Robust chaos synchronization of fractional-order chaotic systems with unknown parameters and uncertain perturbations

Chaotic systems in practice are always influenced by some uncertainties and external disturbances. This paper investigates the problem of practical synchronization of fractional-order chaotic systems. Based on Lyapunov stability theory and a fractional-order differential inequality, a modified adaptive control scheme and adaptive laws of parameters are developed to robustly synchronize coupled fractional-order chaotic systems with unknown parameters and uncertain perturbations. This synchronization approach is simple, global and theoretically rigorous. Simulation results for two fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

Non-observable chaos in piecewise smooth systems

In the present paper, we discuss bifurcations of chaotic attractors in piecewise smooth one-dimensional maps with a high number of switching manifolds. As an example, we consider models of DC/AC power electronic converters (inverters). We demonstrate that chaotic attractors in the considered class of models may contain parts of a very low density, which are unlikely to be observed, neither in physical experiments nor in numerical simulations. We explain how the usual bifurcations of chaotic attractors (merging, expansion and final bifurcations) in piecewise smooth maps with a high number of switching manifolds occur in a specific way, involving low-density parts of attractors, and how this leads to an unusual shape of the bifurcation diagrams.     

Non-existence of Shilnikov chaos in continuous-time systems

In this paper,a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional,continuous-time,and smooth systems is obtained.Based on this result and an elementary example,it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation(ODE) systems characterized by the nonexistence of homoclinic and heteroclinic orbits.     

Impulsive stabilization of chaos in fractional-order systems

Nonlinear Dynamics - This paper considers a class of nonlinear impulsive Caputo differential equations of fractional order, which models chaotic systems. Computer-assisted proof of chaos...     

Bifurcations and chaos in parametrically excited single-degree-of-freedom systems

The behavior of single-degree-of-freedom systems possessing quadratic and cubic nonlinearities subject to parametric excitation is investigated. Both fundamental and principal parametric resonances are considered. A global bifurcation diagram in the excitation amplitude and excitation frequency domain is presented showing different possible stable steady-state solutions (attractors). Fractal basin maps for fundamental and principal parametric resonances when three attractors coexist are presented in color. An enlargement of one region of the map for principal parametric resonance reveals a Cantor-like set of fractal boundaries. For some cases, both periodic and chaotic attractors coexist.     

Nonlinear dynamics and chaos in shape memory alloy systems

Smart material systems and structures have remarkable properties responsible for their application in different fields of human knowledge. Shape memory alloys, piezoelectric ceramics, magnetorheological fluids, and magnetostritive materials constitute the most important materials that belong to the smart materials category. Shape memory alloys (SMAs) are metallic alloys usually employed when large forces and displacements are required. Applications in aerospace structures, rotordynamics and several bioengineering devices are investigated nowadays. In terms of applied dynamics, SMAs are being used in order to exploit adaptive dissipation associated with hysteresis loop and the mechanical property changes due to phase transformations. This paper presents a general overview of nonlinear dynamics and chaos of smart material systems built with SMAs. Oscillators, vibration absorbers, impact systems and structural systems are of concern. Results show several possibilities where SMAs can be employed for dynamical applications.     

Parametric open-plus-closed-loop control of chaos in continuous dynamical systems

This paper presents a parametric open-plus-closed-loop control approach to controlling chaos in continuous dynamical systems. As an example, chaos in the Lorenz model is controlled to demonstrate its application. Finally, the relations between the parametric open-plus-closed-loop control and the former control methods, such as the open-plus-closed-loop control and the parametric entrainment control, are discussed.Supported by the Science Foundation of the State Education Commission for Doctorate Program, and the Applied Science Foundation of the State Ministry of Metallurgical Industry.     

Complete identification of chaos of nonlinear nonholonomic systems

The chaos of nonholonomic systems with two external nonlinear nonholonomic constraints where the magnitude of velocity is a constant and the magnitude of the velocity is a constant with a periodic disturbance, respectively, is completely identified for the first time. The scope of the chaos study is extended to nonlinear nonholonomic systems. By applying the nonlinear nonholonomic form of Lagrange’s equations, the dynamic equation is expressed. The existence of chaos in these two nonlinear nonholonomic systems is first wholly proved by all numerical criteria of chaos, i.e., the most reliable Lyapunov exponents, phase portraits, Poincaré maps, and bifurcation diagrams. Furthermore, it is found that the Feigenbaum number still holds for nonlinear nonholonomic systems.     

Chaos in three-dimensional hybrid systems and design of chaos generators

The existence of horseshoes is proved in a class of 3-dim piecewise linear systems, in which a homoclinic orbit connecting the origin to itself is explicitly given. Based on these results, a mathematically rigorous methodology for design of chaos generators is proposed. Implementation of such chaos generators by circuit is easy and the chaotic attractor is robust under small perturbations.     

Adaptive finite-time control of chaos in permanent magnet synchronous motor with uncertain parameters

Permanent magnet synchronous motor (PMSM) exhibits chaotic behavior when its parameters are within a certain range which seriously affect the stable work of PMSM. In order to eliminate the chaos, many approaches have been proposed. Most of them considered asymptotic stability of the system, while finite-time stability makes more sense in practice. In addition, parameters of PMSM may be uncertain because of some external factors, then adaptive control is a good method to be considered. In this paper, adaptive finite-time stabilization problem is considered to eliminate the chaos in PMSM system with uncertain parameters. To show the effectiveness of the proposed method, some simulation results are provided.     

Motion types and chaos of multi-body systems vibrating with impacts

In this paper, the limit sets theory for an autonomous dynamical system is generalized to a multi-body system vibrating with impacts. We discover that if every motion of the system is bounded, it has only four different types: periodic motion γ1, non·periodic recurrent motion γ2, and non-Poisson stable motions γ3 and γ4 approaching γ1 and γ2, respectively. γ2 is the source of chaos. It is very interesting that chaotic motions seem stochastic but possess the character of recurrence. By way of example, we discuss chaotic motions of a small ball bouncing vertically on a massive vibrating table. The result obtained by us is different from that obtained by Holmes. The project supported by National Natural Science Foundation of China     

Experimental and numerical verification of bifurcations and chaos in cam-follower impacting systems

In this paper, we present the design, modelling and experimental validation of a novel experimental cam-follower rig for the analysis of bifurcations and chaos in piecewise-smooth dynamical systems with impacts. Experimental results are presented for a cam-follower system characterized by a radial cam and a flat-faced follower. Under variation of the cam rotational speed, the follower is observed to detach from the cam and then show the emergence of periodic impacting behaviour characterized by many impacts and chattering. Further variations of the cam speed cause the sudden transition to seemingly aperiodic behaviour. These results are compared with the numerical simulation of a mathematical model of the system which shows the same qualitative behaviour. Excellent quantitative agreement is found between the numerical and experimental results.     

Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems

The global homoclinic bifurcation and transition to chaotic behavior of a nonlinear gear system are studied by means of Melnikov analytical analysis. It is also an effective approach to analyze homoclinic bifurcation and detect chaotic behavior. A generalized nonlinear time varying (NLTV) dynamic model of a spur gear pair is formulated, where the backlash, time varying stiffness, external excitation, and static transmission error are included. From Melnikov method, the threshold values of the control parameter for the occurrence of homoclinic bifurcation and onset of chaos are predicted. Additionally, the numerical bifurcation analysis and numerical simulation of the system including bifurcation diagrams, phase plane portraits, time histories, power spectras, and Poincare sections are used to confirm the analytical predictions and show the transition to chaos.     

Stabilization of stochastic cycles and chaos suppression for nonlinear discrete-time systems

In this paper we consider a nonlinear discrete-time control system with regular and chaotic dynamics forced by stochastic disturbances. The problem addressed is the design of the feedback regulator which stabilizes a limit cycle of the closed-loop deterministic system and synthesizes a required dispersion of random states for the corresponding stochastic system. To solve this problem, we propose a new method based on the stochastic sensitivity function technique. This function approximates a dispersion of random states distributed around deterministic cycle. Explicit formulas for the intercoupling between stochastic sensitivity function and considered system parameters are worked out. The problem of the design of the required stochastic sensitivity function for cycles by feedback regulators is solved. Coefficients of the feedback regulator are constructed and corresponding attainability sets are described. The effectiveness of the proposed approach is demonstrated on the stochastic Verhulst model. It is shown that constructed regulators provide a low level of sensitivity and suppress chaotic oscillations.     

非嵌入式多项式混沌法在爆轰产物JWL参数评估中的应用

介绍了非嵌入多项式混沌法的数学模型,给出了非嵌入式多项式混沌法进行不确定度量化的主要步骤。采用此方法研究了平面、散心爆轰问题数值模拟中, JWL模型参数R₁、R₂服从均匀分布的随机变量时所引起的爆轰过程计算结果的不确度性,着重分析了爆轰传播过程中压力与密度的统计特性。研究结果表明,非嵌入式多项式混沌法可以为模型输入参数不确定性的传播对输出结果响应量的影响建立一种有效不确定度评估方法,为使用JWL模型时选取参数提供参考。     

Analysis method of chaos and sub-harmonic resonance of nonlinear system without small parameters

The Melnikov method is important for detecting the presence of transverse homoclinic orbits and the occurrence of homoclinic bifurcations.Unfortunately,the traditional Melnikov methods strongly depend on small parameters,which do not exist in most practical systems.Those methods are limited in dealing with the systems with strong nonlinearities.This paper presents a procedure to study the chaos and sub-harmonic resonance of strongly nonlinear practical systems by employing a homotopy method that is used ...     

Controlling hyperchaos in planar systems by adjusting parameters

IntroductionIn 1 990 ,Ott,GrebogiandYorke (OGY)introducedtheconceptofcontrolofchaosandgaveamethodforcontrollingchaos,knownastheOGYmethod[1].AfterOGY ,manyothermethodsforcontrollingchaoshavebeendeveloped[2 - 8].Otherrelatedtopics,suchasnoiseeffectsandoptimizationofthecontrol,havealsobeeninvestigated[9- 12 ].TheOGYmethodconsidersthemapξn+ 1=Fp(ξn) ,ξ∈R2 ,wherep∈Risaparameter,whichhasasaddleatacertainvalueoftheparameter.Thereforethereareaone_dimensionalstablemanifoldandaone_dimension…