Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems

In this paper, unstable dynamics is considered for the models of vibro-impact systems with linear differential equations coupled to an impact map. To provide a skeleton for the organization of chaotic attractors, we propose a method for detecting unstable periodic orbits embedded in chaotic attractors through a combination of unconstrained optimization technique and Poincaré map. Three numerical examples from different vibro-impact models demonstrate that the strategy can efficiently detect unstable periodic orbits in chaotic attractors. In order to explore the mechanism responsible for the creation of multi-dimensional tori attractors, we also present another method to detect unstable quasiperiodic orbits embedded multi-dimensional tori attractors by examining a specially transient time series. The upper bound and lower bound of the transient time series (in the Poincaré map) can be obtained by analyzing transient Lyapunov exponent and transient Lyapunov dimension. Some examples verify the effectiveness of the numerical algorithm.     

Impulsive control of chaotic attractors in nonlinear chaotic systems

Based on the study from both domestic and abroad, an impulsive control scheme on chaotic attractors in one kind of chaotic system is presented. By applying impulsive control theory of the universal equation, the asymptotically stable condition of impulsive control on chaotic attractors in such kind of nonlinear chaotic system has been deduced, andwith it, the upper bond of the impulse interval for asymptotically stable control was given.Numerical results are presented, which are considered with important reference value for control of chaotic attractors.     

Chaos disappearance in a piecewise linear Bonhoeffer–van der Pol dynamics with a bistability of stable focus and stable relaxation oscillation under weak periodic perturbation

This study elucidates the bifurcation structure causing chaos disappearance in a four-segment piecewise linear Bonhoeffer–van der Pol oscillator with a diode under a weak periodic perturbation. The parameter values of this oscillator are chosen such that stable focus and stable relaxation oscillation can coexist in close proximity in the phase plane if no perturbation is applied. Chaos disappearance occurs through a previously unreported novel and unconventional bifurcation mechanism. To rigorously analyze these phenomena, the diode in this oscillator is assumed to operate as a switch. In this case, the governing equation is represented as a constraint equation, and the Poincaré map is constructed as an one-dimensional map. By analyzing the Poincaré map, we clearly demonstrate why the stable relaxation oscillation that exists when no perturbation is applied disappears via chaotic oscillation when an extremely weak perturbation is applied.     

Topological horseshoe analysis and the circuit implementation for a four-wing chaotic attractor

The paper first analyzes a newly reported three-dimensional four-wing chaotic attractor, and observes all kinds of attractors, including periodic and chaotic, by numerical simulation. Then, the chaotic characteristic of the system is proved by investigating the existence of a topological horseshoe in it, based on the topological horseshoe theory. At last, an electronic circuit is designed to implement the chaotic system. The results of circuit experiment coincided well with those of numerical simulation.     

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.     

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.     

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.     

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.      

Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation

A three-dimensional autonomous chaotic system is presented and physically implemented. Some basic dynamical properties and behaviors of this system are described in terms of symmetry, dissipative system, equilibria, eigenvalue structures, bifurcations, and phase portraits. By tuning the parameters, the system displays chaotic attractors of different shapes. For specific parameters, the system exhibits periodic and chaotic bursting oscillations which resemble the conventional heart sound signals. The existence of Shilnikov type of heteroclinic orbit in the three-dimensional system is proven using the undetermined coefficients method. As a result, Shilnikov criterion guarantees that the three-dimensional system has the horseshoe chaos. The corresponding electronic circuit is designed and implemented, exhibiting experimental chaotic attractors in accord with numerical simulations.     

Synchronization and circuit simulation of a new double-wing chaos

A new three-dimensional double-wing chaotic system with three quadratic terms was proposed. And the parameters which can induce the system are analyzed. The system with five equilibrium points has sophisticated dynamical behaviors and it is further investigated in details, including phase trajectory, Lyapunov exponent spectrum, Poincaré map, spectrogram map and dissipativity analysis. The circuit simulation results of the chaotic attractors are in agreement with numerical simulations. Furthermore, numerical simulations indicate that mismatch synchronization can be achieved and circuit simulations of the system synchronization are also presented.     

Analysis of a new three-dimensional system with multiple chaotic attractors

In this paper, a new three-dimensional autonomous system with complex dynamical behaviors is reported. This new system has three quadratic nonlinear terms and one constant term. One remarkable feature of the system is that it can generate multiple chaotic and multiple periodic attractors in a wide range of system parameters. The presence of coexisting chaotic and periodic attractors in the system is investigated. Moreover, it is easily found that the new system also can generate four-scroll chaotic attractor. Some basic dynamical behaviors of the system are investigated through theoretical analysis and numerical simulation.     

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.     

Symmetry, cusp bifurcation and chaos of an impact oscillator between two rigid sides

Both the symmetric period n-2 motion and asymmetric one of a one-degree- of-freedom impact oscillator are considered.The theory of bifurcations of the fixed point is applied to such model,and it is proved that the symmetric periodic motion has only pitchfork bifurcation by the analysis of the symmetry of the Poincarémap.The numerical simulation shows that one symmetric periodic orbit could bifurcate into two antisymmet- ric ones via pitchfork bifurcation.While the control parameter changes continuously, the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences,and bring about two antisymmetric chaotic attractors subse- quently.If the symmetric system is transformed into asymmetric one,bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp, and the pitchfork changes into one unbifurcated branch and one fold branch.     

Chaotic dynamics in a neural network under electromagnetic radiation

In this paper, a small Hopfield neural network with three neurons is studied, in which one of the three neurons is considered to be exposed to electromagnetic radiation. The effect of electromagnetic radiation is modeled and considered as magnetic flux across membrane of the neuron, which contributes to the formation of membrane potential, and a feedback with a memristive type is used to describe coupling between magnetic flux and membrane potential. With the electromagnetic radiation being considered, the previous steady neural network can present abundant chaotic dynamics. It is found that hidden attractors can be observed in the neural network under different conditions. Moreover, periodic motion and chaotic motion appear intermittently with variations in some system parameters. Particularly, coexistence of periodic attractor, quasiperiodic attractor, and chaotic strange attractor, coexistence of bifurcation modes and transient chaos can be observed. In addition, an electric circuit of the neural network is implemented in Pspice, and the experimental results agree well with the numerical ones.     

Elucidating the escape dynamics of the four hill potential

This paper treads discontinuous bifurcation in piecewise smooth systems of Filippov type. These bifurcations occur when a fixed point or a periodic orbit crosses with the border between two regions of smooth behavior. A detailed analysis of generalization Poincaré map and monodromy matrix which are related shows that subfamily of system with invariant cone-like objects is foliated by periodic orbits and determines its stability. In addition, we introduce a theoretical framework for analyzing 3D perturbed nonlinear piecewise smooth systems and give necessary conditions so that different types of bifurcations occur. The analysis identifies criteria for the existence of a novel bifurcation based on sensitively the location of the return map. Moreover, the piecewise smooth Melnikov function and sufficient conditions of the existence of the periodic orbits for nonlinear perturbed system are explicitly obtained.     

Kawakami映射的超混沌行为研究

通过对一类平面二维映射系统非线性动力学行为的分析,发现该系统存在一个奇怪吸引子,该吸引子具有两个正Lyapunov指数和分数维。通过该系统不动点的分析揭示了该吸引子的吸引域边界结构,即不稳定第二类结点与不稳定偶数周期点在吸引域边界上的相间排列。     

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.     

参数激励耦合系统的复杂动力学行为分析

分析了耦合van der Pol振子参数共振条件下的复杂动力学行为．基于平均方程，得到了参数平面上的转迁集，这些转迁集将参数平面划分为不同的区域，在各个不同的区域对应于系统不同的解．随着参数的变化，从平衡点分岔出两类不同的周期解，根据不同的分岔特性，这两类周期解失稳后，将产生概周期解或3—D环面解，它们都会随参数的变化进一步导致混吨．发现在系统的混沌区域中，其混吨吸引子随参数的变化会突然发生变化，分解为两个对称的混吨吸引子．值得注意的是，系统首先是由于2—D环面解破裂产生混吨，该混吨吸引子破裂后演变为新的混吨吸引子，却由倒倍周期分岔走向3—D环面解，也即存在两条通向混沌的道路：倍周期分岔和环面破裂，而这两种道路产生的混吨吸引子在一定参数条件下会相互转换．     

Experimental characterization of nonlinear systems: a real-time evaluation of the analogous Chua’s circuit behavior

This paper presents an experimental characterization of the behavior of an analogous version of the Chua’s circuit. The electronic circuit signals are captured using a data acquisition board (DAQ) and processed using LabVIEW environment. The following aspects of the time series analysis are analyzed: time waveforms, phase portraits, frequency spectra, Poincaré sections, and bifurcation diagram. The circuit behavior is experimentally mapped with the parameter variations, where are identified equilibrium points, periodic and chaotic attractors, and bifurcations. These analysis techniques are performed in real-time and can be applied to characterize, with precision, several nonlinear systems.     

Antimonotonicity,chaos and multiple attractors in a novel autonomous memristor-based jerk circuit

A novel memristor-based oscillator derived from the autonomous jerk circuit (Sprott in IEEE Trans Circuits Syst II Express Briefs 58:240–243, 2011) is proposed. A first-order memristive diode bridge replaces the semiconductor diode of the original circuit. The complex behavior of the oscillator is investigated in terms of equilibria and stability, phase space trajectories plots, bifurcation diagrams, graphs of Lyapunov exponents, as well as frequency spectra. Antimonotonicity (i.e. concurrent creation and destruction of periodic orbits), chaos, periodic windows and crises are reported. More interestingly, one of the main features of the novel memristive jerk circuit is the presence of a region in the parameters’ space in which the model develops hysteretic behavior. This later phenomenon is marked by the coexistence of four different (periodic and chaotic) attractors for the same values of system parameters, depending solely on the choice of initial conditions. Basins of attractions of various competing attractors display complex basin boundaries thus suggesting possible jumps between coexisting solutions in experiment. Compared to previously published jerk circuits with similar behavior, the novel system distinguishes by the presence of a single equilibrium point and a relatively simpler structure (only off-the-shelf electronic components are involved). Results of theoretical analyses are perfectly traced by laboratory experimental measurements.