Influence of Initial Conditions on the Postcritical Behavior of a Nonlinear Aeroelastic System

The behavior of a nonlinear, non-Hamiltonian system in the postcritical (flutter) domain is studied with special attention to the influence of initial conditions on the properties of attractors situated at a certain point of the control parameter space. As a prototype system, an elastic panel is considered that is subjected to a combination of supersonic gas flow and quasistatic loading in the middle surface. A two natural modes approximation, resulting in a four-dimensional phase space and several control parameters is considered in detail. For two fixed points in the control parameter space, several plane sections of the four-dimensional space of initial conditions are presented and the asymptotic behavior of the final stationary responses are identified. Amongst the latter there are stable periodic orbits, both symmetric and asymmetric with respect to the origin, as well as chaotic attractors. The mosaic structure of the attraction basins is observed. In particular, it is shown that even for neighboring initial conditions can result in distinctly different nonstationary responses asymptotically approach quite different types of attractors. A number of closely neighboring periodic attractors are observed, separated by Hopf bifurcations. Periodic attractors also are observed under special initial conditions in the domains where chaotic behavior is usually expected.     

齿轮传动系统共存吸引子的不连续分岔

大量的多吸引子共存是引起齿轮传动系统具有丰富动力学行为的一个重要因素.多吸引子共存时,运动工况的变化以及不可避免的扰动都可能导致齿轮传动系统在不同运动行为之间跳跃变换,对整个机器产生不良的影响.目前,一些隐藏的吸引子没有被发现,共存吸引子的分岔演化规律没有被完全揭示.考虑单自由度直齿圆柱齿轮传动系统,构建由局部映射复合的Poincaré映射,给出Jacobi矩阵特征值计算的半解析法.应用数值仿真、延拓打靶法和Floquet特征乘子求解共存吸引子的稳定性与分岔,应用胞映射法计算共存吸引子的吸引域,讨论啮合频率、阻尼比和时变激励幅值对系统动力学的影响,揭示齿轮传动系统倍周期型擦边分岔、亚临界倍周期分岔诱导的鞍结分岔和边界激变等不连续分岔行为.倍周期分岔诱导的鞍结分岔引起相邻周期吸引子相互转迁的跳跃与迟滞,使倍周期分岔呈现亚临界特性.鞍结分岔是共存周期吸引子出现或消失的主要原因.边界激变引起混沌吸引子及其吸引域突然消失,对应周期吸引子的分岔终止.     

Experimental study of regular and chaotic transients in a non-smooth system

This paper focuses on thoroughly exploring the finite-time transient behaviors occurring in a periodically driven non-smooth dynamical system. Prior to settling down into a long-term behavior, such as a periodic forced oscillation, or a chaotic attractor, responses may exhibit a variety of transient behaviors involving regular dynamics, co-existing attractors, and super-persistent chaotic transients. A simple and fundamental impacting mechanical system is used to demonstrate generic transient behavior in an experimental setting for a single degree of freedom non-smooth mechanical oscillator. Specifically, we consider a horizontally driven rigid-arm pendulum system that impacts an inclined rigid barrier. The forcing frequency of the horizontal oscillations is used as a bifurcation parameter. An important feature of this study is the systematic generation of generic experimental initial conditions, allowing a more thorough investigation of basins of attraction when multiple attractors are present. This approach also yields a perspective on some sensitive features associated with grazing bifurcations. In particular, super-persistent chaotic transients lasting much longer than the conventional settling time (associated with linear viscous damping) are characterized and distinguished from regular dynamics for the first time in an experimental mechanical system.     

Numerical and geometrical analysis of bifurcation and chaos for an asymmetric elastic nonlinear oscillator

An asymmetric nonlinear oscillator representative of the finite forced dynamics of a structural system with initial curvature is used as a model system to show how the combined use of numerical and geometrical analysis allows deep insight into bifurcation phenomena and chaotic behaviour in the light of the system global dynamics.Numerical techniques are used to calculate fixed points of the response and bifurcation diagrams, to identify chaotic attractors, and to obtain basins of attraction of coexisting solutions. Geometrical analysis in control-phase portraits of the invariant manifolds of the direct and inverse saddles corresponding to unstable periodic motions is performed systematically in order to understand the global attractor structure and the attractor and basin bifurcations.     

双同步混沌吸引子动力系统中的筛形域

解析证明耦合映射混沌同步系统中的两个同步混沌吸引子的吸引域是筛形域．在特定耦合参数区间中，解析证明这两个同步混沌吸引子的吸引域不仅被无穷远吸引子的吸引域筛形，还通过数值证明它们的吸引域彼此互相筛形，展示出类似于Wada性质的特征．但进一步的讨论表明这种复杂的被两个(或更多)吸引域共同筛形的结构并不是Wada域，而是由于筛形分岔和筛形域局部—全局分岔导致的．     

A new chaotic attractor from hybrid optical bistable system

In this work, a new three-dimensional autonomous chaotic system has been introduced by modifying a hybrid optical system. The single quadratic nonlinearity is replaced by a single cubic nonlinearity; the new system can display two 1-scroll chaotic attractors simultaneously or one 2-scroll chaotic attractor. The bifurcation diagram is obtained and Lyapunov spectrum is calculated for the proposed system. The results show that the new system exhibits rich complexity features such as stable, periodic, and chaotic dynamics.     

Dynamics of a close-loop controlled MEMS resonator

The dynamics of a close-loop electrostatic MEMS resonator, proposed as a platform for ultra sensitive mass sensors, is investigated. The parameter space of the resonator actuation voltage is investigated to determine the optimal operating regions. Bifurcation diagrams of the resonator response are obtained at five different actuation voltage levels. The resonator exhibits bi-stability with two coexisting stable equilibrium points located inside a lower and an upper potential wells. Steady-state chaotic attractors develop inside each of the potential wells and around both wells. The optimal region in the parameter space for mass sensing purposes is determined. In that region, steady-state chaotic attractors develop and spend most of the time in the safe lower well while occasionally visiting the upper well. The robustness of the chaotic attractors in that region is demonstrated by studying their basins of attraction. Further, regions of large dynamic amplification are also identified in the parameter space. In these regions, the resonator can be used as an efficient long-stroke actuator.     

Nonlinear oscillations and chaotic motions in a road vehicle system with driver steering control

The nonlinear dynamics of a differential system describing the motion of a vehicle driven by a pilot is examined. In a first step, the stability of the system near the critical speed is analyzed by the bifurcation method in order to characterize its behavior after a loss of stability. It is shown that a Hopf bifurcation takes place, the stability of limit cycles depending mainly on the vehicle and pilot model parameters. In a second step, the front wheels of the vehicle are assumed to be subjected to a periodic disturbance. Chaotic and hyperchaotic motions are found to occur for some range of the speed parameter. Numerical simulations, such as bifurcation diagrams, Poincaré maps, Fourier spectrums, projection of trajectories, and Lyapunov exponents are used to establish the existence of chaotic attractors. Multiple attractors may coexist for some values of the speed, and basins of attraction for such attractors are shown to have fractal geometries.     

On the hyperchaotic complex Lü system

The aim of this paper is to introduce the new hyperchaotic complex Lü system. This system has complex nonlinear behavior which is studied and investigated in this work. Numerically the range of parameter values of the system at which hyperchaotic attractors exist is calculated. This new system has a whole circle of equilibria and three isolated fixed points, while the real counterpart has only three isolated ones. The stability analysis of the trivial fixed point is studied. Its dynamics is more rich in the sense that our system exhibits both chaotic and hyperchaotic attractors, as well as periodic and quasi-periodic solutions and solutions that approach fixed points. The nonlinear control method based on Lyapunov function is used to synchronize the hyperchaotic attractors. The control of these attractors is studied. Different forms of hyperchaotic complex Lü systems are constructed using the state feedback controller and complex periodic forcing.     

A Simple Chaotic Circuit with Impulsive Switch Depending on Time and State

This paper presents a simple chaotic circuit consisting of two capacitors, one linear two-port VCCS and one time-state-controlled impulsive switch. The impulsive switch causes rich chaotic and periodic behavior. The circuit dynamics can be simplified into a one-dimensional return map that is piecewise linear and piecewise monotone. Using the return map, we clarify parameter conditions for existence of chaotic and periodic attractors and coexistence state of attractors.     

Analysis of a new 3D smooth autonomous system with different wing chaotic attractors and transient chaos

This letter proposes a new 3D quadratic autonomous chaotic system which displays an extremely complicated dynamical behavior over a large range of parameters. The new chaotic system has five real equilibrium points. Interestingly, this system can generate one-wing, two-wing, three-wing and four-wing chaotic attractors and periodic motion with variation of only one parameter. Besides, this new system can generate two coexisting one-wing and two coexisting two-wing attractors with different initial conditions. Furthermore, the transient chaos phenomenon happens in the system. Some basic dynamical behaviors of the proposed chaotic system are studied. Furthermore, the bifurcation diagram, Lyapunov exponents and Poincaré mapping are investigated. Numerical simulations are carried out in order to demonstrate the obtained analytical results. The interesting findings clearly show that this is a special strange new chaotic system, which deserves further detailed investigation.     

The effects of fractional order on a 3-D quadratic autonomous system with four-wing attractor

In this paper, a fractional 3-dimensional (3-D) 4-wing quadratic autonomous system (Qi system) is analyzed. Time domain approximation method (Grunwald–Letnikov method) and frequency domain approximation method are used together to analyze the behavior of this fractional order chaotic system. It is found that the decreasing of the system order has great effect on the dynamics of this nonlinear system. The fractional Qi system can exhibit chaos when the total order less than 3, although the regular one always shows periodic orbits in the same range of parameters. In some fractional order, the 4 wings are decayed to a scroll using the frequency domain approximation method which is different from the result using time domain approximation method. A surprising finding is that the phase diagrams display a character of local self-similar in the 4-wing attractors of this fractional Qi system using the frequency approximation method even though the number and the characteristics of equilibria are not changed. The frequency spectrums show that there is some shrinking tendency of the bandwidth with the falling of the system states order. However, the change of fractional order has little effect on the bandwidth of frequency spectrum using the time domain approximation method. According to the bifurcation analysis, the fractional order Qi system attractors start from sink, then period bifurcation to some simple periodic orbits, and chaotic attractors, finally escape from chaotic attractor to periodic orbits with the increasing of fractional order α in the interval [0.8,1]. The simulation results revealed that the time domain approximation method is more accurate and reliable than the frequency domain approximation method.     

Optimal Control of Nonregular Dynamics in a Duffing Oscillator

A method for controlling nonlinear dynamics and chaos previouslydeveloped by the authors is applied to the classical Duffing oscillator.The method, which consists in choosing the best shape of externalperiodic excitations permitting to avoid the transverse intersection ofthe stable and unstable manifolds of the hilltop saddle, is firstillustrated and then applied by using the Melnikov method foranalytically detecting homoclinic bifurcations. Attention is focused onoptimal excitations with a finite number of superharmonics, because theyare theoretically performant and easy to reproduce. Extensive numericalinvestigations aimed at confirming the theoretical predictions andchecking the effectiveness of the method are performed. In particular,the elimination of the homoclinic tangency and the regularization offractal basins of attraction are numerically verified. The reduction ofthe erosion of the basins of attraction is also investigated in detail,and the paper ends with a study of the effects of control on delayingcross-well chaotic attractors.     

COMPLEX RELAXATION OSCILLATION TRIGGERED BY BOUNDARY CRISIS IN THE DISCRETE DUFFING MAP

多时间尺度问题具有广泛的工程与科学研究背景，慢变参数则是多时间尺度问题的典型标志之一.然而现有文献所报道的慢变参数问题，其展现出的振荡形式及内部分岔结构，大多较为单一，此外少有文献涉及到混沌激变的现象.本文以含慢变周期激励的达芬映射为例，探讨了一类具有复杂分岔结构的张弛振荡.快子系统的分岔表现为S形不动点曲线，其上、下稳定支可经由倍周期分岔通向混沌.而在一定的参数条件下，存在着导致混沌吸引子突然消失的一对临界参数值.当分岔参数达到此临界值时，混沌吸引子可能与不稳定不动点相接触，也可能与之相距一定距离.对快子系统吸引域分布的模拟，表明存在着导致边界激变（boundary crisis）的临界值，在这些值附近，经由延迟倍周期分岔演化而来的混沌吸引子可与2ⁿ（n=0，1，2，…）周期轨道乃至混沌吸引子共存.当慢变量周期地穿过临界点后，双稳态的消失导致原本处于混沌轨道的轨线对称地向此前共存的吸引子转迁，从而使系统出现了不同吸引子之间的滞后行为，由此产生了由边界激变所诱发的多种对称式张弛振荡.本文的结果丰富了对离散系统的多时间尺度动力学机理的认识.     

周期激励下广义蔡氏电路混沌运动中的概周期行为

由于广义蔡氏电路存在2个对称的稳定平衡点,周期激励可能导致系统出现相应于不同初值的2种共存的分岔模式. 概周期解由环面破裂进入混沌,混沌吸引子从相位不同步逐渐演化为同步,并进一步随着参数的变化,产生分裂现象. 分裂后的2个相互对称的混沌吸引子仍存在相位同步效应,这2个混沌吸引子再次相互作用后形成扩大了的混沌吸引子,并交替围绕2个子混沌结构来回振荡. 同时,在混沌过程中,其轨迹在相当长的一段时间内严格按照概周期行为振荡,即混沌结构中存在局部概周期行为,这种局部概周期行为随参数的变化会逐步减弱,直至消失.      

Modified post-bifurcation dynamics and routes to chaos from double-Hopf bifurcations in a hyperchaotic system

In order to understand the onset of hyperchaotic behavior recently observed in many systems, we study bifurcations in the modified Chen system leading from simple dynamics into chaotic regimes. In particular, we demonstrate that the existence of only one fixed point of the system in all regions of parameter space implies that this simple point attractor may only be destabilized via a Hopf or double Hopf bifurcation as system parameters are varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The normal form immediately following double Hopf bifurcations is constructed analytically by the method of multiple scales. Analysis of this generalized double Hopf normal form along standard lines reveals possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. However, considering these more carefully, we find that only certain combinations or sequences of these dynamical regimes are possible, while others derived and considered in earlier work are in fact mathematically impossible. We also discuss the post-bifurcation dynamics in the context of two intermittent routes to chaos (routes following either (i) subcritical or (ii) supercritical Hopf or double Hopf bifurcations). In particular, the route following supercritical bifurcations is somewhat subtle. Such behavior following repeated Hopf bifurcations is well-known and widely observed, including in the classical Ruelle?CTakens and quasiperiodic routes to chaos. However, to the best of our knowledge, it has not been considered in the context of the double-Hopf normal form, although it has been numerically observed and tracked in the post-double Hopf regime. Numerical simulations are employed to corroborate these various predictions from the normal form. They reveal the existence of stable periodic and toroidal attractors in the post-supercritical-Hopf cases, and either attractors at infinity or bounded chaotic dynamics following subcritical Hopf bifurcations. Future work will map out the remainder of the routes into the chaotic regimes, including further bifurcations of the post-supercritical-Hopf two- and three-tori via either torus doubling or breakdown.     

Periodic orbits,basins of attraction and chaotic beats in two coupled Kerr oscillators

Kerr oscillators are model systems which have practical applications in nonlinear optics. Optical Kerr effect, i.e., interaction of optical waves with nonlinear medium with polarizability χ ⁽³⁾ is the basic phenomenon needed to explain, for example, the process of light transmission in fibers and optical couplers. In this paper, we analyze the two Kerr oscillators coupler and we show that there is a possibility to control the dynamics of this system, especially by switching its dynamics from periodic to chaotic motion and vice versa. Moreover, the switching between two different stable periodic states is investigated. The stability of the system is described by the so-called maps of Lyapunov exponents in parametric spaces. Comparison of basins of attractions between two Kerr couplers and a single Kerr system is also presented.     

Forward and backward motion control of a vibro-impact capsule system

A capsule system driven by a harmonic force applied to its inner mass is considered in this study. Four various friction models are employed to describe motion of the capsule in different environments taking into account Coulomb friction, viscous damping, Stribeck effect, pre-sliding, and frictional memory. The non-linear dynamics analysis has been conducted to identify the optimal amplitude and frequency of the applied force in order to achieve the motion in the required direction and to maximize its speed. In addition, a position feedback control method suitable for dealing with chaos control and coexisting attractors is applied for enhancing the desirable forward and backward capsule motion. The evolution of basins of attraction under control gain variation is presented and it is shown that the basin of the desired attractors could be significantly enlarged by slight adjustment of the control gain improving the probability of reaching such an attractor.     

Partially unstable attractors in networks of forced integrate-and-fire oscillators

The asymptotic attractors of a nonlinear dynamical system play a key role in the long-term physically observable behaviors of the system. The study of attractors and the search for distinct types of attractor have been a central task in nonlinear dynamics. In smooth dynamical systems, an attractor is often enclosed completely in its basin of attraction with a finite distance from the basin boundary. Recent works have uncovered that, in neuronal networks, unstable attractors with a remote basin can arise, where almost every point on the attractor is locally transversely repelling. Herewith we report our discovery of a class of attractors: partially unstable attractors, in pulse-coupled integrate-and-fire networks subject to a periodic forcing. The defining feature of such an attractor is that it can simultaneously possess locally stable and unstable sets, both of positive measure. Exploiting the structure of the key dynamical events in the network, we develop a symbolic analysis that can fully explain the emergence of the partially unstable attractors. To our knowledge, such exotic attractors have not been reported previously, and we expect them to arise commonly in biological networks whose dynamics are governed by pulse (or spike) generation.