Existence of horseshoe maps in voltage-mode controlled buck converters

In this paper, chaos in a voltage-mode controlled buck converter is studied. The existence of chaos is proven theoretically in this system. The proof consists of showing that the dynamics of the system is semiconjugate to that of a one-sided shift map, which implies positive entropy of the system and hence chaotic behaviour. The essential tool is the horseshoe hypotheses proposed by Kennedy and Yorke, which will be reviewed prior to the discussion of the main finding.     

Routes to chaos,universality and glass formation

We review recent results obtained for the dynamics of incipient chaos. These results suggest a common picture underlying the three universal routes to chaos displayed by the prototypical logistic and circle maps. Namely, the period doubling, intermittency, and quasiperiodicity routes. In these situations the dynamical behavior is exactly describable through infinite families of Tsallis’ q-exponential functions. Furthermore, the addition of a noise perturbation to the dynamics at the onset of chaos of the logistic map allows to establish parallels with the behavior of supercooled liquids close to glass formation. Specifically, the occurrence of two-step relaxation, aging with its characteristic scaling property, and subdiffusion and arrest is corroborated for such a system.     

Feedback Induced Instability and Chaos in Semiconductor Lasers and Their Applications

Semiconductor laser with feedback is an excellent model for nonlinear optical system which shows chaotic dynamics. It is interesting not only from the fundamental physical study but also application standpoints of view. The dynamics of feedback induced instability and chaos, especially for optical feedback, and their applications are reviewed in this paper. The model of such a system is described by the laser rate equations. At first the dynamic behaviors of feedback induced instability and chaos in semiconductor lasers are discussed on the basis of the theory and experiments. Instability and chaos may be stabilized by the method of chaos control. Then we apply the method to suppress the noise induced by the feedback in a semiconductor laser. The synchronization of chaos between two similar systems is also an important issue in chaos applications and we discuss secure communications based on chaos synchronization. Some other examples of applications of feedback induced chaos are also described.     

de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

A recently developed unified theory of classical and quantum chaos, based on the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigorously defined to be chaotic if the Lyapunov number, associated with the quantum trajectories in de Broglie-Bohm phase space, is positive definite. This definition of quantum chaos which under classical conditions goes over to the well-known definition of classical chaos in terms of positivity of Lyapunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positive Lyapunov numbers in a simple quantum system is given analytically, proving the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated. It is shown that the rigorous deterministic quantum chaos provides an intrinsic mechanism towards irreversibility of the Schrodinger evolution of the wave function, without invoking ‘wave function collapse’ or ‘measurements’     

DC-DC变换器的信息熵分析

为确定不同反馈系数k下DC-DC变换器系统的行为,结合系统处于周期状态时的稳定性和系统处于混沌时不会重复经过每一点的特点,提出了一种采用极限思想和信息熵来估计DC-DC变换器非线性行为的方法.该方法能准确分析系统处于周期状态和混沌状态的熵值,量化了DC-DC变换器倍周期分叉和混沌行为.以一阶电压反馈DCM Boost变换器和DCM Buck变换器为例进行仿真.研究结果表明,所提出的信息熵可以准确反映分叉点、周期数及产生混沌的位置,完善了该类变换器非线性动力学分析的理论和方法.     

Non-standard finite-difference time-domain method for solving the Schrödinger equation

The literature on chaos has highlighted several chaotic systems with special features. In this work, a novel chaotic jerk system with non-hyperbolic equilibrium is proposed. The dynamics of this new system is revealed through equilibrium analysis, phase portrait, bifurcation diagram and Lyapunov exponents. In addition, we investigate the time-delay effects on the proposed system. Realisation of such a system is presented to verify its feasibility.     

Chaos, fractals, and atomic flights in cavities

A semiclassical study is carried out of the nonlinear interaction dynamics between two-level atoms and a standing-wave field in a high-finesse cavity. As a result of atomic movement or wave amplitude modulation, a dynamic local instability occurs in a strongly coupled atom-field system. The appearance of dynamical Hamiltonian chaos, fractals, and Lévy flights is demonstrated for the models of two experimental devices: a (micro)maser with thermal Rydberg atoms and a microlaser with cold atoms. Numerical simulation showed that the manifestations of classical chaos, atomic fractals, and flights can be observed in the appropriate real experiments. Attention is drawn to the prospects provided by work on the atom-field systems in the coupling-modulated high-finesse cavities for further investigation of the quantum-classical correspondence, quantum chaos, and decoherence.     

Wideband chaos generation using a delayed oscillator and a two-dimensional nonlinearity induced by a quadrature phase-shift-keying electro-optic modulator

We present a new optoelectronic architecture intended for chaotic optical intensity generation. The principle relies on an electro-optic nonlinear delay dynamics, where the nonlinearity originates from an integrated four-wave optical interferometer, involving two independent electro-optic modulation inputs. Consequently, the setup involves both two-dimensional nonlinearity and dual-delay feedback dynamics, which results in enhanced chaos complexity of particular interest in chaos encryption schemes. The generated chaos observed with large feedback gains has a bandwidth ranging from 30 kHz to 13 GHz and is confirmed by numerical simulations of the proposed dynamical model and bifurcation diagram calculation.     

Hidden attractors in a new fractional–order discrete system: Chaos,complexity, entropy, and control

This paper studies the dynamics of a new fractional-order discrete system based on the Caputo-like difference operator.This is the first study to explore a three-dimensional fractional-order discrete chaotic system without equilibrium. Through phase portrait, bifurcation diagrams, and largest Lyapunov exponents, it is shown that the proposed fractional-order discrete system exhibits a range of different dynamical behaviors. Also, different tests are used to confirm the existence of chaos,such as 0–1 test and C0 complexity. In addition, the quantification of the level of chaos in the new fractional-order discrete system is measured by the approximate entropy technique. Furthermore, based on the fractional linearization method, a one-dimensional controller to stabilize the new system is proposed. Numerical results are presented to validate the findings of the paper.     

Chaos control in biological system using recursive backstepping sliding mode control

This paper puts forward the control of chaos in the biological system. A new controller based on recursive backstepping sliding mode control is proposed such that it can control the chaotic dynamics in the biological system to stabilize at any position or to track any trajectory that is a smooth function of time. A proportional integral switching surface is proposed to achieve the stability condition of the error dynamics. Unlike the open loop and open plus closed loop control techniques, the design of proposed controller does not require the parameter perturbation. The required stability condition is derived based on Lyapunov stability theory. Simulation is achieved in MATLAB environment. Numerical simulation results are presented in order to show the effective verification of the proposed controller design. Simulation results correspond that the objective of chaos control is achieved successfully.     

Analysis,synchronization and circuit design of a novel butterfly attractor

This research paper introduces a novel three-dimensional autonomous system, whose dynamics support periodic and chaotic butterfly attractors as certain parameters vary. A special case of this system, exhibiting reflectional symmetry, is amenable to analytical and numerical analysis. Qualitative properties of the new chaotic system are discussed in detail. Adaptive control laws are derived to achieve global chaotic synchronization of the new chaotic system with unknown parameters. Furthermore, a novel electronic circuit realization of the new chaotic system is presented, examined and realized using Orcad-PSpice program and physical components. The proposed novel butterfly chaotic attractor is very useful for the deliberate generation of chaos in applications.     

Analysis of Chaotic Dynamics by the Extended Entropic Chaos Degree

The Lyapunov exponent is the most-well-known measure for quantifying chaos in a dynamical system. However, its computation for any time series without information regarding a dynamical system is challenging because the Jacobian matrix of the map generating the dynamical system is required. The entropic chaos degree measures the chaos of a dynamical system as an information quantity in the framework of Information Dynamics and can be directly computed for any time series even if the dynamical system is unknown. A recent study introduced the extended entropic chaos degree, which attained the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. Moreover, an improved calculation formula for the extended entropic chaos degree was recently proposed to obtain appropriate numerical computation results for multidimensional chaotic maps. This study shows that all Lyapunov exponents of a chaotic map can be estimated to calculate the extended entropic chaos degree and proposes a computational algorithm for the extended entropic chaos degree; furthermore, this computational algorithm was applied to one and two-dimensional chaotic maps. The results indicate that the extended entropic chaos degree may be a viable alternative to the Lyapunov exponent for both one and two-dimensional chaotic dynamics.     

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The dynamics of cold atoms in conservative optical lattices obviously depends on the geometry of the lattice. But very similar lattices may lead to deeply different dynamics. In a 2D optical lattice with a square mesh, it is expected that the coupling between the degrees of freedom leads to chaotic motions. However, in some conditions, chaos remains marginal. The aim of this paper is to understand the dynamical mechanisms inhibiting the appearance of chaos in such a case. As the quantum dynamics of a system is defined as a function of its classical dynamics – e.g. quantum chaos is defined as the quantum regime of a system whose classical dynamics is chaotic – we focus here on the dynamical regimes of classical atoms inside a well. We show that when chaos is inhibited, the motions in the two directions of space are frequency locked in most of the phase space, for most of the parameters of the lattice and atoms. This synchronization, not as strict as that of a dissipative system, is nevertheless a mechanism powerful enough to explain that chaos cannot appear in such conditions.     

Noise can prevent onset of chaos in spatiotemporal population dynamics

Many theoretical approaches predict the dynamics of interacting populations to be chaotic but that has very rarely been observed in ecological data. It has therefore risen a question about factors that can prevent the onset of chaos by, for instance, making the population fluctuations synchronized over the whole habitat. One such factor is stochasticity. The so-called Moran effect predicts that a spatially correlated noise can synchronize the local population dynamics in a spatially discrete system, thus preventing the onset of spatiotemporal chaos. On the whole, however, the issue of noise has remained controversial and insufficiently understood. In particular, a well-built nonspatial theory infers that noise enhances chaos by making the system more sensitive to the initial conditions. In this paper, we address the problem of the interplay between deterministic dynamics and noise by considering a spatially explicit predator-prey system where some parameters are affected by noise. Our findings are rather counter-intuitive. We show that a small noise (i.e. preserving the deterministic skeleton) can indeed synchronize the population oscillations throughout space and hence keep the dynamics regular, but the dependence of the chaos prevention probability on the noise intensity is of resonance type. Once chaos has developed, it appears to be stable with respect to a small noise but it can be suppressed by a large noise. Finally, we show that our results are in a good qualitative agreement with some available field data.     

Period doubling cascade and deterministic chaos in a laser self-mode-locked by the combination of inertial negative and positive feedbacks

We proposed a simple laser system that demonstrates the nonlinear dynamics typical for the logistic map: period doubling cascade, periodicity windows, intermittency, and deterministic chaos on the time scale of the cavity round trip.     

Generality of deterministic chaos, exponential spectra, and Lorentzian pulses in magnetically confined plasmas

The dynamics of transport at the edge of magnetized plasmas is deterministic chaos. The connection is made by a previous survey [M. A. Pedrosa et al., Phys. Rev. Lett. 82, 3621 (1999)] of measurements of fluctuations that is shown to exhibit power spectra with exponential frequency dependence over a broad range, which is the signature of deterministic chaos. The exponential character arises from Lorentzian pulses. The results suggest that the generalization to complex times used in studies of deterministic chaos is a representation of Lorentzian pulses emerging from the chaotic dynamics.     

基于反演自适应动态滑模的FitzHugh-Nagumo神经元混沌同步控制

本文采用反演自适应动态滑模控制实现耦合FitzHugh-Nagumo (FHN) 神经元混沌同步. 该方法将自适应技术与反演控制方法相结合, 通过设计新型切换函数, 采用动态滑模控制律, 实现了带有不确定参数的耦合FHN神经元混沌放电同步. 研究表明该方法可以有效地削弱系统的抖振, 从而避免破坏神经元的本质特性, 且响应速度快. 仿真结果证明了该控制方法的有效性. 关键词： 自适应 动态滑模控制 FitzHugh-Nagumo神经元 混沌同步     

基于线性矩阵不等式的一类新羽翼倍增混沌分析与控制

在对已有的混沌系统分析和研究的基础上,将一个二次混沌系统第三个方程关于x的线性项引入到第二个方程中,通过对该系统第二个等式中的线性项x作绝对值运算,提出了一类新的二次非线性系统.采用非线性动力学方法分析了系统参数变化时所经历的稳定、准周期、混沌的过渡过程,模拟电路实验结果与Matlab数值仿真结果相一致.分析发现混沌态时绝对值运算后的系统比原系统的Lyapunov指数更大,并可将原系统的混沌吸引子由两个翼的拓扑结构变为四翼的拓扑结构,从而实现羽翼倍增.针对该混沌特性更强的羽翼倍增混沌系统,基于Takagi-Sugeno(T-S)模糊模型和线性矩阵不等式(LMI),设计出使该羽翼倍增混沌系统渐近稳定的鲁棒模糊控制器.仿真结果证实了所提出定理和设计控制器的有效性.     

Extensive chaos in the Lorenz-96 model

We explore the high-dimensional chaotic dynamics of the Lorenz-96 model by computing the variation of the fractal dimension with system parameters. The Lorenz-96 model is a continuous in time and discrete in space model first proposed by Lorenz to study fundamental issues regarding the forecasting of spatially extended chaotic systems such as the atmosphere. First, we explore the spatiotemporal chaos limit by increasing the system size while holding the magnitude of the external forcing constant. Second, we explore the strong driving limit by increasing the external forcing while holding the system size fixed. As the system size is increased for small values of the forcing we find dynamical states that alternate between periodic and chaotic dynamics. The windows of chaos are extensive, on average, with relative deviations from extensivity on the order of 20%. For intermediate values of the forcing we find chaotic dynamics for all system sizes past a critical value. The fractal dimension exhibits a maximum deviation from extensivity on the order of 5% for small changes in system size and the deviation from extensivity decreases nonmonotonically with increasing system size. The length scale describing the deviations from extensivity is consistent with the natural chaotic length scale in support of the suggestion that deviations from extensivity are due to the addition of chaotic degrees of freedom as the system size is increased. We find that each wavelength of the deviation from extensive chaos contains on the order of two chaotic degrees of freedom. As the forcing is increased, at constant system size, the dimension density grows monotonically and saturates at a value less than unity. We use this to quantify the decreasing size of chaotic degrees of freedom with increased forcing which we compare with spatial features of the patterns.     

Origin of transient and intermittent dynamics in spatiotemporal chaotic systems

Nonattracting chaotic sets (chaotic saddles) are shown to be responsible for transient and intermittent dynamics in an extended system exemplified by a nonlinear regularized long-wave equation, relevant to plasma and fluid studies. As the driver amplitude is increased, the system undergoes a transition from quasiperiodicity to temporal chaos, then to spatiotemporal chaos. The resulting intermittent time series of spatiotemporal chaos displays random switching between laminar and bursty phases. We identify temporally and spatiotemporally chaotic saddles which are responsible for the laminar and bursty phases, respectively. Prior to the transition to spatiotemporal chaos, a spatiotemporally chaotic saddle is responsible for chaotic transients that mimic the dynamics of the post-transition attractor.