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.     

Occurrence of multiple attractor bifurcations in the two-dimensional piecewise linear normal form map

Multiple attractor bifurcations occurring in piecewise smooth dynamical systems may lead to potentially damaging situations. In order to avoid these in physical systems, it is necessary to know their conditions of occurrence. Using the piecewise-linear 2D normal form, we investigate which types of multiple attractor bifurcations may occur and where in the parameter space they can be expected. For piecewise smooth maps, multiple attractor bifurcations will be expected to occur if the condition we identified for the piecewise-linear 2D normal form are satisfied in the close neighborhood of the border.     

Lyapunov and LMI analysis and feedback control of border collision bifurcations

Feedback control of piecewise smooth discrete-time systems that undergo border collision bifurcations is considered. These bifurcations occur when a fixed point or a periodic orbit of a piecewise smooth system crosses or collides with the border between two regions of smooth operation as a system parameter is quasistatically varied. The class of systems studied is piecewise smooth maps that depend on a parameter, where the system dimension n can take any value. The goal of the control effort in this work is to replace the bifurcation so that in the closed-loop system, the steady state remains locally attracting and locally unique (“nonbifurcation with persistent stability”). To achieve this, Lyapunov and linear matrix inequality (LMI) techniques are used to derive a sufficient condition for nonbifurcation with persistent stability. The derived condition is stated in terms of LMIs. This condition is then used as a basis for the design of feedback controls to eliminate border collision bifurcations in piecewise smooth maps and to produce the desirable behavior noted earlier. Numerical examples that demonstrate the effectiveness of the proposed control techniques are given.     

Border collision bifurcations in 3D piecewise smooth chaotic circuit

A variety of border collision bifurcations in a three-dimensional (3D) piecewise smooth chaotic electrical circuit are investigated. The existence and stability of the equilibrium points are analyzed. It is found that there are two kinds of non-smooth fold bifurcations. The existence of periodic orbits is also proved to show the occurrence of non-smooth Hopf bifurcations. As a composite of non-smooth fold and Hopf bifurcations, the multiple crossing bifurcation is studied by the generalized Jacobian matrix. Some interesting phenomena which cannot occur in smooth bifurcations are also considered.     

Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

In this paper we investigate the bifurcations and the chaos of a piecewise linear discontinuous (PWLD) system based upon a rig-coupled SD oscillator, which can be smooth or discontinuous (SD) depending on the value of a system parameter, proposed in [18], showing the equilibrium bifurcations and the transitions between single, double and triple well dynamics for smooth regions. All solutions of the perturbed PWLD system, including equilibria, periodic orbits and homoclinic-like and heteroclinic-like orbits, are obtained and also the chaotic solutions are given analytically for this system. This allows us to employ the Melnikov method to detect the chaotic criterion analytically from the breaking of the homoclinic-like and heteroclinic-like orbits in the presence of viscous damping and an external harmonic driving force. The results presented here in this paper show the complicated dynamics for PWLD system of the subharmonic solutions, chaotic solutions and the coexistence of multiple solutions for the single well system, double well system and the triple well dynamics.     

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.     

Periodic and chaotic dynamics of a sliding driven oscillator with dry friction

We have performed a numerical study of the dynamics of a harmonically forced sliding oscillator with two degrees of freedom and dry friction. The study of the four-dimensional dynamical system corresponding to the two non-linear motion equations can be reduced, in this case, to the study of a three-dimensional Poincaré map. The behaviour of the system has been investigated calculating bifurcation diagrams, time series, periodic and chaotic attractors and basins of attraction. Furthermore, a systematic study of the stability of periodic solutions and their bifurcations has been carried out applying the Floquet theory. The results show rich dynamics being very sensitive to the changes in forcing amplitudes (control parameter), where periodic and chaotic states alternatively appear. It is shown how the system exhibits different types of bifurcational phenomena (saddle-node, symmetry-breaking, period-doubling cascades and intermittent transitions to chaos) into relatively narrow intervals of the control parameter. Moreover, a collection of chaotic attractors was computed to show the evolution of the chaotic regime. Finally, basins of attraction were calculated. In all the cases studied, the basins exhibit fractal structure boundaries and, when more of two attractors are coexisting, we have found Wada basin boundaries.     

Wada分形吸引域边界上的混沌鞍

应用广义胞映射图论（Generalized Cell Mapping Digraph)方法，数值地研究Thompson的逃逸方程在最佳逃逸点附近的分岔。发现了嵌入在Wada分形吸引域边界上的混沌鞍，混沌鞍是状态空间不稳定（非吸引）的混沌不变集合。Wada分形吸引域边界是具有Wada性质的边界，即吸引域边界上的任意点也同时是至少两个其它吸引域的边界点，称为Wada域边界。我们证明Wada域边界上的混沌鞍导致局部鞍结分岔具有全局不确定性结局，研究了Wada域边界上混沌鞍的形成与演化，证明最终的逃逸分岔是混沌吸引子碰撞混沌鞍的边界激变。     

Global bifurcations in externally excited autoparametric systems

We consider an autoparametric system consisting of an oscillator coupled with an externally excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in primary resonance. The method of second-order averaging is used to obtain a set of autonomous equations of the second-order approximations to the externally excited system with autoparametric resonance. The Šhilnikov-type homoclinic orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Šhilnikov-type homoclinic orbits in the averaged equations. The results obtained above mean the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the externally excited system with autoparametric resonance. Furthermore, a detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Nine branches of dynamic solutions are found. Two of these branches emerge from two Hopf bifurcations and the other seven are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, intermittency chaos and homoclinic explosions are also observed.     

Nonlinear aeroelastic analysis of an airfoil-store system with a freeplay by precise integration method

The aeroelastic system of an airfoil-store configuration with a pitch freeplay is investigated using the precise integration method (PIM). According to the piecewise feature, the system is divided into three linear sub-systems. The sub-systems are separated by switching points related to the freeplay nonlinearity. The PIM is then employed to solve the sub-systems one by one. During the solution procedures, one challenge arises when determining the vibration state passing the switching points. A predictor-corrector algorithm is proposed based on the PIM to tackle this computational obstacle. Compared with exact solutions, the PIM can provide solutions to the precision in the order of magnitude of 10⁻¹². Given the same step length, the PIM results are much more accurate than those of the Runge–Kutta (RK) method. Moreover, the RK method might falsely track limit cycle oscillations (LCOs), bifurcation charts or chaotic attractors; even the step length is chosen much smaller than that for the PIM. Bifurcations and LCOs are obtained and analyzed by the PIM in detail. Interestingly, it is found that multiple LCOs and chaotic attractors can exist simultaneously. With this magnitude of precision and efficiency, the PIM could become a solution technique with excellent potential for piecewise nonlinear aeroelastic systems.     

Bifurcations in impact oscillations

Models of impact oscillators using an instantaneous impact law are by their very nature discontinuous. These discontinuities geve rise to bifurcations which cannot be classified using the usual tools of bifurcation analysis. However, we present numerical evidence which suggests that these discontinuous bifurcations are just the limits (in some sense) of standard bifurcations of smooth dynamical systems as the impact is hardened. Finally we show how one dimensional maps of the interval with essentially similar characteristics can exhibit the same kinds of bifurcational behaviour, and how these bifurcations are related to standard bifurcations.     

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.     

On the existence of low-period orbits in <Emphasis Type="Italic">n</Emphasis>-dimensional piecewise linear discontinuous maps

Discontinuous maps occur in many practical systems, and yet bifurcation phenomena in such maps is quite poorly understood. In this paper, we report some important results that help in analyzing the border collision bifurcations that occur in n-dimensional discontinuous maps. For this purpose, we use the piecewise linear approximation in the neighborhood of the plane of discontinuity. Earlier, Feigin had made a similar analysis for general n-dimensional piecewise smooth continuous maps. In this paper, we extend that line of work for maps with discontinuity to obtain the general conditions of existence of period-1 and period-2 fixed points before and after a border collision bifurcation. The application of the method is then illustrated using a specific example of a two-dimensional discontinuous map. This work was supported in part by the BRNS, Department of Atomic Energy (DAE), Government of India under project no. 2003/37/11/BRNS.     

Global bifurcation analysis of Rayleigh–Duffing oscillator through the composite cell coordinate system method

In this paper, a new conception of composite cell coordinate system is presented by dividing the continuous state space into the cell state space with different scales. For a dynamical system, attractors, basins of attraction, basin boundaries, saddles, and invariant manifolds can be easily obtained, and any region of the state space can be refined by this method. The global bifurcations, such as crisis and metamorphosis, of the Rayleigh?CDuffing oscillator are studied by the composite cell coordinate system method. According to the sudden changes in shapes of the chaotic attractor and the chaotic saddle, we find that three types of crises can all occur, including boundary crisis, interior crisis, and attractor emerging crisis. In addition, the basin boundary metamorphoses, such as fractal-Wada, Wada-Wada, and Wada-fractal, are analyzed through observing the shapes of basin boundaries. These results demonstrate the efficiency and validity of this method in analyzing dynamical systems.     

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.     

Partial synchronization in a system of globally coupled maps

We study the appearance of a chaotic partial synchronization in a system of globally coupled maps. We analyze the structure of cluster zones for small values of the coupling parameter and conditions for the formation of chaotic attractors on cluster manifolds. We find a formula that describes the relationship between the transversal and longitudinal Lyapunov numbers for trajectories on the manifold and necessary conditions for the transversal stability of these trajectories.Translated from Neliniini Kolyvannya, Vol. 7, No. 2, pp. 229–240, April–June, 2004.     

一双峰混沌系统非线性动力学行为

通过对一双峰混沌系统的非线性动力学行为的研究,发现随着系统参数的变化,双峰混沌系统由混沌状态开始,经阵发性混沌、不动点、倍周期分岔到受初始值的影响两个混沌吸引子,而后又收敛为另一个不动点,最后再次进入混沌状态。该系统呈现出复杂的非线性动力学行为。     

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.     

Bifurcation analysis of attitude control systems with switching-constrained actuators

On-off thrusters are frequently used as actuators for attitude control and are typically subject to switching constraints. In systems with switching actuators, different types of persistent motions may be found, and in the presence of model uncertainties, the occurrence of bifurcations in such systems can seriously affect performance. In this paper the nature of persistent motions in an attitude control system with actuators subject to switching-time restrictions is examined to provide useful information for control design in the presence of uncertainty. The main tools used are bifurcation diagrams, Poincaré maps and Lyapunov spectrum. Border-collision type bifurcations are characterized in this piecewise affine system, as well as unusual patterns of persistent motion. Multistability and complex-switching sequences are also observed, revealing the existence of motions with sensitive dependence on initial conditions.     

Gaussian mixture model and receding horizon control for multiple UAV search in complex environment

Intriguing as the discovery of new chaotic maps is, some new maps also bring new nonlinear phenomena of iterative map behavior. In this paper, we present a simple two-dimensional chaotic map which has three totally separated regions. The twin regions, creating strange and interesting attractors, are close to each other and vertically reflected however not identical in shape, while the distant region, generating a Hénon-like attractor, starts with period-doubling until complete chaos. Given the unusual behavior of the map introduced in this paper, we initially presented linear stability and bifurcation analysis per regions, with Lyapunov exponents and largest exponent computation. Besides the standardized calculations, what we focus here is to find out how a simple map can exhibit different chaotic behaviors in different regions.