Relative Entropy and Relative Entropy of Entanglement for Infinite-Dimensional Systems

In this paper, firstly, we derive some inequalities about the relative entropy for infinite-dimensional quantum systems. Secondly, we propose a new measurement based on the relative entropy of entanglement for infinite-dimensional systems with bounded mean energy, and give a lower bound on this entanglement measure. Lastly, we generalize this measure to multi-partite quantum systems.     

Recovery of parameters of delayed-feedback systems from chaotic time series

Methods for reconstructing a delay differential equation from the time series of an observable quantity are proposed for various classes of time-delay systems. The methods rely on knowledge of the distributions of extrema of the time series of observed oscillations and projection of the infinite-dimensional phase spaces of time-delay systems onto special low-dimensional subspaces. The effectiveness of the proposed methods is demonstrated by reconstructing delay differential equations from their chaotic solutions, including those corrupted by noise, and by constructing models of real time-delay systems from chaotic time series.     

A Hopf-like equation and perturbation theory for differential delay equations

We extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite dimensional, phase-space densities for these systems are functionals. We derive a Hopf-like functional differential equation governing the evolution of these densities. The functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory. A necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given. Finally, we show that the evolution equation for the density functional is the Fourier transform of the infinite-dimensional version of the Kramers-Moyal expansion.     

Laser stabilization with multiple-delay feedback control

Stabilization of chaotic intensity fluctuations of intracavity frequency-doubled solid-state (Nd: YAG) lasers using multiple-delay feedback control (MDFC) is demonstrated by numerical simulations. It is shown that MDFC not only provides stable (cw) output for constant pump rates but also works with slowly varying pump currents, resulting in corresponding (nonchaotic) intensity modulations.     

On deformations of quasi-Miura transformations and the Dubrovin–Zhang bracket

In our recent paper, we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of partial differential equations (PDEs) associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin and Zhang, and the bracket is the first Poisson structure of their hierarchy.     

Entropy Change in Quantum Measurements for Infinite-Dimensional Quantum Systems

International Journal of Theoretical Physics - In this paper, first, we prove the projective measurement increases the entropy in infinite-dimensional quantum systems, and discuss some useful...     

Characterization of chaos

We give a brief introduction to chaos and its characterization. We examine some standard systems in detail from the perspective of chaos and review their properties. Concepts necessary to understand them, such as dimension, are also reviewed. To illustrate the main ideas, we choose three examples which have served as paradigms for the study of chaos in physical systems, namely, the Hénon discrete mapping, the Lorenz system of coupled ordinary differential equations, and the Mackey-Glass infinite-dimensional delay differential equation.     

An expansion of system with time delayed feedback control into spatio-temporal state space

Time delayed feedback control is well known as an effective continuous control method for stabilizing the unstable periodic orbit embedded in chaotic attractors. As for the system with time delay, the solution is in a function space and shows characteristics governed by an infinite dimension. Therefore it is difficult to understand the system behavior analytically. In this paper, it is shown that, when the state space for the system with time delay is expanded into the spatio-temporal state space, the solution propagates in the space as a wave theoretically and numerically. The dynamic behavior experimentally obtained in the sinusoidally excited magneto-elastic beam system under time delayed feedback control is also discussed by the transformation into the expanded spatio-temporal state space. (c) 1999 American Institute of Physics.     

Nano-pattern stabilization by multiplicative noise

Within the reaction-diffusion framework, a one-component system was confined by means of a multiplicative noise into the attraction basin of a patterning attractor in its (infinite-dimensional) configuration space. In this way, inhomogeneities that otherwise would have been “expelled” from this basin, and subsequently eliminated through the diffusive process, were stabilized. For the present study, a model describing the physics of adsorbed particles on a metallic surface has been used. In particular, an underlying deterministic inhomogeneity-building mechanism was exploited, that acts driven by lateral interactions among the adsorbed particles. This process cannot by itself sustain and stabilize the inhomogeneities, but together with the contribution of a particular form of multiplicative noise, it is able to confine the system into the region of configuration space where this mechanism is enabled, hence stabilizing the pattern. Although the proposal could be applied to more general situations, for the particular model studied here we have found that nanopatterns that without the indicated noise source would be eliminated by diffusion, under its effect can grow and be stabilized.     

Controlling chaos using Takagi--Sugeno fuzzy model and adaptive adjustment

In this paper, an approach to the control of continuous-time chaotic systems is proposed using the Takagi--Sugeno (TS) fuzzy model and adaptive adjustment. Sufficient conditions are derived to guarantee chaos control from Lyapunov stability theory. The proposed approach offers a systematic design procedure for stabilizing a large class of chaotic systems in the literature about chaos research. The simulation results on R\"{o}ssler's system verify the effectiveness of the proposed methods.     

Symmetries and casimir of an extended classical long wave system

In this paper, we derive Lie point, generalized, master and time-dependent symmetries of a dispersionless equation, which is an extension of a classical long wave system. This equation also admits an infinite-dimensional Lie algebraic structure of Virasoro-type, as in the dispersive integrable systems. We discuss the construction of a sequence of negative ranking symmetries through the property of uniformity in rank. More interestingly, we obtain the conserved quantities directly from the casimir of Poisson pencil.     

Instantaneous optimal method for vibration control of linear sampled-data systems with time delay in control

In an actively controlled system, time delay exists inevitably. Neglecting time delay may cause degradation of control performance or even induce instability to the dynamic system. In this paper, instantaneous optimal control method for vibration control of linear sampled-data systems with time delay in control is investigated. By a peculiar integral transformation, the first order state equation with time delay is transformed into the standard first order state equation, which contains no time delay. Then the optimal controller is designed based on the numerical algorithm of the regular fourth order Runge-Kutta method. Since the obtained controller contains integral term, which is not practical for control implementation, the numerical algorithm for this integral term is investigated too. Since the controller is deduced directly from the time-delay differential equation, the control method presented is prone to guarantee system stability. Thus the presented control method can be applicable to the case of large time delay. The performance of the control method is demonstrated by numerical simulation. Simulation results indicate that this control method is feasible and is an attractive strategy for dealing with the time delay in vibration control systems and is effective in suppressing maximum structural responses. Instability in structural responses may occur if the systems with time delay are controlled using the controller designed in the case of no time delay.     

A separation principle for dynamical delayed output feedback control of chaos

In this Letter, a dynamical delayed output-feedback (DDOF) control strategy is proposed for stabilizing unstable periodic orbits (UPOs) of chaotic systems. Using the Floquet theory, a separation principle is established which gives a necessary and sufficient stability condition for DDOF UPO stabilizing control systems. The new principle shows that the so-called “odd number limitation” for delayed state-feedback control systems also applies to DDOF control.     

Progress in the control of chaos

In this review, we summarize several of the key contributions made over the past 5 years to the control of chaotic dynamical systems. The idea that chaotic systems can in fact be controlled may be counter-intuitive; after all they are unpredictable in the long term. Nevertheless, numerous theorists have independently developed methods which can be applied to chaotic systems, and many experimentalists have demonstrated that physical chaotic systems respond well to both simple and sophisticated control strategies. The great bulk of these researchers have restricted their study to low-dimensional systems, and correspondingly we critique this work at length. Most recently, a few researchers have proposed control techniques for application to high- or infinite-dimensional systems, and we describe this work in some detail as well.     

Nambu-Poisson dynamics of superintegrable systems

After an introduction to Nambu-Poisson dynamics (NPD), some applications of NPD in finite-dimensional (superintegrable) and infinite-dimensional (extended quantum mechanics and hydrodynamics) systems are considered. The text was submitted by the author in English.     

Design of a negative group delay filter via reservoir computing approach: Real-time prediction of chaotic signals

An optimal arbitrary order filter with negative group delay is proposed for a real-time prediction of complex band-limited signals. The filter consists of a set of second-order linear dissipative oscillators and is determined by a rational transfer function with fixed poles and optimized zeros. The filter design and optimization algorithm is based on a reservoir computing approach. The predictive properties of the filter are demonstrated numerically for two chaotic model systems — the finite-dimensional Rössler system and the infinite-dimensional Mackey-Glass system, as well as for the real biological signal of the fingertip photoplethysmogram.     

Switch control for piecewise affine chaotic systems

Switch control can be imposed naturally on the piecewise affine system, where the control action switches from an affine subsystem to another according to switch conditions depending on the system states. In this paper we present such piecewise feedback control for stabilizing unstable equilibrium points of piecewise affine systems. The noise effect on the stabilization is also investigated. The original Chua's circuit is used to illustrate our results.     

基于时延补偿机理的网络化输出反馈控制器设计

本文考虑前向和反馈通道均存在网络时延的网络化控制系统,提出了一种新的动态输出反馈控制器的设计方法.针对状态可测和不可测两种情况,整个设计过程采用不同的时延补偿机理,来主动地消除网络时延的影响.同时,讨论了闭环网络化控制系统的稳定性.最后的仿真实例表明了该方法的有效性.     

Chaotic dynamics of the fractional-order Ikeda delay system and its synchronization

In this paper we numerically investigate the chaotic behaviours of the fractional-order Ikeda delay system. The results show that chaos exists in the fractional-order Ikeda delay system with order less than 1. The lowest order for chaos to be able to appear in this system is found to be 0.1. Master--slave synchronization of chaotic fractional-order Ikeda delay systems with linear coupling is also studied.