3-scroll and 4-scroll chaotic attractors generated from a new 3-D quadratic autonomous system

This article introduces a new chaotic system of 3-D quadratic autonomous ordinary differential equations, which can display 2-scroll chaotic attractors. Some basic dynamical behaviors of the new 3-D system are investigated. Of particular interest is that the chaotic system can generate complex 3-scroll and 4-scroll chaotic attractors. Finally, bifurcation analysis shows that the system can display extremely rich dynamics. The obtained results clearly show that this is a new chaotic system which deserves further detailed investigation.     

Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system

In this paper, we introduce a new chaotic complex nonlinear system and study its dynamical properties including invariance, dissipativity, equilibria and their stability, Lyapunov exponents, chaotic behavior, chaotic attractors, as well as necessary conditions for this system to generate chaos. Our system displays 2 and 4-scroll chaotic attractors for certain values of its parameters. Chaos synchronization of these attractors is studied via active control and explicit expressions are derived for the control functions which are used to achieve chaos synchronization. These expressions are tested numerically and excellent agreement is found. A Lyapunov function is derived to prove that the error system is asymptotically stable.     

Optimizing the maximum Lyapunov exponent and phase space portraits in multi-scroll chaotic oscillators

This investigation introduces the application of the nondominated sorting genetic algorithm to optimize two characteristics of multiscroll chaotic oscillators: (a) Maximizing the values of the maximum Lyapunov exponent (MLE), and (b) minimizing the dispersions of the phase space portraits (PSP) among all scrolls in an attractor. As shown in this study, these two oscillator’s characteristics are in conflict and must be considered at the same time. The cases of study are two multiscroll chaotic oscillators based on piecewise-linear functions, namely: saturated function series and Chua’s diode (negative slopes). Basically, a very new procedure to measure the PSP coverture among all generated scrolls is introduced in the optimization loop for each feasible solution maximizing the MLE. The best optimized results are compared with traditional values of the coefficients of the equations describing the oscillators. Finally, we list the values of the optimized MLE and their corresponding PSP when generating from 2- to 6-scroll attractors.     

Complex Dynamics of a Pyranose Ring Structure Molecule Attached to an Atomic Force Microscope

Dynamic numerical simulations were performed for a pyranose ring structure molecule attached to an Atomic Force Microscope (AFM) using a standard semiempirical potential energy surface model. The fundamental static force-extension behavior was first determined using a slow pulling base excitation at the AFM probe. The static force-extension curve displays a stiffness nonlinearity, both softening and hardening, that depends upon level of the pulling force. For the dynamic analysis, a single harmonic base excitation is applied to the AFM probe. A typical evolution process from periodic to aperiodic or chaotic motion obtained by varying the excitation frequency and amplitude is discussed. A strong chaotic response motion was generated for certain system parameters. The numerical analysis shows this chaotic response arises from a molecular structure conformational change.     

A novel method for designing nonlinear component for block cipher based on TD-ERCS chaotic sequence

A substitution box is used to induce nonlinearity in plaintext for encryption systems. Recently, the application of chaotic maps to encryption applications has resulted in some interesting nonlinear transformations. In this paper, we propose an efficient method to design nonlinear components for block ciphers that are based on TD-ERCS chaotic sequence. The new substitution box is analyzed for nonlinearity, bit independence, strict avalanche criterion, generalized majority logic criterion, and differential and linear approximation probabilities. The results show high resistance to differential and linear cryptanalysis in comparison to some recently proposed chaotic substitution boxes.     

Adaptive sliding mode control of uncertain chaotic systems with input nonlinearity

This paper is concerned with the stabilization problem of uncertain chaotic systems with input nonlinearity. The slope parameters of this nonlinearity are unmeasured. A new sliding function is designed, then an adaptive sliding mode controller is established such that the trajectory of the system converges to the sliding surface in a finite time and finite-time reachability is theoretically proved. Using a virtual state feedback control technique, sufficient condition for the asymptotic stability of sliding mode dynamics is derived via linear matrix inequality (LMI). Then the results can be extended to uncertain chaotic systems with disturbances and adaptive sliding mode H _∞ controllers are designed. Finally, a simulation example is presented to show the validity and advantage of the proposed method.     

Observer-based model reference adaptive control for unknown time-delay chaotic systems with input nonlinearity

Existence of unknown time-delay in the systems is a drastic restriction that it can menace the stability criteria and even deteriorate the performance system. This undesired case would be more intensified if that the uncertain input nonlinearity effects are also considered. To handle the input nonlinearities effects (results in dead-zone and/or hysteresis phenomena) and also unknown time-delay in the chaotic systems, this paper presents an observer-based Model Reference Adaptive Control (MRAC) scheme for a class of unknown time-delay chaotic systems with disturbances. This new method is a delay-independent variable-structure control method which is integrated with an observer system. The main task of the proposed approach is to accomplish a perfect tracking procedure such that unknown parameters are adapted via output estimation error. Furthermore, stability of the closed-loop system is achieved by means of the Lyapunov stability theory. Finally, the proposed methods are applied to some famous chaotic systems to verify the effectiveness of the proposed methods.     

Design and analysis of a first order time-delayed chaotic system

The present paper reports the design and analysis of a new time-delayed chaotic system and its electronic circuit implementation. The system is described by a first-order nonlinear retarded type delay differential equation with a closed form mathematical function describing the nonlinearity. We carry out stability and bifurcation analysis to show that with the suitable delay and system parameters the system shows sustained oscillation through supercritical Hopf bifurcation. It is shown through numerical simulations that the system depicts bifurcation and chaos for a certain range of the system parameters. The complexity and predictability of the system are characterized by Lyapunov exponents and Kaplan?CYork dimension. It is shown that, for some suitably chosen system parameters, the system shows hyperchaos even for a small or moderate delay. Finally, we set up an experiment to implement the proposed system in electronic circuit using off-the-shelf circuit elements, and it is shown that the behavior of the time delay chaotic electronic circuit agrees well with our analytical and numerical results.     

含间隙裂缝的RC结构滞回系统混沌振动分析

研究了含间隙裂缝的钢筋混凝土结构对称滞回非线性问题. 建立了 一种分段线性的对称滞回模型, 利用一次谐波线性方法求解结构系统的等效阻尼和 等效刚度系数,得到了对称滞回非线性系统的等价线性方程. 通过数值分析比较了考虑和不考虑间隙与碰撞 影响的两种情况下系统的混沌动力特性,研究表明: 不考虑间隙与碰撞影响的系统出现周期运 动, 考虑间隙与碰撞影响系统更容易出现混沌运动; 在特定的参数范围内系统一定会出现无 序的混沌运动.     

ANALYSIS OF THE CHAOTIC MOTION FOR A ROTOR SYSTEM WITH A TRANSVERSE CRACK

The chaotic motion of an elastic shaft-disk rotor system including the geometricnonlinearity of the shaft with a transverse crack is investigated.The critical condition for thesystem to enter chaotic state is given by the Melnikov method.Meanwhile,whether the chaosoccurs is determined by the Poincaré map,phase portrait and time-displacement history diagram.     

Synchronization for chaotic Lur’e systems with sector-restricted nonlinearities via delayed feedback control

In this paper, the effects of time delay on chaotic master–slave synchronization scheme are considered. Using delayed feedback control scheme, a delay-dependent stability criterion is derived for the synchronization of chaotic systems that are represented by Lur’e system with sector-restricted nonlinearities. The derived criterion is a sufficient condition for absolute stability of error dynamics between the master and the slave system. Using a convex representation of the nonlinearity, the stability condition based on the Lyapunov–Krasovskii functional is obtained via LMI formulation. The proposed delay-dependent synchronization criterion is less conservative than the existing ones. The effectiveness of our work is verified through numerical examples.     

亚音速气流作用下薄板结构混沌运动的时滞反馈控制

研究了亚音速气流下非线性二维薄板结构在横向周期载荷作用下的混沌运动及控制问题。基于von Karman大变形板理论和分离变量法,建立了亚音速下薄板结构的运动控制方程。对于未控系统,采用Melnikov方法判断其混沌运动阈值,并用Runge-Kutta法进行数值验证。对处于混沌运动状态的系统,采用时滞反馈控制方法对混沌运动进行控制。结果表明,Melnikov方法可以有效地预测系统的混沌运动行为,时滞反馈控制方法可以有效地将系统的混沌运动转化为周期运动。     

Multi-pulse chaotic dynamics of six-dimensional non-autonomous nonlinear system for a composite laminated piezoelectric rectangular plate

Global bifurcations and multi-pulse chaotic dynamics are studied for a four-edge simply supported composite laminated piezoelectric rectangular plate under combined in-plane, transverse, and dynamic electrical excitations. Based on the von Karman type equations for the geometric nonlinearity and Reddy’s third-order shear deformation theory, the governing equations of motion for a composite laminated piezoelectric rectangular plate are derived. The Galerkin method is employed to discretize the partial differential equations of motion to a three-degree-of-freedom nonlinear system. The six-dimensional non-autonomous nonlinear system is simplified to a three-order standard form by using the method of normal form. The extended Melnikov method is improved to investigate the six-dimensional non-autonomous nonlinear dynamical system in mixed coordinate. The global bifurcations and multi-pulse chaotic dynamics of the composite laminated piezoelectric rectangular plate are studied by using the improved extended Melnikov method. The multi-pulse chaotic motions of the system are found by using numerical simulation, which further verifies the result of theoretical analysis.     

A novel strange attractor with a stretched loop

This short paper introduces a new 3D strange attractor topologically different from any other known chaotic attractors. The intentionally constructed model of three autonomous first-order differential equations derives from the coupling-induced complexity of the well-established 2D Lotka?CVolterra oscillator. Its chaotification process via an anti-equilibrium feedback allows the exploration of a new domain of dynamical behavior including chaotic patterns. To focus a rapid presentation, a fixed set of parameters is selected linked to the widest range of dynamics. Indeed, the new system leads to a chaotic attractor exhibiting a double scroll bridged by a loop. It mutates to a single scroll with a very stretched loop by the variation of one parameter. Indexes of stability of the equilibrium points corresponding to the two typical strange attractors are also investigated. To encompass the global behavior of the new low-dimensional dissipative dynamical model, diagrams of bifurcation displaying chaotic bubbles and windows of periodic oscillations are computed. Besides, the dominant exponent of the Lyapunov spectrum is positive reporting the chaotic nature of the system. Eventually, the novel chaotic model is suitable for digital signal encryption in the field of communication with a rich set of keys.     

A control approach for vibrations of a nonlinear microbeam system in multi-dimensional form

Microbeams are widely seen in micro-electro-mechanical systems and their engineering applications. An active control strategy based on the fuzzy sliding mode control is developed in this research for controlling and stabilizing the nonlinear vibrations of a micro-electro-mechanical beam. An Euler-Bernoulli beam with a fixed-fixed boundary is employed to represent the microbeam, and the geometric nonlinearity of the beam and loading nonlinearity from the electrostatic force are considered. The governing equation of the microbeam is established and transformed into a multi-dimensional dynamic system with the third-order Galerkin method. A stability analysis is provided to show the necessity of the derived multi-dimensional dynamic system, and a chaotic motion is discovered. Then, a control approach is proposed, including a control strategy and a two-phase control method. For describing the application of the control approach developed, control of a chaotic motion of the microbeam is presented. The effectiveness of the active control approach is demonstrated via controlling and stabilizing the nonlinear vibration of the microbeam.     

The post-Hopf-bifurcation response of an airfoil in incompressible two-dimensional flow

A bifurcation analysis of a two-dimensional airfoil with a structural nonlinearity in the pitch direction and subject to incompressible flow is presented. The nonlinearity is an analytical third-order rational curve fitted to a structural freeplay. The aeroelastic equations-of-motion are reformulated into a system of eight first-order ordinary differential equations. An eigenvalue analysis of the linearized equations is used to give the linear flutter speed. The nonlinear equations of motion are either integrated numerically using a fourth-order Runge-Kutta method or analyzed using the AUTO software package. Fixed points of the system are found analytically and regions of limit cycle oscillations are detected for velocities well below the divergent flutter boundary. Bifurcation diagrams showing both stable and unstable periodic solutions are calculated, and the types of bifurcations are assessed by evaluating the Floquet multipliers. In cases where the structural preload is small, regions of chaotic motion are obtained, as demonstrated by bifurcation diagrams, power spectral densities, phase-plane plots and Poincaré sections of the airfoil motion; the existence of chaos is also confirmed via calculation of the Lyapunov exponents. The general behaviour of the system is explained by the effectiveness of the freeplay part of the nonlinearity in a complete cycle of oscillation. Results obtained using this reformulated set of equations and the analytical nonlinearity are in good agreement with previously obtained finite difference results for a freeplay nonlinearity.     

Chaotic belt phenomena in nonlinear elastic beam

IntroductionThechaoticphenomenainsolidmechanicsfieldsbringmoreandmoreinterest.In 1 998,F .C .Moon[1]analyzedthechaoticbehaviorsofbeamsexperimentallyfirst.Thenhestudiedthedynamicsresponseoflinearelasticbeamsubjectedtransverseperiodicload .Thechaoticmotionsoflineardampingbeamshavebeenstudiedbymanyscholarsathomeandabroadinrecentyears[2 ,3].ThedynamicbehaviorsofnonlineardampingbeamssubjectedtotransverseloadP=δP0 (f+cosωt)sin(πx/l)arestudiedinthispaper.Thecriticconditionsthatchaosoccursinthes…     

Sliding mode control with adaptive fuzzy dead-zone compensation for uncertain chaotic systems

The dead-zone nonlinearity is frequently encountered in many industrial automation equipments and its presence can severely compromise control system performance. In this work, an adaptive variable structure controller is proposed to deal with a class of uncertain nonlinear systems subject to an unknown dead-zone input. The adopted approach is primarily based on the sliding mode control methodology but enhanced by an adaptive fuzzy algorithm to compensate the dead-zone. Using Lyapunov stability theory and Barbalat??s lemma, the convergence properties of the closed-loop system are analytically proven. In order to illustrate the controller design methodology, an application of the proposed scheme to a chaotic pendulum is introduced. A comparison between the stabilization of general orbits and unstable periodic orbits embedded in chaotic attractor is carried out showing that the chaos control can confer flexibility to the system by changing the response with low power consumption.     

Localization of compact invariant sets of a new nonlinear finance chaotic system

The application of the nonlinear chaotic dynamic system in economics and finance has expanded rapidly in the last decades. This paper considers the localization of all compact invariant sets of a new three-dimensional autonomous nonlinear finance chaotic system. The boundedness of the new finance chaotic system is the first time being investigated. Based on the iteration theorem and the first-order extremum theorem, a new method is proposed, too. The comparison of our method with the traditional method is presented as well. More specifically, the compact invariant sets are analyzed in three aspects: First of all, a localization of the new finance chaotic system by two frusta and an ellipsoidal used by traditional methods is discussed. Second, a localization of the new finance chaotic system by two frusta and a parabolic cylinder is provided. Third, localization of the new finance chaotic system according to superposition of the ellipsoidal, parabolic cylinder, and two frusta are presented, and the boundedness of the chaotic attracter is smaller than in the classical methods. Numerical simulations are given to indicate the effectiveness of the proposed method.     

Dynamics analysis and hybrid function projective synchronization of a new chaotic system

This paper introduces a novel three-dimensional autonomous chaotic system by adding a quadratic cross-product term to the first equation and modifying the state variable in the third equation of a chaotic system proposed by Cai et al. (Acta Phys. Sin. 56:6230, 2007). By means of theoretical analysis and computer simulations, some basic dynamical properties, such as Lyapunov exponent spectrum, bifurcations, equilibria, and chaotic dynamical behaviors of the new chaotic system are investigated. Furthermore, hybrid function projective synchronization (HFPS) of the new chaotic system is studied by employing three different synchronization methods, i.e., adaptive control, system coupling and active control. The proposed approaches are applied to achieve HFPS between two identical new chaotic systems with fully uncertain parameters, HFPS in coupled new chaotic systems, and HFPS between the integer-order new chaotic system and the fractional-order Lü chaotic system, respectively. Corresponding numerical simulations are provided to validate and illustrate the analytical results.