Exact Derivation of Global Form of All Periodic Solutions for Zero Stiffness Impact Oscillator using Initial Value Correction Periodize Method (Case of Viscous Damped System)

We study the dynamics of a two-degree-of-freedom nonlinear system consisting of a linear oscillator with an essentially nonlinear attachment. For the undamped system, we perform a numerical study based on non-smooth temporal transformations to determine its periodic solutions in a frequency-energy plot. It turns out that there is a sequence of periodic solutions bifurcating from the main backbone curve of the plot. We then study analytically the periodic orbits of the undamped system using the complexification / averaging technique in order to determine the frequency contents of the various branches of solutions, and to understand the types of oscillation performed by the system at the different regimes of the motion. The transient responses of the weakly damped system are then examined, and numerical wavelet transforms are used to study the time evolutions of their harmonic components. We show that the structure of periodic orbits of the undamped system greatly influences the damped dynamics, as it causes complicated transitions between modes in the damped transient motion. In addition, there is the possibility of strong passive energy transfer from the linear oscillator to the nonlinear attachment if certain periodic orbits of the undamped dynamics are excited by the initial conditions.

Download Full-text

Exact Determination of Anti-Symmetric Periodic Solutions for Impact Oscillator

The Proceedings of Ibaraki District Conference ◽

10.1299/jsmeibaraki.2002.37 ◽

2002 ◽

Vol 2002 (0) ◽

pp. 37-38

Author(s):

Hitoshi IMAMURA

Keyword(s):

Periodic Solutions ◽

Impact Oscillator ◽

Exact Determination

Download Full-text

A case of boundedness in Littlewood's problem on oscillatory differential equations

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700024862 ◽

1976 ◽

Vol 14 (1) ◽

pp. 71-93 ◽

Cited By ~ 95

Author(s):

G.R. Morris

Keyword(s):

Positive Integer ◽

Small Parameter ◽

Periodic Solutions ◽

Almost Periodic Solutions ◽

Almost Periodic ◽

Initial Value ◽

Closed Curves ◽

Essential Step ◽

Period 2 ◽

All Solutions

It is shown that all solutions of ẍ + 2x3 = p(t) are bounded, the notation indicating that p is periodic. It is not necessary to have a small parameter multiplying p.The essential step is to show by appeal to Moser's theorem that, under the mapping (of the initial-value plane) which corresponds to the equation, there are invariant simple closed curves. This implies also that there is an uncountable infinity of almost-periodic solutions and, for each positive integer m, an infinity of periodic solutions of least period 2mπ (2π being taken as the least period of p ).It is suggested that for a large class of equations the same attack would show all solutions of ẍ + g(x) = p(t) bounded. However, in order to show the method clearly, no generalisation is attempted here.

Download Full-text

On periodic solutions to a class of delay differential variational inequalities

Evolution Equations and Control Theory ◽

10.3934/eect.2021045 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Nguyen Thi Van Anh

Keyword(s):

Variational Inequalities ◽

Periodic Solutions ◽

Delay Differential Equations ◽

Conditional Stability ◽

Sufficient Condition ◽

Initial Value ◽

Periodic Problems ◽

Existence Of Periodic Solutions ◽

Delay Differential ◽

Differential Variational Inequalities

<p style='text-indent:20px;'>In this paper, we introduce and study a class of delay differential variational inequalities comprising delay differential equations and variational inequalities. We establish a sufficient condition for the existence of periodic solutions to delay differential variational inequalities. Based on some fixed point arguments, in both single-valued and multivalued cases, the solvability of initial value and periodic problems are proved. Furthermore, we study the conditional stability of periodic solutions to this systems.</p>

Download Full-text

A MONOTONE-ITERATIVE METHOD FOR FINDING PERIODIC SOLUTION OF AN IMPULSIVE DIFFERENTIAL EQUATIONS AND APPLICATION

International Journal of Biomathematics ◽

10.1142/s1793524508000205 ◽

2008 ◽

Vol 01 (02) ◽

pp. 247-256

Author(s):

JIAWEI DOU

Keyword(s):

Differential Equations ◽

Periodic Solutions ◽

Upper And Lower Solutions ◽

Impulsive Differential Equations ◽

Iteration Process ◽

Impulsive System ◽

Monotone Iterative Method ◽

Initial Value ◽

Monotone Iterative ◽

Monotone Iterations

In this paper, using the method of upper and lower solutions and its associated monotone iterations, we establish a new monotone-iterative scheme for finding periodic solutions of an impulsive differential equations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for impulsive differential equations initial value problem. This method is constructive and can be used to develop a computational algorithm for numerical solution of the periodic impulsive system. Our existence result improves a result established in [1]. The result is applied to a model of mutualism of Lotka–Volterra type which involves interactions among a mutualist-competitor, a competitor and a mutualist, the existence of positive periodic solutions of the model is obtained.

Download Full-text