Grazing Bifurcation Research Articles

Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators

By introducing a switching scheme related to the state and time, a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system. Shooting methods to locate the limit cycle and specify bifurcation sets are described by defining an appropriate Poincaré map. Different types of multiple-Focus/Cycle and single-Focus/Cycle period oscillations in the system can be observed. Symmetry-breaking, period-doubling, and grazing bifurcation curves are obtained in the plane of bifurcation parameters, dividing the parameters plane into several regions corresponding to different kinds of oscillations. Meanwhile, based on the numerical simulation and bifurcation analysis, the mechanisms of several typical dynamical behaviors observed in different regions are presented.

Higher order discontinuity mapping for double grazing bifurcations in a slender rigid block confined between side-walls

We investigate the near grazing dynamics in a slender rigid block confined between two side-walls. We obtain conditions of the existence of double grazing orbits of the system, namely, periodic orbits that have two grazing points. By extending the method of discontinuity mapping for grazing orbit with a single grazing point, we compute the lower and the higher order approximations of the Poincaré map respectively near the double grazing orbit. The results of computing Monte Carlo bifurcation diagrams obtained by the lower and the higher order maps respectively are compared with those from direct simulations of the original system. We find that there are large disagreements between the lower order map and the original system. Thus the lower order map is not accurate enough to study double grazing bifurcations. On the other hand, the higher order map can effectively reduce such disagreements and we expect that it plays an important role in the study of the near grazing dynamics of the system.

Two-parameter non-smooth bifurcations of period-one motions in a plastic impacting oscillator

A 2-DOF plastic impacting oscillator is considered. The discontinuity boundaries of the system phase space are defined as well as the local maps and Poincaré map. Four cases of sliding bifurcation of limit cycles in impact system are described schematically. The period-one motion modes and occurrence regions of the system are identified in the two-parameter space by numerical simulation. One-parameter bifurcation diagrams and trajectories are computed to explain the evolutions of adjacent periodic motions and the non-smooth bifurcation behaviors including grazing and four types of sliding bifurcation. Due to the discontinuous grazing bifurcation there appear a narrow hysteresis domain in the two-parameter evolution between neighboring non-impacting and impacting motions. Grazing, switching–sliding and multi-sliding bifurcations of periodic sticking motions are all manifested as rising phenomenon of the time trajectory, but the different locations of the rises in the sticking phase correspond to the different types of non-smooth bifurcations. The grazing induced dynamics and the characteristics of rising events are discussed. Notably, the codimension-2 bifurcation points are detected and investigated by numerical continuation algorithm.

Existence of Impact Period-1 Orbit After Grazing Bifurcation for a Soft Impacting System

In the paper, the theoretical study on some experimental and numerical results of grazing bifurcation for a soft impacting system are analyzed. After the conditions under which nonimpact period-1 orbit and grazing bifurcation exist are given, we prove that an impact period-1 orbit exists by the implicit function theorem. The details of these periodic orbits are also investigated, which also helps us numerically find them. The method in this paper is still efficient for the multiperiodic orbits with single impact.

Discontinuity-induced bifurcations in a piecewise-smooth capsule system with bidirectional drifts

Discontinuous friction and impact exist in a variety of piecewise-smooth dynamical systems that exhibit rich and complex nonlinear phenomena. This paper aims to study the discontinuity-induced grazing and adding-sliding bifurcations in a piecewise-smooth capsule system subjected to bidirectional drifts. For this purpose, analytical and semi-analytical onset conditions of the impact-induced grazing and the friction-induced adding-sliding bifurcations are investigated by solving explicitly the underlying piecewise-linear model. In this way, parametric relations revealing the occurrence mechanisms of transitions of the system from stationary to forward progression are derived. This analytical approach is verified numerically using path-following techniques for piecewise-smooth dynamical systems. The numerical investigation includes a parametric study of the observed capsule's average velocity and power consumption suggesting optimum parameters for system operation.

Invariant torus and its destruction for an oscillator with dry friction

Mechanical systems with dry friction are typical Filippov systems. Such class of systems have complicated dynamical behaviors due to the existence of sliding motion. In this work, we consider a one-degree-of-freedom oscillator with dry friction force. The phase map is derived to reduce the system to a circle map, and then, the existence of forward invariant torus is proved under suitable assumptions. Moreover, the typical resonance phenomenon and the grazing bifurcation of invariant torus are discussed. We find that the destruction of invariant tori is due to a loss of transversality for sufficiently large perturbation, which is different from the usual smooth tori dynamics.

Global asymptotical stability and sliding bifurcation analysis of a general Filippov-type predator-prey model with a refuge

In this paper, a general Filippov-type predator-prey model with a refuge is presented. We propose a discontinuous predator-prey model incorporating a threshold policy by extending a general continuous predator-prey model. By employing the qualitative analysis theory related to Filippov systems, the necessary and sufficient conditions for the global asymptotical stability of a standard cycle, a touching cycle and a sliding cycle are obtained respectively. Furthermore, the sliding cycle is globally finite-time stable. Especially, several kinds of sliding bifurcations including boundary node bifurcation, boundary focus bifurcation and grazing bifurcation are studied. Moreover, two specific models are provided to verify the main results obtained from the general model.

Dynamic Behaviors of a Two-Degree-of-Freedom Impact Oscillator with Two-Sided Constraints

The dynamic model of a vibroimpact system subjected to harmonic excitation with symmetric elastic constraints is investigated with analytical and numerical methods. The codimension-one bifurcation diagrams with respect to frequency of the excitation are obtained by means of the continuation technique, and the different types of bifurcations are detected, such as grazing bifurcation, saddle-node bifurcation, and period-doubling bifurcation, which predicts the complexity of the system considered. Based on the grazing phenomenon obtained, the zero-time-discontinuity mapping is extended from the single constraint system presented in the literature to the two-sided elastic constraint system discussed in this paper. The Poincare mapping of double grazing periodic motion is derived, and this compound mapping is applied to obtain the existence conditions of codimension-two grazing bifurcation point of the system. According to the deduced theoretical result, the grazing curve and the codimension-two grazing bifurcation points are validated by numerical simulation. Finally, various types of periodic-impact motions near the codimension-two grazing bifurcation point are illustrated through the unfolding diagram and phase diagrams.

Discontinuous Dynamic Analysis of a Class of 2-DOF Oscillators With Strong Nonlinearity Under a Periodic Excitation

Starting from the car suspension system, the nonlinear characteristics of a class of two-degree-of-freedom oscillators with strong nonlinearity under a periodic excitation are discussed by the switching theory of flow in discontinuous dynamical systems. Based on the discontinuous forces and different motions of the two masses, the phase plane of each mass is composed of stick domain, nonstick domain (or free domain) and separation boundary in absolute and relative coordinates, respectively. The swithching criteria between the stick and nonstick motions and the conditions of grazing motion in two different regions are developed via the G-functions and switching control laws. The mapping dynamics theory is used to give the four-dimensional transformation set and four-dimensional mapping, and the conditions for periodic motions are explored. In addition, the stick motions, two kinds of grazing motions, periodic motions for this system and a comparison of the velocities, accelerations (or forces responses) of the two masses under the two conditions of control force are simulated numerically. The results show that the stability and comfort of the vehicle can be improved by adjusting the control force, which is generated by the control unit of system or exerted by the external excitation. For further investigating the influence of system parameters on dynamical behaviors, the stick and grazing bifurcation scenarios varying with driving frequency or amplitude are also developed, which can provide useful information for parameter selection of vibration systems with clearance and the optimal design of vehicle suspension systems. This paper also has important reference value for practical applications in other industries or machinery with elastic impacts.

Complex dynamical behavior of modified MLC circuit

Abstract In this paper, we mainly investigate the complex dynamical behavior of modified MLC circuit. When its external excitation doesn’t equal zero, it is nonautonomous and non-smooth, hence we theoretically give the conditions under which grazing bifurcation occurs and the periodic orbits only lie in some one zone. More complex grazing bifurcations and coexisting attractors are also found by numerical simulation, among which is mainly produced by the symmetry. Grazing bifurcations and doubling-period bifurcations that can induce chaotic motion and some basins of attraction are given in this paper. If the external excitation of this system equal zero, it is a non-smooth autonomous system. For this case, we theoretically analysis its non-smooth bifurcations of equilibrium points and limit cycle bifurcation, which is also verified by numerical simulation.

Global dynamics of a non-smooth system with elastic and rigid impacts and dry friction

Global dynamics of a non-smooth dynamic model is discussed under the combined effects of three non-smooth factors of elastic impact, rigid impact and dry friction,which is a rarely discussed topic before.Six different basic phases, nine events and six segments of the system have been described. The stability of periodic motions in the system is analyzed theoretically. The motions distribution and the transition to the sticking-adhesion motions are studied. Influences of the sliding bifurcation and grazing bifurcation in the way to the transition to sticking-adhesion motions are demonstrated. The transition to sticking-adhesion motions composed of one or more of three basic routes is illustrated. The transition law of three kinds of fundamental periodic motions in the low-frequency and small-gap parameter domain is further analyzed so as to delineate the transition mechanism from fundamental periodic motions to sticking-adhesion motions. In high frequency region, the global motions distribution in the system is obtained through the global analysis method in parameter-state space. The transition and coexistence of the motions including stable and unstable periodic motions are further explored. In addition, the motions distribution and transition of large friction force versus forcing amplitude from the static state are discussed. The results will be beneficial for the motions control of the system.

Post-grazing dynamics of a vibro-impacting energy generator

The motion of a forced vibro-impacting inclined energy harvester is investigated in parameter regimes with asymmetry in the number of impacts on the bottom and top of the device. This motion occurs beyond a grazing bifurcation, at which alternating top and bottom impacts are supplemented by a zero velocity impact with the bottom of the device. For periodic forcing, we obtain semi-analytical expressions for the asymmetric periodic motion with a ratio of 2:1 for the impacts on the device bottom and top, respectively. These expressions are derived via a set of nonlinear maps between different pairs of impacts, combined with impact conditions that provide jump discontinuities in the velocity. Bifurcation diagrams for the analytical solutions are complemented by a linear stability analysis around the 2:1 asymmetric periodic solutions, and are validated numerically. For smaller incline angles, a second grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For larger incline angles, period doubling bifurcations precede grazing bifurcations. The converted electrical energy per impact is reduced for the asymmetric motions, and therefore less desirable under this metric.

Global behavior of a simplified model for the micro-vibration molding machine in parameter-state space

Global behavior of a simplified non-smooth dynamic model for a micro-vibration molding machine in parameter-state space is explored. Solutions for the periodic motions of the impact oscillator are obtained by the semi-analytical method. Conditions for the grazing motions and sticking motions of the model are deduced. The motions distribution and transition in the fundamental periodic motion regions and the periodic inclusions areas are studied. Transition of the motions due to the singularity of the grazing bifurcation is analyzed. Polymorphic coexistence under the effect of the grazing bifurcation and the saddle-node bifurcation are revealed. Through the global analysis method in parameter-state space, the global distribution map in high frequency region can be acquired, which is rarely considered before. Under the guidance of the global motions distribution, the evolution of the polymorphic coexistence areas can be further discussed. The results can provide some references and ideas for the dynamic design and motion control of the micro-vibration molding machine.

Two-parameter bifurcations of an impact system under different damping conditions

Abstract The periodic motion patterns and (ω, δ)-parameter domains of an impact system are identified by means of constructing two Poincare maps. The influence of the damping ratio ζ on the transition characteristics between adjacent p/1 (p ≥ 0) motions is analyzed, and the correlations between impact velocities of 1/n (n ≥ 1) motions, occurrence domains and system damping parameters are discussed. The transition between adjacent fundamental motions is continuous and reversible for the case of large ζ. However, such transition is irreversible for the case of small ζ. In the cases of small ζ and very small ζ, there exist two types of transition zones, namely ligulate and hysteresis zones, which lie between the (ω, δ)-parameter domains of adjacent p/1 (p ≥ 0) motions. We give a detailed analysis of dynamics in the ligulate and hysteresis zones. As ζ is small enough, a subharmonic inclusions zone appears in the bottom region of p/1 fundamental motion. In the subharmonic inclusions zones, the system presents rich dynamic behaviors, including periodic bubble, period-doubling cascade, grazing and saddle-node bifurcations, as well as hysteresis phenomena and chaos. An additional discovery is that a narrow hysteresis interval exists near the period-doubling bifurcation and the period-doubling bifurcation boundary locates at different positions under different initial conditions.

Periodic Orbit Bifurcations in Planar Hysteretic Systems without Equilibria

This paper is devoted to the analysis of bidimensional piecewise linear systems with hysteresis coming from 3D systems with slow–fast dynamics. We focus our attention on the symmetric case without equilibria, determining the existence of periodic orbits as well as their stability, and possible bifurcations. New analytical characterizations of bifurcations in these hysteretic systems are obtained. In particular, bifurcations of periodic orbits from infinity, grazing and saddle-node bifurcations of periodic orbits are studied in detail and the corresponding bifurcation sets are provided. Finally, the study of the hysteretic systems is shown to be useful in detecting periodic orbits for some [Formula: see text]D piecewise linear systems.

Explanation of the onset of bouncing cycles in isotropic rotor dynamics; a grazing bifurcation analysis.

The dynamics associated with bouncing-type partial contact cycles are considered for a 2 degree-of-freedom unbalanced rotor in the rigid-stator limit. Specifically, analytical explanation is provided for a previously proposed criterion for the onset upon increasing the rotor speed Ω of single-bounce-per-period periodic motion, namely internal resonance between forward and backward whirling modes. Focusing on the cases of 2 : 1 and 3 : 2 resonances, detailed numerical results for small rotor damping reveal that stable bouncing periodic orbits, which coexist with non-contacting motion, arise just beyond the resonance speed Ω p:q . The theory of discontinuity maps is used to analyse the problem as a codimension-two degenerate grazing bifurcation in the limit of zero rotor damping and Ω = Ω p:q . An analytic unfolding of the map explains all the features of the bouncing orbits locally. In particular, for non-zero damping ζ, stable bouncing motion bifurcates in the direction of increasing Ω speed in a smooth fold bifurcation point that is at rotor speed beyond Ω p:q . The results provide the first analytic explanation of partial-contact bouncing orbits and has implications for prediction and avoidance of unwanted machine vibrations in a number of different industrial settings.

Control of Near-Grazing Dynamics in the Two-Degree-of-Freedom Vibroimpact System with Symmetrical Constraints

The stability of grazing bifurcation is lost in three ways through the local analysis of the near-grazing dynamics using the classical concept of discontinuity mappings in the two-degree-of-freedom vibroimpact system with symmetrical constraints. For this instability problem, a control strategy for the stability of grazing bifurcation is presented by controlling the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Discrete-in-time feedback controllers designed on two Poincare sections are employed to retain the existence of an attractor near the grazing trajectory. The implementation relies on the stability criterion under which a local attractor persists near a grazing trajectory. Based on the stability criterion, the control region of the two parameters is obtained and the control strategy for the persistence of near-grazing attractors is designed accordingly. Especially, the chaos near codimension-two grazing bifurcation points was controlled by the control strategy. In the end, the results of numerical simulation are used to verify the feasibility of the control method.

A New Route to Strange Nonchaotic Attractors in an Interval Map

We identify an unusual route to the creation of a strange nonchaotic attractor (SNA) in a quasiperiodically forced interval map. We find that the smooth quasiperiodic torus becomes nonsmooth due to the grazing bifurcation of the torus. The nonsmooth points on the torus increase more and more with the change of control parameter. Finally, the torus gets extremely fractal and becomes a SNA which is termed the grazing bifurcation route to the SNA. We characterize the SNA by maximal Lyapunov exponents and their variance, phase sensitivity exponents and power spectra. We also describe the transition between a torus and a SNA by the recurrence analysis. A remarkable feature of the route to SNAs is that the positive tails decay linearly and the negative tails exhibit recurrent fluctuations in the distribution of the finite-time Lyapunov exponents.

Dynamic characteristics of a mechanical impact oscillator with a clearance

A two-degree-of-freedom mechanical impact oscillator with a clearance is considered. Pattern types, occurrence and stability domains and bifurcation characteristics of periodic motions are investigated through two-parameter co-simulation analysis. With varying of the bifurcation parameters in two directions, forward and backward, two-parameter grazing, period-doubling and saddle-node bifurcations of impactless, fundamental and subharmonic motions are calculated and analyzed. There exist two types of transition zones, hysteresis and liguliform zones, in mutual transition between neighboring motions p/1 and (p + 1)/1 (p ≥ 0). Dynamics in the hysteresis and liguliform zones is introduced in detail. Attracting domains and Poincaré mapping diagrams of coexisting motions in the neighborhood of grazing bifurcations are discussed. Pattern types and bifurcations of subharmonic motions in liguliform zones show regularity. In the two-parameter plane, two grazing bifurcation curves of p/1 (p ≥ 0) motion and period-doubling and saddle-node bifurcation curves of (np + 1)/n (n ≥ 1) motion intersect at a singular point along the upper boundary of liguliform zones LZp/1∩(p + 1)/1. Consequently, the singular point is a point of double-grazing bifurcation of p/1 motion and codimension-2 flip-fold bifurcation of (np + 1)/n motion. The grazing bifurcation of neighboring motions is continuous and reversible only at the singular points, and displacement amplitudes of the impact oscillator vary continuously when the varying parameter passes through the grazing bifurcation boundary.

On discontinuous dynamics of a class of friction-influenced oscillators with nonlinear damping under bilateral rigid constraints

This article is concerned with the discontinuous dynamical behaviors of a class of single degree of freedom oscillators with linear and nonlinear springs and viscous dampers in the co-existence of bilateral rigid impact and friction through the theory of flow switchability and flow barrier in discontinuous dynamical systems, where considering the inequality of static and kinetic friction coefficients. By the analysis of vector fields and G-functions on the corresponding discontinuous boundaries for such an oscillator, the analytical conditions of all possible motions are established, which are in the form of a set of inequalities or equalities and can be used for the selection of control parameters in this kind of system. Additionally, based on mapping dynamics, the two-dimensional basic mappings are defined, the analytical predictions and stability analysis of different periodic motions are accomplished. Finally, our theoretical results are further demonstrated by numerical simulations by means of dimensional and dimensionless parameters with graphical illustrations of the time histories of displacement, velocity, G-function and the trajectories in phase space for the passable motion, stuck motion, impact motion, grazing motion and periodic motion etc., and stick and grazing bifurcation scenarios varying with excitation frequency and amplitude are also given.