Dynamics of fractional-order sinusoidally forced simplified Lorenz system and its synchronization

In this paper, dynamical behaviors of the fractional-order sinusoidally forced simplified Lorenz are investigated by employing the time-domain solution algorithm of fractional-order calculus. The system parameters and the fractional derivative orders q are treated as bifurcation parameters. The range of the bifurcation parameters in which the system generates chaos is determined by bifurcation, phase portrait, and Poincaré section, and different bifurcation motions are visualized by virtue of a systematic numerical analysis. We find that the lowest order of this system to yield chaos is 3.903. Based on fractional-order stability theory, synchronization is achieved by using nonlinear feedback control method. Simulation results show the scheme is effective and a chaotic secure communication scheme is present based on this synchronization.     

Limit cycles for extended bistable stochastic resonance system

Bistable system is a typical model in stochastic resonance (SR) researching. And it has been known that the deterministic dynamics of a noised system plays an important role in making SR phenomena occur. By embedding a non-autonomous system onto a cylinder, or extending the one-dimensional non-autonomous system as a three-dimensional autonomous system restricted on a cylinder, this paper theoretically analyzes the dynamics of the well-known periodically driven bistable system. After three limit cycles (equivalent to the periodic states of the un-extended system) are proved existent for subthreshold case, they are shown to be preserved even in the suprathreshold case with large driving frequency. Moreover, it is also proved in suprathreshold case that the three ones keep touching each other and combine into one larger globally stable limit cycle as the driving frequency decreases to some certain degree.     

Chaos and limit cycles in the Lorenz model

The Lorenz model is interpreted as a damping motion under a time-dependent force. The range of the Rayleigh number r in which limit cycles exist is studied by numerical simulation. The shape of the limit cycle is given.     

Limit cycles in quantum theories

Renormalization group limit cycles and more chaotic behavior may be commonplace for quantum Hamiltonians requiring renormalization, in contrast to experience based on classical models with critical behavior, where fixed points are far more common. We discuss the simplest quantum model Hamiltonian identified so far that exhibits a renormalization group with both limit cycle and chaotic behavior. The model is a discrete Hermitian matrix with two coupling constants, both governed by a nonperturbative renormalization group equation that involves changes in only one of these couplings and is soluble analytically.     

Limit Cycles near Stationary Points in the Lorenz System

The limit cycles in the Lorenz system near the stationary points are analysed numerically. A plane in phase space of the linear Lorenz system is used to locate suitable initial points of trajectories near the limit cycles. The numerical results show a stable and an unstable limit cycle near the stationary point. The stable limit cycle is smaller than the unstable one and has not been previously reported in the literature. In addition, all the limit cycles in the Lorenz system are theoreticallv Proven not to be planar.     

Limit cycles in self-excited multi-degree-of-freedom systems

A possibility of generating stable multi-frequency almost-periodic limit cycles in n-degree-of-freedom self-excited systems is investigated. Analytical approximate methods are used. Systems with non-linear forces described by analytical functions are considered. Several examples of two-degree-of-freedom systems with van der Pol terms are analyzed in detail. The effect of non-linear restoring forces is also considered. The possibility of occurrence of multi-frequency limit cycles is proved by means of analytical methods and confirmed by analogue computer results.     

Limit cycles and chaos in realistic models of the Belousov-Zhabotinskii reaction system

Effects of spatial variation in the Belousov-Zhabotinskii reaction is studied numerically by adopting the Field-Noyes kinetics (Oregonator) and the Zhabotinskii-Zaikin-Korzukhin-Kreitser kinetics. This is carried out for a spatially-discrete model composed ofN equivalent cells interacting through gradient coupling. When the system is near the boundary at which a uniform steady state bifurcates into a limit cycle, it is found with the aid of a perturbation expansion that the above models withN=3 exhibit various types of oscillations depending on the interaction strength between cells. Chaotic characteristics are also observed for a certain region of parameters. It is shown that the ZZKK model withN=2 exhibits a different kind of chaos when the size of the limit cycle becomes sensitive to external parameters, e.g., the concentrations of bromate ion or bromomalonic acid. Although each cell is equivalent, symmetry about cell numbers usually breaks down in a periodic state. It is found, however, that symmetry is recovered for the former kind of chaos, while the latter kind of chaos, there exists an asymmetric chaos as well as symmetric chaos. This has been examined by the time evolution of a certain concentration variable and by its Lorenz plot. In the asymmetric chaos, the Lorenz plot constitutes approximately a one-dimensional map. Furthermore, possible connections of the present limit cycles and chaos with the experiments of Zhabotinskii and Vavilin-Zhabotinskii-Zaikin are suggested.     

Global stabilization of a Lorenz system

In this paper,using feedback linearizing technique,we show that a Lorenz system can be considered as a cascade system.Moreover,this system satisfies the assumptions of global stabilization of casecade systems.Thus coninuous state feedback control laws are proposed to globally stabilize the Lorenz system at the equilibrium point.Simulation results are presented to verify our method.This method can be further generalized to other chaotic systems such as Chen system,coupled dynamos system,etc.     

超混沌Lorenz系统

对Lorenz系统添加一个非线性控制器，使之构成四维超混沌Lorenz系统.利用分岔图、Lyapunov指数谱及相图分析方法，研究了超混沌Lorenz系统的运动规律.数值模拟结果表明：新引入参数处于不同取值范围时，超混沌Lorenz系统可以分别呈现收敛、发散、周期、混沌及超混沌动力学行为. 关键词： Lorenz系统 超混沌 Lyapunov指数 分岔     

A generalized Lorenz system

A 14-dimensional generalized Lorenz system of ordinary differential equations is constructed and its bifurcation sequence is then studied numerically. Several fundamental differences are found which serve to distinguish this model from Lorenz's original one, the most unexpected of which is a family of invariant two-tori whose ultimate bifurcation leads to a strange attractor. The strange attractor seems to have many of the gross features observed in Lorenz's model and therefore is an excellent candidate for a higher dimensional analogue.On leave from Department of Mathematics, Howard University, Washington, DC, USAThe National Center for Atmospheric Research is sponsored by the National Science Foundation     

Limit cycles for a class of second order differential equations

We study the limit cycles of a wide class of second order differential equations, which can be seen as a particular perturbation of the harmonic oscillator. In particular, by choosing adequately the perturbed function we show, using the averaging theory, that it is possible to obtain as many limit cycles as we want.     

Limit cycles and detailed balance in Fokker-Planck equations

A class of non-linear Fokker Planck equations with exactly known steady state solution is investigated and its relation with the models considered earlier in the literature is discussed. A characterisation of the absence of detailed balance and possible existence of limit cycles is given. The implications of detailed balance on the existence and the character of limit cycle behaviour are studied. It is shown that detailed balance does not determine the existence or non-existence of limit cycles but rather their character.     

Limit cycles and detailed balance in Fokker-Planck equations

A class of non-linear Fokker Planck equations with exactly known steady state solution is investigated and its relation with the models considered earlier in the literature is discussed. A characterisation of the absence of detailed balance and possible existence of limit cycles is given. The implications of detailed balance on the existence and the character of limit cycle behaviour are studied. It is shown that detailed balance does not determine the existence or non-existence of limit cycles but rather their character.     

Projective synchronization of a new hyperchaotic Lorenz system

This Letter mainly concerns projective synchronization (PS) of a new hyperchaotic Lorenz system. PS with both identical and different scaling factors between two hyperchaotic Lorenz systems are realized. A general sufficient condition for PS in a certain class of chaotic (hyperchaotic) system with uncertainties is obtained by using adaptive control. Numerical simulations are performed to verify and illustrate the analytical results.     

一个临界系统与Lorenz系统和Chen系统的异结构同步

对最近提出的一个临界系统，设计了一个非线性控制器，使得系统的第一个状态信号以指数收敛速度追踪任意给定的参考信号；利用Active控制实现了这个临界系统与Lorenz系统以及Chen系统的异结构同步.大量数值实验验证了理论结果. 关键词： 追踪控制 异结构同步 混沌系统     

Fluctuational escape from a quasi-hyperbolic attractor in the Lorenz system

Noise-induced escape from the basin of attraction of a quasi-hyperbolic chaotic attractor in the Lorenz system is considered. The investigation is carried out in terms of the theory of large fluctuations by experimentally analyzing the escape prehistory. The optimal escape trajectory is shown to be unique and determined by the saddle-point manifolds of the Lorenz system. We established that the escape process consists of three stages and that noise plays a fundamentally different role at each of these stages. The dynamics of fluctuational escape from a quasi-hyperbolic attractor is shown to differ fundamentally from the dynamics of escape from a nonhyperbolic attractor considered previously [1]. We discuss the possibility of analytically describing large noise-induced deviations from a quasi-hyperbolic chaotic attractor and outline the range of outstanding problems in this field.     

Hyperchaos and quasiperiodicity from a four-dimensional system based on the Lorenz system

This paper reports on numerically computed parameter plane plots for a dynamical system modeled by a set of five-parameter, four autonomous first-order nonlinear ordinary differential equations. The dynamical behavior of each point, in each parameter plane, is characterized by Lyapunov exponents spectra. Each of these diagrams indicates parameter values for which hyperchaos, chaos, quasiperiodicity, and periodicity may be found. In fact, each diagram shows delimited regions where each of these behaviors happens. Moreover, it is shown that some of these parameter planes display organized periodic structures embedded in quasiperiodic and chaotic regions.