Grazing Bifurcation Research Articles

Grazing and Hopf Bifurcations of a Periodic Forced System with Soft Impacts

Based on the research of a periodic forced system with soft impacts, the piecewise properties of the soft-impacts system, such as asymmetric motion and singularity, were analyzed by using the Poincaré map and Runge-Kutta numerical simulation method. The routes from periodic motions to chaos, via Hopf bifurcation and grazing bifurcation, were investigated thoroughly. In the case of large constraint stiffness, the Hopf bifurcation is observed in the periodic forced system with soft impacts. The clearances of the system are the main reasons for influencing the chaotic motion. For small clearances, the grazing bifurcations bring about asymmetric motion and singularity. The steady 1-1-1 period orbits will exist within a wideband frequency range when appropriate system parameters are chosen.

Nonsmooth Bifurcations in a Cracked Rotor-Active Magnetic Bearings System

A cracked rotor-active magnetic bearings (AMB) system with the time-varying stiffness is modeled by a piecewise smooth system due to the breath of crack in a rotating shaft. The governing nonlinear equations of motion for the nonsmooth system are established and solved with the numerical method. The simulation results show that a grazing bifurcation, period-double bifurcation and chaotic motions exist in the response. These nonsmooth bifurcations can give rise to jumps between periodic motions, quasi-periodic motions and chaos.

MODELING OF DYNAMIC SYSTEMS WITH VIBRO-IMPACT INTERACTION AND DISCONTINUITY MAPPING

ABSTRACTThis paper presents an overview of selected modeling techniques of vibro-impactdynamics. Vibro-impact dynamics has occupied a wide spectrum of studies bydynamicists, physicists, and mathematicians. These studies may be classified intothree main categories: modeling, mapping and applications. The main techniquesused in modeling of vibro-impact systems include phenomenological modeling,Hertzian models, and non-smooth coordinate transformations developed byZhuravlev and Ivanov. One of the most critical situations impeded in vibro-impactsystems is the grazing bifurcation. Grazing bifurcation is usually studied throughdiscontinuity mapping techniques, which are useful to uncover the rich dynamics inthe process of impact interaction. Complex dynamic phenomena of vibro-impactsystems such as sub-harmonic oscillations, chaotic motion, and coexistence ofdifferent attractors for the same excitation and system parameters but under differentinitial conditions will be discussed.

Double Grazing Periodic Motions and Bifurcations in a Vibroimpact System with Bilateral Stops

The double grazing periodic motions and bifurcations are investigated for a two-degree-of-freedom vibroimpact system with symmetrical rigid stops in this paper. From the initial condition and periodicity, existence of the double grazing periodic motion of the system is discussed. Using the existence condition derived, a set of parameter values is found that generates a double grazing periodic motion in the considered system. By extending the discontinuity mapping of one constraint surface to that of two constraint surfaces, the Poincaré map of the vibroimpact system is constructed in the proximity of the grazing point of a double grazing periodic orbit, which has a more complex form than that of the single grazing periodic orbit. The grazing bifurcation of the system is analyzed through the Poincaré map with clearance as a bifurcation parameter. Numerical simulations show that there is a continuous transition from the chaotic band to a period-1 periodic motion, which is confirmed by the numerical simulation of the original system.

Grazing bifurcation in aeroelastic systems with freeplay nonlinearity

A nonlinear analysis is performed to characterize the effects of a nonsmooth freeplay nonlinearity on the response of an aeroelastic system. This system consists of a plunging and pitching rigid airfoil supported by a linear spring in the plunge degree of freedom and a nonlinear spring in the pitch degree of freedom. The nonsmooth freeplay nonlinearity is associated with the pitch degree of freedom. The aerodynamic loads are modeled using the unsteady formulation. Linear analysis is first performed to determine the coupled damping and frequencies and the associated linear flutter speed. Then, a nonlinear analysis is performed to determine the effects of the size of the freeplay gap on the response of the aeroelastic system. To this end, two different sizes are considered. The results show that, for both considered freeplay gaps, there are two different transitions or sudden jumps in the system’s response when varying the freestream velocity (below linear flutter speed) with the appearance and disappearance of quadratic nonlinearity induced by discontinuity. It is demonstrated that these sudden transitions are associated with a tangential contact between the trajectory and the freeplay boundaries (grazing bifurcation). At the first transition, it is demonstrated that increasing the freestream velocity is accompanied by the appearance of a superharmonic frequency of order 2 of the main oscillating frequency. At the second transition, the results show that an increase in the freestream velocity is followed by the disappearance of the superharmonic frequency of order 2 and a return to a simple periodic response (main oscillating frequency).

Through the Looking-Glass of the Grazing Bifurcation: Part I - Theoretical Framework

It is well-known and demonstrated in many papers that for vibro-impact systems an existence of a periodic solution with a low-velocity impact may yield a complex behavior of solutions. The limit case, i.e. existence of a periodic solution with a zerovelocity impact is called grazing. Slightly changing the parameters of a system with grazing, we may obtain a periodic solution which passes near the delimiter without touching it. In this paper we show that sometimes existence of these solutions can provide chaos.

SINGULARITY, SWITCHABILITY AND BIFURCATIONS IN A 2-DOF, PERIODICALLY FORCED, FRICTIONAL OSCILLATOR

In this paper, the analytical dynamics for singularity, switchability, and bifurcations of a 2-DOF friction-induced oscillator is investigated. The analytical conditions of the domain flow switchability at the boundaries and edges are developed from the theory of discontinuous dynamical systems, and the switchability conditions of boundary flows from domain and edge flows are presented. From the singularity and switchability of flow to the boundary, grazing, sliding and edge bifurcations are obtained. For a better understanding of the motion complexity of such a frictional oscillator, switching sets and mappings are introduced, and mapping structures for periodic motions are adopted. Using an eigenvalue analysis, the stability and bifurcation analysis of periodic motions in the friction-induced system is carried out. Analytical predictions and parameter maps of periodic motions are performed. Illustrations of periodic motions and the analytical conditions are completed. The analytical conditions and methodology can be applied to the multi-degrees-of-freedom frictional oscillators in the same fashion.

Stochastic Regular Grazing Bifurcations

A grazing bifurcation corresponds to the collision of a periodic orbit with a switching manifold in a piecewise-smooth ODE system and often generates complicated dynamics. The lowest order terms of the induced Poincare map expanded about a regular grazing bifurcation constitute a Nordmark map. In this paper we study a normal form of the Nordmark map in two dimensions with additive Gaussian noise of amplitude $\varepsilon$. We show that this particular noise formulation arises in a general setting and consider a harmonically forced linear oscillator subject to compliant impacts to illustrate the accuracy of the map. Numerically computed invariant densities of the stochastic Nordmark map can take highly irregular forms or, if there exists an attracting period-$n$ solution when $\varepsilon=0$, be well approximated by the sum of $n$ Gaussian densities centered about each point of the deterministic solution, and scaled by $\frac{1}{n}$, for sufficiently small $\varepsilon>0$. We explain the irregular forms an...

Weak Allee effect, grazing, and S-shaped bifurcation curves

We study a one-dimensional reaction-diffusion model arising in population dynamics where the growth rate is a weak Allee type. In particular, we consider the effects of grazing on the steady states and discuss the complete evolution of the bifurcation curve of positive solutions as the grazing parameter varies. We obtain our results via the quadrature method and Mathematica computations. We establish that the bifurcation curve is S-shaped for certain ranges of the grazing parameter. We also prove this occurrence of an S-shaped bifurcation curve analytically.

Period-three route to chaos induced by a cyclic-fold bifurcation in passive dynamic walking of a compass-gait biped robot

This paper presents a study of the passive dynamic walking of a compass-gait biped robot as it goes down an inclined plane. This biped robot is a two-degrees-of-freedom mechanical system modeled by an impulsive hybrid nonlinear dynamics with unilateral constraints. It is well-known to possess periodic as well as chaotic gaits and to possess only one stable gait for a given set of parameters. The main contribution of this paper is the finding of a window in the parameters space of the compass-gait model where there is multistability. Using constraints of a grazing bifurcation on the basis of a shooting method and the Davidchack–Lai scheme, we show that, depending on initial conditions, new passive walking patterns can be observed besides those already known. Through bifurcation diagrams and Floquet multipliers, we show that a pair of stable and unstable period-three gait patterns is generated through a cyclic-fold bifurcation. We show also that the stable period-three orbit generates a route to chaos.

Investigation of the near-grazing behavior in hard-impact oscillators using model-based TS fuzzy approach

An impact oscillator is a non-smooth dy- namical system with discontinuous state jumps whose dynamical behavior illustrates a variety of non-linear phenomena including a grazing bifurcation. This spe- cific phenomenon is difficult to analyze because it co- incides with an infinite stretching of the phase space in the neighborhood of the grazing orbit, resulting in the well-known problem of the square-root singular- ity of the Jacobian of the discrete-time map. A novel Takagi-Sugeno fuzzy model-based approach is pre- sented in this paper to model a hard impacting system as a non-smooth dynamical system including discon- tinuous jumps. Employing non-smooth Lyapunov the- ory, the structural stability of the system is analyzed to predict the onset of the destabilizing chaotic be- havior. The proposed stability results, formulated as a Linear Matrix Inequality (LMI) problem, demonstrate how the new method can detect the loss of stability just before the grazing bifurcation.

Noise-induced Phenomena in Two Strongly Pulse-coupled Spiking Neurons

In this study, we address noise-induced phenomena in a system of two pulse-coupled resonate-and-fire neuron models as a piecewise smooth dynamical system. The model is a spiking neuron model that has second-order membrane dynamics with a firing threshold, and it exhibits subthreshold oscillation of membrane potential, resulting in the sensitivity to the timing of inputs. The pulse-coupled system shows the grazing bifurcation, a specific feature of a piecewise smooth dynamical system, as well as the saddle-node bifurcation. We numerically investigated mixture bifurcation structure of the system, leading to complex behaviors, such as burst synchronization and chaotic phenomena. Moreover, we found that a novel type of noise-induced order and chaos occur in the system when additive noise are given as a perturbation on the firing threshold. Finally, we will discuss a considerable mechanism of such noise-induced phenomena and the relationship to discontinuity-induced bifurcation in view of the piecewise smooth dynamical system theory.

544 歯車を模擬した衝突振動系に生じる動的挙動

Gear system is one of impact vibrating systems in the field of mechanical engineering. Generally, various dynamical behaviors or bifurcations are arised on the impact vibrating system. These behaviors and bifurcations produce the noise or abrasion on gear system. The two-mass impact model is proposed as the simplest model of gear system. However, there is little experimental study for it. In this paper, we make an experimental device of the two-mass impact model and study the dynamical behaviors or bifurcations arised on it. On the other hand, we calculate equations of motion of this model and compare the numerical results and our experimental results. As a result, it is found that the grazing bifurcation arised on this system when the gap between masses is varied.

Topology of vibro-impact systems in the neighborhood of grazing

The grazing bifurcation is considered for the Newtonian model of vibro-impact systems. A brief review on the conditions, sufficient for the existence of a grazing family of periodic solutions, is given. The properties of these periodic solutions are discussed. A plenty of results on the topological structure of attractors of vibro-impact systems is known. However, since the considered system is strongly nonlinear, these attractors must be very sensitive to changes of parameters of the system. On the other hand, they are observed in experiments and numerical simulations. We offer (Theorem 2) an approach which allows to explain this contradiction and give a new robust mathematical model of the non-hyperbolic dynamics in a neighborhood of grazing.

Numerical Computation of Grazing Bifurcation for Periodic Orbit

In this paper, a numerical algorithm for computing grazing bifurcation of periodic orbit in autonomous planar systems is derived and analyzed. We propose an analytical condition which to ensure the orbit quadratically intersects the switching manifold. Based upon this, we take into a new phase condition which ensure there is a branch of periodic orbits. The nondegenerate condition with respect to its bifurcation parameter is presented to ensure the defining equation well posed. A numerical simulation is carried out for the bifurcation diagram of the van der pol system, which agrees with analytical results very well.

Nonsmooth dynamics in spiking neuron models

Large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue. One approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models. An alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies. In this latter case the single neuron model of choice is often a variant of the classic integrate-and-fire model, which is described by a nonsmooth dynamical system. In this paper we review some of the more popular spiking models of this class and describe the types of spiking pattern that they can generate (ranging from tonic to burst firing). We show that a number of techniques originally developed for the study of impact oscillators are directly relevant to their analysis, particularly those for treating grazing bifurcations. Importantly we highlight one particular single neuron model, capable of generating realistic spike trains, that is both computationally cheap and analytically tractable. This is a planar nonlinear integrate-and-fire model with a piecewise linear vector field and a state dependent reset upon spiking. We call this the PWL-IF model and analyse it at both the single neuron and network level. The techniques and terminology of nonsmooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model, as well as to develop the notion of Lyapunov exponents. We also show how to construct the phase response curve for this system, emphasising that techniques in mathematical neuroscience may also translate back to the field of nonsmooth dynamical systems. The stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times. At the network level we consider linear coupling between voltage variables, as would occur in neurobiological networks with gap-junction coupling, and show how to analyse the properties (existence and stability) of both the asynchronous and synchronous states. In the former case we use a phase-density technique that is valid for any large system of globally coupled limit cycle oscillators, whilst in the latter we develop a novel technique that can handle the nonsmooth reset of the model upon spiking. Finally we discuss other aspects of neuroscience modelling that may benefit from further translation of ideas from the growing body of knowledge on nonsmooth dynamics.

Interactions between global and grazing bifurcations in an impacting system

It is well known that the locus of boundary crises in smooth systems contains gaps that give rise to periodic windows. We show that this phenomenon can also be observed in an impacting system, and that the mechanism by which these gaps are created is different. Namely, here gaps are created and disappear at points along the branches of boundary crises where they are intersected by branches of grazing bifurcations. We locate a novel type of double-crisis vertex which we call a grazing-crisis vertex. Additionally, we illustrate several types of basin-boundary metamorphosis that are intricately related with grazing bifurcations.

An Extended Continuation Problem for Bifurcation Analysis in the Presence of Constraints

This paper presents an extended formulation of the basic continuation problem for implicitly defined, embedded manifolds in Rn. The formulation is chosen so as to allow for the arbitrary imposition of additional constraints during continuation and the restriction to selective parametrizations of the corresponding higher-codimension solution manifolds. In particular, the formalism is demonstrated to clearly separate between the essential functionality required of core routines in application-oriented continuation packages, on the one hand, and the functionality provided by auxiliary toolboxes that encode classes of continuation problems and user definitions that narrowly focus on a particular problem implementation, on the other hand. Several examples are chosen to illustrate the formalism and its implementation in the recently developed continuation core package COCO and auxiliary toolboxes, including the continuation of families of periodic orbits in a hybrid dynamical system with impacts and friction as well as the detection and constrained continuation of selected degeneracies characteristic of such systems, such as grazing and switching-sliding bifurcations.

Existence and stability of the grazing periodic trajectory in a two-degree-of-freedom vibro-impact system

This paper investigates the existence and stability of the grazing periodic trajectory in a two-degree-of-freedom vibro-impact system. The criterion for existence of grazing period-n motion is presented. A local analysis based on the discontinuity-mapping approach is employed to derive a normal form Poincaré mapping near the grazing trajectory. Based on the above approach, a condition of stability can be formulated, such that a grazing trajectory will be discontinuous if the condition is unfulfilled. The predicted grazing bifurcations are in agreement with the numerical results. In particular, comparison of the grazing bifurcation diagrams of the normal form Poincaré mapping and the simulation diagrams of the original differential equation illustrates the validity of the discontinuity-mapping approach.

On the global dynamics of chatter in the orthogonal cuttingmodel

The large-amplitude motions of a one degree-of-freedom model of orthogonal cutting are analysed. The model takes the form of a delay differential equation which is non-smooth at the instant at which the tool loses contact with the workpiece, and which is coupled to an algebraic equation that stores the profile of the cut surface whilst the tool is not in contact. This system is approximated by a smooth delay differential equation without algebraic effects which is analysed with numerical continuation software. The grazing bifurcation that defines the onset of chattering motion is thus analysed as are secondary (period-doubling, etc.) bifurcations of chattering orbits, and convergence of the bifurcation diagrams is established in the vanishing limit of the smoothing parameters. The bifurcation diagrams of the smoothed system are then compared with initial value simulations of the full non-smooth delay differential algebraic equation. These simulations mostly validate the smoothing technique and show in detail how chaotic chattering dynamics emerge from the non-smooth bifurcations of periodic orbits.