Oscillations in Differential Equations with State-Dependent Delays

We study the existence of oscillating solutions of delay differential equations with delay depending directly on a state. Necessary and sufficient conditions for oscillations are established.     

Slowly oscillating solutions in a class of second-order discontinuous delayed systems

Nonlinear Dynamics - This paper investigates the dynamics of oscillations for a class of second-order discontinuous differential equations with time delay. First, by analyzing an implicit function...     

Z n equivariant in delay coupled dissipative Stuart–Landau oscillators

We consider a coupled dissipative Stuart?CLandau oscillator models. As the propagation time delay in the coupling varies, stability switches for the trivial solution are found. We discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. The existence of multiple branches of bifurcating periodic solution is obtained. We also found that the spatio-temporal patterns of bifurcating periodic oscillations alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of dissipative Stuart?CLandau oscillators. Some numerical simulations support our analysis results.     

Estimating Critical Hopf Bifurcation Parameters for a Second-Order Delay Differential Equation with Application to Machine Tool Chatter

Nonlinear time delay differential equations are well known to havearisen in models in physiology, biology and population dynamics. Theyhave also arisen in models of metal cutting processes. Machine toolchatter, from a process called regenerative chatter, has been identifiedas self-sustained oscillations for nonlinear delay differentialequations. The actual chatter occurs when the machine tool shifts from astable fixed point to a limit cycle and has been identified as arealized Hopf bifurcation. This paper demonstrates first that a class ofnonlinear delay differential equations used to model regenerativechatter satisfies the Hopf conditions. It then gives a precisecharacterization of the critical eigenvalues on the stability boundaryand continues with a complete development of the Hopf parameter, theperiod of the bifurcating solution and associated Floquet exponents.Several cases are simulated in order to show the Hopf bifurcationoccurring at the stability boundary. A discussion of a method ofintegrating delay differential equations is also given.     

Hopf bifurcations through delay in pilot reaction in a longitudinal flight

The paper is devoted to the study of pilot induced oscillations in the landing transition between the approach task and flare to touch-down. These oscillations are proved to appear in a longitudinal flight model when the delay in pilot’s reactions exceeds a certain threshold for which the stability of equilibria is lost and a Hopf bifurcation appears. The formulae needed to compute the Lyapunov coefficient and an approximation of the solution are developed for the delay differential equations that model the pilot–vehicle interaction in landing task. These are applied for a concrete model.     

Rich dynamics in a non-local population model over three patches

We consider a system of nonlinear delay differential equations that describes the growth of the mature population of a species with age-structure living over three patches. We analyze existence of non-negative homogeneous equilibria and their stability and discuss possible Hopf bifurcation from these equilibria. More precisely, by employing both the standard Hopf bifurcation theory and the symmetric bifurcation theory for functional differential equations, we obtain very rich dynamics for the system, including bistable equilibria, transient oscillations, synchronous periodic solutions, phase-locked periodic solutions, mirror-reflecting waves and standing waves.     

Hopf bifurcation and spatio-temporal patterns in?delay-coupled van der Pol oscillators

In this paper, the dynamics of a pair of van der Pol oscillators with delayed velocity coupling is studied by taking the time delay as a bifurcation parameter. We first investigate the stability of the zero equilibrium and the existence of Hopf bifurcations induced by delay, and then study the direction and stability of the Hopf bifurcations. Then by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups, we investigate the spatio-temporal patterns of Hopf bifurcating periodic oscillations. We find that there are different in-phase and anti-phase patterns as the coupling time delay is increased. The analytical theory is supported by numerical simulations, which show good agreement with the theory.     

Finite amplitude oscillations of a hyperelastic spherical cavity

We present numerical results for the finite oscillations of a hyperelastic spherical cavity by employing the governing equations for finite amplitude oscillations of hyperelastic spherical shells and simplifying it for a spherical cavity in an infinite medium and then applying a fourth-order Runge-Kutta numerical technique to the resulting non-linear first-order differential equation.The results are plotted for Mooney-Rivlin type materials for free and forced oscillations under Heaviside type step loading. The results for Neo-Hookean materials are also discussed. Dependence of the amplitudes and frequencies of oscillations on different parameters of the problem is also discussed in length.     

Regularization of Neutral Delay Differential Equations with Several Delays

For neutral delay differential equations the right-hand side can be multi-valued, when one or several delayed arguments cross a breaking point. This article studies a regularization via a singularly perturbed problem, which smooths the vector field and removes the discontinuities in the derivative of the solution. A low-dimensional dynamical system is presented, which characterizes the kind of generalized solution that is approximated. For the case that the solution of the regularized problem has high frequency oscillations around a codimension-2 weak solution of the original problem, a new stabilizing regularization is proposed and analyzed.     

Delay equations with fluctuating delay related to the regenerative chatter

The mathematical models representing machine tool chatter dynamics have been cast as differential equations with delay. In this paper, non-linear delay differential equations with periodic delays which model the machine tool chatter with continuously modulated spindle speed are studied. The explicit time-dependent delay terms, due to spindle speed modulation, are replaced by state-dependent delay terms by augmenting the original equations. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. The reduced bifurcation equation is obtained by making use of Lyapunov-Schmidt Reduction method. By using the reduced bifurcation equations, the periodic solutions are determined to analyze the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions near the new stability boundary.     

Topological Dynamics for Monotone Skew-Product Semiflows with Applications

Classical techniques of topological dynamics are used to prove a flow extension result for linearly stable minimal sets in monotone and differentiable skew-product semiflows. Moreover, motivated by the field of delay equations, a new version of the concept of continuous separation is introduced and studied in an abstract setting. The application of these results to the skew-product semiflows defined by almost periodic ordinary differential equations, delay equations and parabolic partial differential equations permits us to extend previous results guaranteeing the presence of almost automorphic minimal sets.     

基于积分本构黏弹性带的横向非线性动力学

研究了黏弹性传动带在1:1内共振时的横向非平面非线性动力学特性. 首先,利用Hamilton原理建立了黏弹性传动带横向非平面非线性动力学方程. 然后综合应用多尺度法和Galerkin离散法对偏微分形式的动力学方程进行摄动分析,得到了四维平均方程. 对平均方程的稳定性进行了分析,从理论上讨论了动力系统解的稳定性变化情况. 最后数值模拟结果表明黏弹性传动带系统存在混沌运动、概周期运动和周期运动.      

Analysis of a class of undesired multimode oscillations

The behavior of a negative-resistance circuit in which the parameters permit self-sustained oscillations to occur is discussed. In such a circuit, undesired oscillations of higher frequency caused by parasitic elements can exist in addition to normal oscillations. Parasitic oscillations are described by second-order simultaneous non-linear differential equations, taking into account of existence of the shunt capacitance, lead inductance and resistance of a negative-resistance element. Assuming that the frequency of the parasitic oscillations is sufficiently high compared with that of the desired normal oscillations, the approximate periodic solutions are obtained by using a method of averaging. In addition, the theoretical results are compared with the observed behavior of an experimental oscillator having similar parameters.     

A Hilbert Space Perspective on Ordinary Differential Equations with Memory Term

We discuss ordinary differential equations with delay and memory terms in Hilbert spaces. By introducing a time derivative as a normal operator in an appropriate Hilbert space, we develop a new approach to a solution theory covering integro-differential equations, neutral differential equations and general delay differential equations within a unified framework. We show that reasonable differential equations lead to causal solution operators.     

Oscillation of Higher-Order Delay Differential Equation with Applications to Hyperbolic Equations

In this paper, we consider an odd-order delay differential equation with positive and negative coefficients. New sufficient conditions that guarantee the oscillation of all solutions are presented. Our results extend and improve some known results. Next, these results are used for establishing oscillation criteria for hyperbolic delay differential equations with positive and negative coefficients corresponding to three sets of boundary conditions.     

Robust stability of uncertain neutral linear stochastic differential delay system

The LaSalle-type theorem for the neutral stochastic differential equations with delay is established for the first time and then applied to propose algebraic criteria of the stochastically asymptotic stability and almost exponential stability for the uncertain neutral stochastic differential systems with delay. An example is given to verify the effectiveness of obtained results.     

The impact of delayed feedback on the pulsating oscillations of class-B lasers

Class-B laser systems can be described by a set of slow-fast differential equations with a small parameter, and they exhibit pulsating oscillations in common. In this paper, the impact of the delayed feedback on the pulsating solutions is investigated. At first, a careful analysis of the local stability and bifurcation shows the existence of a series of Hopf bifurcations and double Hopf bifurcations when the strength of the delayed feedback or the delay increases, by means of stability switches. Then, an application of the geometric singular perturbation theory reveals the evolutional mechanism of the pulsating solutions from transients to stationary states, and the effect of the delayed feedback upon the pulsating solutions is examined.     

Attractors of a rotating viscoelastic beam

We investigate the non-linear oscillations of a rotating viscoelastic beam with variable pitch angle. The governing equations of motion are two coupled partial differential equations for the longitudinal and transversal displacements. Using a perturbation technique and Galerkin's projection, we reduce the equations of motion to a non-autonomous ordinary differential equation. Our regular perturbation technique is based on the expansion of longitudinal displacement and the amplitude of first transversal mode in terms of a small parameter. We numerically generate the Poincaré maps of the reduced equations and reveal that the system exhibits regular and chaotic attractors. The regular attractors are stable limit-cycles that are relevant to stable, short-period oscillations of the beam. A bifurcation analysis has also been performed when the pitch angle is constant.     

Wide Oscillation Finite Time Blow Up for Solutions to Nonlinear Fourth Order Differential Equations

We give sufficient conditions for local solutions to some fourth order semilinear ordinary differential equations to blow up in finite time with wide oscillations, a phenomenon not visible for lower order equations. The result is then applied to several classes of semilinear partial differential equations in order to characterize the blow up of solutions including, in particular, its applications to a suspension bridge model. We also give numerical results which describe this oscillating blow up and allow us to suggest several open problems and to formulate some related conjectures.     

Generalized equivalence transformations and group classification of systems of differential equations

The notion of generalized equivalence transformations for which the equivalence transformations considered by Ovsyannikov are universal equivalence transformations is introduced for a system of differential equations. An algorithm of the group classification of the system of differential equations with the help of these generalized equivalence transformations is proposed. The efficiency and advantages of this algorithm are demonstrated by examples of equations of gas dynamics and equations of nonlinear longitudinal oscillations of a viscoelastic bar in the Kelvin model.