scholarly journals  Van der Pol Model in Two-Delay Differential Equation Representation

2021 ◽  
Author(s):  
M. A. Elfouly ◽  
M. A. Sohaly
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Delay Model ◽  
Cubic Nonlinearity ◽  
Van Der Pol ◽  
Order Ordinary Differential Equation ◽  
Two Delays ◽  
Van Der Pol Model ◽  
Delay Differential ◽  
The Mathematical Model

Abstract The Van der Pol equation is the mathematical model of a second-order ordinary differential equation with cubic nonlinearity. In this paper, the differential equation of the Van der Pol model and RLC (resistor - inductor-capacitor) circuit are deduced from a delay differential equation. The Van der Pol delay model contains two delays, which opens the way for the re-use of its applications. Also, the model for Parkinson's disease modification is described as the Van der Pol model.

Fold Bifurcation in the State-Dependent Delay Model of Milling: Analytical and Numerical Solutions

2011 ◽  
Author(s):  
Daniel Bachrathy ◽  
Gabor Stepan
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Delay Differential Equation ◽  
Milling Process ◽  
The State ◽  
Delay Model ◽  
State Dependent ◽  
Fold Bifurcation ◽  
State Dependent Delay ◽  
Delay Differential

The standard models of the milling process describe the surface regeneration effect by a delay-differential equation with constant time delay. In this study, an improved two degree of freedom model is presented for milling process where the regenerative effect is described by an improved state dependent time delay model. The model contains exact nonlinear screen functions describing the entrance and exit positions of the cutting edges of the milling tool. This model is valid in case of large amplitude forced vibrations close to the near-resonant spindle speeds. The periodic motions of this nonlinear system are calculated by a shooting method. The stability calculation is based on the linearization of the state-dependent delay differential equation around these periodic solutions by means of the semi-discretization method. The results are validated by an advanced numerical time domain simulation where the chip thickness is calculated by means of Boolean algebra.

Complex dynamics in a delay differential equation with two delays in tick growth with diapause

2020 ◽  
Vol 269 (12) ◽  
pp. 10937-10963 ◽  
Author(s):  
Hongying Shu ◽  
Wanxiao Xu ◽  
Xiang-Sheng Wang ◽  
Jianhong Wu
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Complex Dynamics ◽  
Two Delays ◽  
Delay Differential

scholarly journals Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 105
Author(s):  
Lokesh Singh ◽  
Dhirendra Bahuguna
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Invariant Manifold ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stable Manifold ◽  
Exponential Dichotomy ◽  
Delay Differential ◽  
Stable Invariant ◽  
Nonuniform Exponential Dichotomy

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

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