Stick motions and grazing flows in an inclined impact oscillator

In this paper, the dynamics of an inclined impact oscillator under periodic excitation are investigated using the flow switchability theory of the discontinuous dynamical systems. Different domains and boundaries for such system are defined according to the impact discontinuity. Based on above domains and boundaries, the analytical conditions of the stick motions and grazing motions for the inclined impact oscillator are obtained mathematically, from which it can be seen that such oscillator has more complicated and rich dynamical behaviors. The numerical simulations are given to illustrate the analytical results of complex motions, and several period-1 motions period-2 motion and chaotic motion of the ball in the inclined impact oscillator are also presented. There are more theories about such impact pair to be discussed in future.     

Controlling bifurcation and chaos of a plastic impact oscillator

A two-degree-of-freedom plastic impact oscillator is considered. Based on the analysis of sticking and non-sticking impact motions of the system, we introduce a three-dimensional impact Poincaré map with dynamical variables defined at the impact instants. The plastic impacts complicate the dynamic responses of the impact oscillator considerably. Consequently, the piecewise property and singularity are found to exist in the three-dimensional map. The piecewise property is caused by the transitions of free flight and sticking motions of two masses immediately after impact, and the singularity of the map is generated via the grazing contact of two masses and the instability of their corresponding periodic motions. The nonlinear dynamics of the plastic impact oscillator is analyzed by using the Poincaré map. The simulated results show that the dynamic behavior of this system is very complex under parameter variation, varying from different types of sticking or non-sticking periodic motions to chaos. Suppressing bifurcation and chaotic-impact motions is studied by using an external driving force, delay feedback and damping control law. The effectiveness of these methods is demonstrated by the presentation of examples of suppressing bifurcations and chaos for the plastic impact oscillator.     

The period function of a delay differential equation and an application

We consider the delay differential equation [(x)\dot](t) = - mx(t) + f(x(t - t))\dot x(t) = - \mu x(t) + f(x(t - \tau )), where μ, τ are positive parameters and f is a strictly monotone, nonlinear C ¹-function satisfying f(0) = 0 and some convexity properties. It is well known that for prescribed oscillation frequencies (characterized by the values of a discrete Lyapunov functional) there exists τ* > 0 such that for every τ > τ* there is a unique periodic solution. The period function is the minimal period of the unique periodic solution as a function of τ > τ*. First we show that it is a monotone nondecreasing Lipschitz continuous function of τ with Lipschitz constant 2. As an application of our theorem we give a new proof of some recent results of Yi, Chen and Wu [14] about uniqueness and existence of periodic solutions of a system of delay differential equations.     

Global analysis of boundary and interior crises in an elastic impact oscillator

The crisis phenomena of a Duffing–Van der Pol oscillator with a one-side elastic constraint are studied by the composite cell coordinate system method in this paper. By computing the global properties such as attractors, basins of attraction and saddles, the vivid evolutionary process of two kinds of crises: boundary crisis and interior crisis are shown. The boundary crisis is resulted by the collision of a chaotic attractor and a periodic saddle on the basin boundary. It is observed that there are two types of interior crises. One is caused by the collision of a chaotic attractor and a chaotic saddle within the interior of basin of attraction. The other one occurs because a period attractor collides with a chaotic saddle within the interior of basin of attraction. The saddles of system play an important role in the crisis process. The results show that this method is an efficient tool to perform the global analysis of elastic impact oscillators.     

Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function

A simple neural network model with discrete time delay is investigated. The linear stability of this model is discussed by analyzing the associated characteristic transcendental equation. For the case with inhibitory influence from the past state, it is found that Hopf bifurcation occurs when this influence varies and passes through a sequence of critical values. The stability of bifurcating periodic solutions and the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Chaotic behavior of a single delayed neuron equation with non-monotonously increasing transfer function has been observed in computer simulation. Some waveform diagrams, phase portraits, power spectra and plots of the largest Lyapunov exponent will also be given.     

Experimental investigation of an oscillator with discontinuous support considering different system aspects

Non-smooth characteristics are, in general, the source of difficulties for the modeling and simulation of natural systems. These characteristics are usually related to either the friction phenomenon or the discontinuous behavior as intermittent contacts. This article develops an experimental investigation concerning non-smooth systems with discontinuous support. An experimental apparatus is developed in order to analyze the nonlinear dynamics of a single-degree of freedom system with discontinuous support. The apparatus is composed by an oscillator constructed by a car, free to move over a rail, connected to an excitation system. The discontinuous support is constructed considering mass–spring systems separated by a gap to the car position. This apparatus is instrumented to obtain all the system state variables. System dynamical behavior shows a rich response, presenting dynamical jumps, bifurcations and chaos. Different configurations of the experimental set up are treated in order to evaluate the influence of the internal impact within the car and also support characteristics in the system dynamics.     

Stochastic auto-oscillations of a nonlinear oscillator with impact energy absorber

A nonlinear Van der Pol oscillator with impact energy absorber is considered. Dependence of all possible forms of the system phase pattern on parameters is determined. Conditions of steady stochastic motion onset are defined. The parameter space is divided in regions that qualitatively correspond to different phase patterns of the system.     

On the period function for a family of complex differential equations

We consider planar differential equations of the form being f(z) and g(z) holomorphic functions and prove that if g(z) is not constant then for any continuum of period orbits the period function has at most one isolated critical period, which is a minimum. Among other implications, the paper extends a well-known result for meromorphic equations, that says that any continuum of periodic orbits has a constant period function.     

First-passage failure of linear oscillator with non-classical inelastic impact

The first-passage failure of linear oscillator with inelastic impact subjected to the additive and multiplicative random noises is investigated. The impact is described by the non-classical inelastic impact model, which is essentially different from the traditional impact model and can provide the whole information of the impact process. First of all, the impact force in the motion equation is replaced by the quasi-linear damping and nonlinear stiffness terms. Then, the stochastic averaging is adopted and the averaged Itô stochastic deferential equation of the total system energy is derived. Last, by solving the established backward Kolmogorov equation and Pontryagin equation from the averaged Itô equation numerically, the conditional reliability, the conditional probability density function (PDF) and the mean time of first-passage failure can be obtained. The comparison between the analytical results and those from Monte-Carlo simulation reveals the proposed procedure is effective. The influences of some system parameters are discussed in detail.     

Dynamics of a delayed epidemic model with non-monotonic incidence rate

A delayed epidemic model with non-monotonic incidence rate which describes the psychological effect of certain serious on the community when the number of infectives is getting larger is studied. The disease-free equilibrium is globally asymptotically stable when R₀<1 and is globally attractive when R₀=1 are derived. On the other hand, The disease is permanent when R₀>1 is also obtained. Numerical simulation results are given to support the theoretical predictions.     

Bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action

In this paper, we study the bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action, which describes the psychological effects of the community on certain serious diseases when the number of infective is getting larger. By carrying out the bifurcation analysis of the model, we show that there exist some values of the model parameters such that numerous kinds of bifurcation occur for the model, such as Hopf bifurcation, Bogdanov–Takens bifurcation.     

Numerical investigation of a developmental pneumatically fed impact pulveriser

The current work has been carried out on a pneumatically fed impact pulveriser. A recent build of the proposed device, and early experiments have shown promising size reduction ratios and energy savings in comparison with conventional milling techniques.     

Stability for a damped nonlinear oscillator

The stability of the null solution of Eq. (1.1) below is discussed. Under some unusual assumptions, we obtain new stability results for this classical oscillator equation. Our approach allows extensions to both the vector case and the case of the whole real line.     

Explanation of appearance and characteristics of intermittency chaos of the impact oscillator

Chaotic motion of an intermittency type of the impact oscillator appears near segments of saddle-node stability boundaries of subharmonic motions with two different impacts in motion period, which is n multiple (n3) of excitation period. Chaotic motion arises due to an additional impact, which interrupts the process of instability. It is proved and shown by numerical simulations of the system motion. More detail characteristics of the intermittency chaos are evaluated. Described phenomena present a non-usual example, when transition cross special segments of saddle-node stability boundaries of subharmonic impact motions is reversible.