio-temporal periodical states analogous to unstable periodic orbits of chaotic dynamics of the systems with few degrees of freedom. We show that this method is effective and allows to achieve desired regular dynamics chosen from a number of possible in the considered system.     

一类多涡卷混沌系统构造方法研究

提出一种基于平移变换的多涡卷混沌系统构造方法.首先选取合适的不稳定线性系统,利用一系列相互平行的分界面对相空间进行划分,然后通过对该线性系统进行平移变换,使新系统作用于相应的空间区域,最后利用基于符号函数的非线性函数来统一表示整个系统.根据该方法可以构造具有任意奇数个涡卷的新型混沌系统,同时也可以对某些现有混沌系统进行改进,使之成为多涡卷系统.数值仿真结果验证了该方法的可行性. 关键词： 多涡卷混沌吸引子 平移变换 涡卷     

Eigenvalues and eigenfunctions of a clover plate

We report a numerical study of the flexural modes of a plate using semi-classical analysis developed in the context of quantum systems. We first introduce the Clover billiard as a paradigm for a system inside which rays exhibit stable and chaotic trajectories. The resulting phase space explored by the ray trajectories is illustrated using the Poincare surface of section, and shows that it has both integrable and chaotic regions. Examples of the stable and the unstable periodic orbits in the geometry are presented. We numerically solve the biharmonic equation for the flexural vibrations of the Clover shaped plate with clamped boundary conditions. The first few hundred eigenvalues and the eigenfunctions are obtained using a boundary elements method. The Fourier transform of the eigenvalues show strong peaks which correspond to ray periodic orbits. However, the peaks corresponding to the shortest stable periodic orbits are not stronger than the peaks associated with unstable periodic orbits. We also perform statistics on the obtained eigenvalues and the eigenfunctions. The eigenvalue spacing distribution P(s) shows a strong peak and therefore deviates from both the Poisson and the Wigner distribution of random matrix theory at small spacings because of the C_4v symmetry of the Clover geometry. The density distribution of the eigenfunctions is observed to agree with the Porter-Thomas distribution of random matrix theory. Received 12 February 2001 and Received in final form 17 April 2001     

Chaotic Entanglement: Entropy and Geometry

In chaotic entanglement, pairs of interacting classically-chaotic systems are induced into a state of mutual stabilization that can be maintained without external controls and that exhibits several properties consistent with quantum entanglement. In such a state, the chaotic behavior of each system is stabilized onto one of the system’s many unstable periodic orbits (generally located densely on the associated attractor), and the ensuing periodicity of each system is sustained by the symbolic dynamics of its partner system, and vice versa. Notably, chaotic entanglement is an entropy-reversing event: the entropy of each member of an entangled pair decreases to zero when each system collapses onto a given period orbit. In this paper, we discuss the role that entropy plays in chaotic entanglement. We also describe the geometry that arises when pairs of entangled chaotic systems organize into coherent structures that range in complexity from simple tripartite lattices to more involved patterns. We conclude with a discussion of future research directions.     

Chaos control via a simple fractional-order controller

In this Letter, we propose a fractional-order controller to stabilize the unstable fixed points of an unstable open-loop system. Also, we show that this controller has strong ability to eliminate chaotic oscillations or reduce them to regular oscillations in the chaotic systems. This controller has simple structure and is designed very easily. To determine the control parameters, one needs only a little knowledge about the plant and therefore, the proposed controller is a suitable choice in the control of uncertain chaotic systems.     

Strange saddles and the dimensions of their invariant manifolds

Nonattracting chaotic sets play a fundamental role in typical dynamical systems. They occur, for example, in the form of chaotic transient sets and fractal basin boundaries. The subject of this paper is the dimensions of these sets and of their stable and unstable manifolds. Numerical experiments are performed to determine these dimensions. The results are consistent with a conjectured formulae expressing the dimensions in terms of Lyapunov exponents and the transient life-time associated with the strange saddle.     

Chaotic bursting as chaotic itinerancy in coupled neural oscillators

We show that chaotic bursting activity observed in coupled neural oscillators is a kind of chaotic itinerancy. In neuronal systems with phase deformation along the trajectory, diffusive coupling induces a dephasing effect. Because of this effect, an antiphase synchronized solution is stable for weak coupling, while an in-phase solution is stable for very strong coupling. For intermediate coupling, a chaotic bursting activity is generated. It is a mixture of three different states: an antiphase firing state, an in-phase firing state, and a nonfiring resting state. As we construct numerically the deformed torus manifold underlying the chaotic bursting state, it is shown that the three unstable states are connected to give rise to a global chaotic itinerancy structure. Thus we claim that chaotic itinerancy provides an alternative route to chaos via torus breakdown.     

Bubbling transition to spatial mode excitation in an extended dynamical system

We investigated the transition to spatio-temporal chaos in spatially extended nonlinear dynamical systems possessing an invariant subspace with a low-dimensional attractor. When the latter is chaotic and the subspace is transversely stable we have a spatially homogeneous state only. The onset of spatio-temporal chaos, i.e. the excitation of spatially inhomogeneous modes, occur through the loss of transversal stability of some unstable periodic orbit embedded in the chaotic attractor lying in the invariant subspace. This is a bubbling transition, since there is a switching between spatially homogeneous and nonhomogeneous states with statistical properties of on-off intermittency. Hence the onset of spatio-temporal chaos depends critically both on the existence of a chaotic attractor in the invariant subspace and its being transversely stable or unstable.     

一种具有隐藏吸引子的分数阶混沌系统的动力学分析及有限时间同步

对于具有隐藏吸引子的混沌系统,既有文献大多只针对整数阶系统进行分析与控制研究.基于Sprott E系统,构建了仅有一个稳定平衡点的分数阶混沌系统,通过相位图、Poincare映射和功率谱等,分析了该系统的基本动力学特征.结果显示,该系统展现出了丰富而复杂的动力学特性,且通过随阶次变化的分岔图可知,系统在不同阶次下呈现出周期运动、倍周期运动和混沌运动等状态,这些动力学特征对于保密通信等实际工程领域有重要的研究价值.针对该具有隐藏吸引子的分数阶系统,应用分数阶系统有限时间稳定性理论设计控制器,对系统进行有限时间同步控制,并通过数值仿真验证了其有效性.     

Crisis and unstable dimension variability in the bailout embedding map

The dynamics of inertial particles in 2-d incompressible flows can be modeled by 4-d bailout embedding maps. The density of the inertial particles, relative to the density of the fluid, is a crucial parameter which controls the dynamical behaviour of the particles. We study here the dynamical behaviour of aerosols, i.e. particles heavier than the flow. An attractor widening and merging crisis is seen in the phase space in the aerosol case. Crisis-induced intermittency is seen in the time series and the laminar length distribution of times before bursts give rise to a power law with the exponent β = −1/3. The maximum Lyapunov exponent near the crisis fluctuates around zero indicating unstable dimension variability (UDV) in the system. The presence of unstable dimension variability is confirmed by the behaviour of the probability distributions of the finite time Lyapunov exponents.