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![The periodic orbit: μ=0.4,f0=0.8,w=0.8512,y0∗=0.8410,t0∗=5.3485\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =0.4, f_{0}=0.8, w=0.8512, y^{*}_{0}=0.8410, t^{*}_{0}=5.3485$$\end{document}](publication/324917581/figure/fig2/AS:779422244274176@1562840152100/The-periodic-orbit.gif)
The periodic orbit: μ=0.4,f0=0.8,w=0.8512,y0∗=0.8410,t0∗=5.3485\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =0.4, f_{0}=0.8, w=0.8512, y^{*}_{0}=0.8410, t^{*}_{0}=5.3485$$\end{document}
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Piecewise linear approach is one of important methods to study nonlinear dynamics of oscillators with complex nonlinear restoring forces, so piecewise-smooth systems with multiple switching manifolds sometimes are ideal mathematical models to analyze the nonlinear dynamics of nonlinear oscillators. In this paper, inspired by the work presented by C...
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Citations
... The mechanical model of the SD oscillator and its rich dynamics subjected to weak damping and weak periodic excitation with U (x) = x 2 − 2 √ x 2 + α 2 + 2α, 0 < α < 1, g(x,ẋ) = δẋ, and f (t) = f cos(ωt): (a) Mechanical model: The original length of the springs is L, whose length after compression in the vertical direction is l; (b) potential energy diagrams for α = 2/3; (c) phase trajectories of the unperturbed systems for δ = f = 0 and α = 2/3; (d) the phase diagram for f = 0.18, δ = 0.1, ω = 0.4, and α = 2/3; (e) the phase diagram for f = 0.26, δ = 0.1, ω = 0.4, and α = 2/3; and (f) the phase diagram for f = 0.34, δ = 0.1, ω = 0.4, and α = 2/3. for smooth dynamical systems, and extended by [Kunze, 2000;Shi et al., 2013;Battelli & Fečkan, 2008, 2010, 2012Tian et al., 2016aTian et al., , 2016bTian et al., 2020;Li et al., 2014Li et al., , 2015Li et al., 2016aLi et al., , 2016bLi et al., 2017Li et al., , 2020Li & Zhao, 2018] to nonsmooth dynamical systems. In recent times, an increasing number of academics have developed a keen interest in these models. ...
This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable, vibro-impact Smooth-and-Discontinuous (SD) oscillator. First, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod, so the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistable vibration characteristics, and two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate the vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable, vibro-impact SD oscillator through studying the persistence of the unique, unperturbed, nonsmooth, homoclinic structure. Second, the general framework of random Melnikov process for a class of bistable, vibro-impact systems contaminated with Gaussian white noise is derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable, vibro-impact SD oscillator. Third, the effectiveness of a semi-analytical prediction by the Melnikov function is verified using the largest Lyapunov exponent, bifurcation series, and 0–1 test. Finally, the sensitivity to the initial values of chaos is verified by the fractal attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show the rich dynamical geometric structures.
... From the perspective of engineering applications, some scholars have paid attention to vibro-impact bistable oscillators by considering the collision constraint, such as the vibro-impact inverted pendulum [3,4,5], the vibroimpact SD oscillator [6,7,8], etc. The basic idea of global vibro-impact dynamics is to extend the non-smooth Melnikov method, which was first proposed by Melnikov [9] and developed in Guckenheimer & Holmes [10] and Wiggins [11] for smooth dynamical systems and extended by Kunze [12], Shi et al. [13], Battelli and Feckan [14,15,16,17], Tian et al. [18,19,20], and Li et al. [21,22,23,24,25,26,27] to non-smooth dynamical systems. In recent times, an increasing number of academics have developed a keen interest in these models. ...
... Proof. Substituting (23)- (26) into (14)- (18) gives ...
This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable vibro-impact SD oscillator. Firstly, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod so that the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistble vibration characteristic, two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable vibro-impact SD oscillator through studying the persistence of the unique unperturbed non-smooth homoclinic structure. Secondly, the general framework of random Melnikov process for a class of bistable vibro-impact systems under Gaussian white noise is simply derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable vibro-impact SD oscillator. Thirdly, the effectiveness of a semi-analytical prediction by the Melnikov function is verified through the lagest Lyapunov exponent, bifurcation series, and 0 − 1 test. In addition, the sensitivity to the initial values of chaos is verified by the fractal of attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show rich dynamical geometric structures.
... Finally, the correctness of the theoretical model is verified by experiment [9]. The Melnikov function of sub-harmonic orbit of piecewise system is studied in reference [10]. Shi considers a gap periodic forced vibration system composed of two-sided symmetric rigid constraints [11]. ...
Considering the mass block system under Piecewise nonlinear constraint, the vibration dynamic model of the system is established according to the generalized dissipative Lagrange principle, and the average method is used to solve the amplitude-frequency response of the vibration system. The influence of system parameters on vibration characteristics is analyzed with amplitude-frequency characteristics, phase plane characteristics, frequency characteristics, bifurcation characteristics ,and so on. The results show that: 1) the reverse of the rate of change of Piecewise nonlinear elastic force will destroy the stability of the system and obtain the relationship of the constraint parameters that need to be satisfied when the system is stable at the piecewise critical point. 2) With the increase in the number of nonlinear constraints, the vibration displacement of the system tends to be chaotic, and the frequency composition becomes more complex and variable, prone to resonance behavior. 3) As the static gap decreases and the dynamic gap amplitude and frequency increase, the unstable frequency range of the system will increase, and the vibration behavior will become chaotic and difficult to predict. 4) The design of a differential sliding mode controller can effectively control the bifurcation behavior of the system.
... In another aspect, subharmonic orbits are peculiar time-periodic orbits which satisfy some rational ratio of its own period to an excitation period, and a number of researches have been done to study the subharmonic orbits for non-smooth dynamical systems with time-periodic perturbations [38][39][40][41][42][43][44][45][46]. For example, In [38,39], Du et al. have presented a general Melnikov analysis of Type I and Type II subharmonic orbits for a class of nonlinear vibro-impact systems. ...
... Then, the existence of subharmonic orbits is presented in the follow theorem, and the similar proof can be found in [42] and omitted here. (12) denoting the period of the periodic orbits in the region inside the homoclinic orbits for different initial conditions. ...
... From the idea in Li et al. [42], we need to find (ȳ 0 ,t 0 ) in the following analysis such that ...
In this paper, subharmonic motions of a bistable vibro-impact oscillator are studied by establishing the theoretical framework of the Melnikov analysis for an abstract non-smooth dynamical system. A ideal impacting map is employed to describe velocity changing during instantaneous collision with the bilateral rigid constraints. The unperturbed system without considering damping and external excitations is supposed to have a pair of homoclinic orbits connecting the origin to itself, and the inner and outer regions separated by the homoclinic orbits are assumed to be fully covered by periodic orbits, whose periods monotonically increase as they approach the homoclinic connections. Furthermore, periodic or homoclinic grazing of the unperturbed system can also occur by adjusting the position of the constraints. When a periodic perturbation is considered, the definitions of the Unilateral subharmonic orbits, the Bilateral subharmonic orbits and the Compound subharmonic orbits for this class of non-smooth systems are given by combining the impacting dynamics. The Melnikov functions for the first two types subharmonic orbits are also obtained and employed to detect the initial conditions for the existence of the corresponding subharmonic orbits. Finally, the bistable vibro-impact oscillator as an example is used to show the effectiveness of the developed Melnikov method for seeking subharmonic motions for this class of bistable vibro-impact oscillator. A numerical integration method is also introduced to overcome the difficulty of the Melnikov integration along the unperturbed periodic orbits.
... Si et al. [42] have extended non-smooth Melnikov method and studied bifurcations and chaos for a piecewise nonlinear roll system of rolling mill. In order to avoid differential items in the Melnikov function and pursuit the intuitiveness of geometry, Li et al. [43][44][45][46][47][48][49] have presented the Melnikov method for many kinds of time-periodic planar (hybrid) systems with simple perturbation techniques. ...
How to model non-smooth dynamical systems and study its non-smooth global dynamics by developing analytical methods is now an important topic. In this paper, a new bistable nonlinear oscillator with bilateral elastic constraints and a three-piecewise nonlinear restoring force is established to study the perturbation of viscous damping and an external harmonic excitation. A five-piecewise linear approach to the restoring force can transform the original dimensionless dynamical equations into a two-dimensional piecewise-defined Hamiltonian systems, separated by four switching manifolds with symmetrical characteristic and subjected to damping dissipation and a periodic excitation. The geometrical structure of the unperturbed system has a pair of symmetrical piecewise-defined homoclinic orbits and each of them transversally and continuously crosses two switching manifolds. The analytical Melnikov method is extended here to fit the theoretical framework for analyzing homoclinic bifurcations and chaotic dynamics for non-smooth oscillators with multiple switching manifolds. A global perturbation technique is presented to calculate the distance between the stable and unstable manifolds and derive the Melnikov function in the form of piecewise-integration. A major innovation here is no need to extend the vector filed near the switching manifolds and carry out rigorous perturbation analysis with geometrical intuition to derive the Hamilton energy differences of trajectories involving infinite time. The developed Melnikov function can be directly used to detect the parameter threshold of homoclinic chaos in the bistable nonlinear oscillator under bilateral elastic collision. Finally, the effectiveness of the Melnikov analysis for homoclinic chaos is verified by simulations.
... The Melnikov method for non-smooth systems is an important analytical method [38][39][40][41][42][43][44] used to analyze the global chaos of piecewise-smooth systems. Hence, based on this mechanism, it is necessary to reduce the dimensions of the EI-BNES from four to two and analyse the sufficient conditions to induce transient chaos by means of the Melnikov function of non-smooth systems. ...
A new highly efficient elastic-impact bistable nonlinear energy sink (EI-BNES) based on magnetic-elastic impacts with negative stiffness and bistability is proposed and optimized through global dynamical analysis. The EI-BNES has better robustness and higher energy dissipation rates with nearly more than 96.5\% for broadband impulsive excitations than the traditional cubic NESs and single bistable NESs. The structure of negative stiffness impacts is realized by reasonable layout of permanent ring magnets and springs. A two-degree-of-freedom (two-DOF) elastic-impact system is established to describe the coupled nonlinear interaction between the main structure and the attached EI-BNES. A global Melnikov reduction analysis (GMRA) is proposed to study global dynamics and homoclinic bifurcations of the reduced two-dimensional subsystem, which is used to explain the mechanism of nonlinear targeted energy transfer (TET) and detect the threshold of impulsive amplitudes of EI-BNES for in-well and compound motions between in-well and cross-well resonance responses. A special type of saddle-center equilibrium points is also found in the non-smooth system of the EI-BNES and can be used to effectively increase the energy dissipation rates. The optimal design criterion of the tuned EI-BNES for better dissipation performance is also first discussed based on the GMRA and numerical techniques for calculating the Melnikov function of the non-smooth systems. The effectiveness of the analytical GMRA is also verified by numerical simulations.
... Li et al. [25,26] further extended the Melnikov method of homoclinic trajectories for a planar hybrid piecewise smooth system with a switching manifold, on which an impact map was applied to represent the impact rule of trajectories. Li et al. [27] presented the analytical Melnikov method for studying subharmonic periodic trajectories in planar piecewise smooth systems with two switching manifolds. ...
A new bistable impact oscillator with bilateral rigid constraints under periodic excitations is established and the global dynamics are studied in detail respectively by the extended analytical Melnikov method for non-smooth systems and dynamical experiments. Firstly, the Melnikov method is extended from a new viewpoint of geometry for an abstract non-smooth dynamical system denoting a class of bistable impact oscillators with bilateral rigid constraints. Secondly, the analytical expression of homoclinic orbits is difficult to obtain, a semi-analytical and semi-numerical method for calculating the Melnikov function is applied to obtain the threshold of parameters for the global bifurcations and chaotic oscillations of the established impact oscillator. Then, numerical simulations and dynamical experiments are carried out together to show intra-well periodic oscillations, inter-well periodic or chaotic oscillations under different amplitudes of periodic excitations, which furthermore validates the reliability of the extended Melnikov method for this class of non-smooth systems.
... Further, the well-known Melnikov method is a technique for investigating the existence of periodic orbits and homoclinic/heteroclinic connection for planar PDS by guaranteeing the existence of simple zeros of a certain function. For instance, in [3,6,14,29,30], the authors investigated the existence of periodic orbits, sliding homoclinic, homoclinic, subharmonic, heteroclinic and chaotic behavior for different types of nonsmoothness planar system by extending the classical method to cover abrupt events of PDS. Furthermore, in dynamical engineering systems, Awrejcewicz and Holicke [4] derived Melnikov's method for PDS with the existence of dry friction for a stick-slip oscillator and their results extended to study the appearance of transversal homoclinic orbits, stick-slip and slip-slip chaotic orbits. ...
... Differential equations with the presence of discontinuities have appeared in many important applications in scientific and engineering research [2,9,23,25,32,33,36]. Hence, the analysis of PDS has been the subject of much ongoing research [18,22,28,30,31]. Close to the codimension − 1 discontinuity surface, the local dynamics in two possible behaviors (transversal crossing or attractive sliding) are well understood, analytically as well as numerically. ...
This paper deals with bifurcation of limit cycles for perturbed piecewise-smooth systems. Concentrating on the case in which the vector fields are defined in four domains and the discontinuity surfaces are codimension-2 manifolds in the phase space. We present a generalization of the Poincar\'{e} map and establish some novel criteria to create a new version of the Melnikov-like function. Naturally, this function is designed corresponding to a system trajectory that interacts with two different discontinuity surfaces. This provides an approach to prove the existence of special type of invariant manifolds enabling the reduction of dynamics of the full system to the 2-dimensional surfaces of the invariant cones. It is shown that there exists a novel bifurcation concerning the existence of multiple invariant cones for such system. Further, our results are then used to control the persistence of limit cycles for 2- and 3-dimensional perturbed systems. The theoretical results of these examples are illustrated by numerical simulations.
... Tian et al. [41] also derived the Melnikov function and studied the chaotic threshold for a non-smooth pendulum with different impulsive excitations realized by collision with a rigid wall. In order to study the global dynamics of discontinuous systems with geometric visualization and overcome the difficulties in calculation and applications, more recently Li et al. [42][43][44][45][46][47] employed a geometrical approach and different proof to systematically derive the Melnikov function in a simple form for trajectories transversally crossing or instantaneously jumping on switching manifolds. ...
In this work, some new effective methods for suppressing homoclinic chaos in a weak periodically excited non-smooth oscillator are studied, and the main idea is to modify slightly the Melnikov function such that the zeros are eliminated. Firstly, a general form of planar piecewise-smooth oscillators is given to approximatively model many nonlinear restoring force of smooth oscillators subjected to all kinds of damping and periodic excitations. In the absence of controls, the Melnikov method for non-smooth homoclinic trajectories within the framework of a piecewise-smooth oscillator is briefly introduced without detailed derivation. This analytical tool is useful to detect the threshold of parameters for the existence of homoclinic chaos in the non-smooth oscillator. After some methods of state feedback control, self-adaptive control and parametric excitations control are, respectively, considered, sufficient criteria for suppressing homoclinic chaos are derived by employing the Melnikov function of non-smooth systems. Finally, the effectiveness of strategies for suppressing homoclinic chaos is analytically and numerically demonstrated through a specific example.
In this paper, a double pendulum model with multi-point collision is established to study the sub-harmonic bifurcation of high-dimensional coupled non-smooth systems. Considering the coupling and non-smoothness of the system, a two-step decoupling method is proposed to detect the sub-harmonic bifurcation of a two-degree-of-freedom non-smooth coupled system. The core view is to introduce energy-time scale transformation to overcome the obstacle of the system coupled term. In the first step, a reversible transformation is introduced to decouple the system. This transformation enables the coupled form of the impact term, which presents novel obstacles to the high-dimensional non-smooth system. By introducing energy-time scale transformation in the second step, the system is expressed as a smooth decoupling form of the energy coordinate, and the trouble of impact term coupled is solved. Furthermore, the sub-harmonic Melnikov function which depends on frequency, amplitude of excitation and impact recovery coefficient is derived by using the two-step decoupling method. Hence, the sub-harmonic Melnikov function is extended to the high-dimensional non-smooth system, which reveals the influence of the impact recovery coefficient on the existence of sub-harmonic periodic orbits. The innovation of this method is that it solves the coupled problem of non-smooth terms, quantifies the impact of impact recovery coefficient on the dynamic behavior of the system, and provides a theoretical basis for the actual parameter design and control in engineering. The obtained theoretical results are verified through the numerical simulations.
