scholarly journals Criticality of Hopf bifurcation in state-dependent delay model of turning processes

2008 ◽  
Vol 43 (2) ◽  
pp. 140-149 ◽  
Author(s):  
Tamás Insperger ◽  
David A.W. Barton ◽  
Gábor Stépán
Keyword(s):  
Hopf Bifurcation ◽  
Delay Model ◽  
State Dependent ◽  
State Dependent Delay

Periodic Solutions for a Class of Functional Differential Equations with State-Dependent Delay Close to Zero

2003 ◽  
Vol 13 (06) ◽  
pp. 807-841 ◽  
Author(s):  
R. Ouifki ◽  
M. L. Hbid
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Delay Equation ◽  
Constant Delay ◽  
State Dependent ◽  
State Dependent Delay ◽  
Existence Of Periodic Solutions ◽  
Functional Differential

The purpose of the paper is to prove the existence of periodic solutions for a functional differential equation with state-dependent delay, of the type [Formula: see text] Transforming this equation into a perturbed constant delay equation and using the Hopf bifurcation result and the Poincaré procedure for this last equation, we prove the existence of a branch of periodic solutions for the state-dependent delay equation, bifurcating from r ≡ 0.

Harvesting on a State-Dependent Time Delay Model

2020 ◽  
Vol 8 (1) ◽  
pp. 82-96
Author(s):  
Ruijun Xie ◽  
Xin Zhang ◽  
Wei Zhang
Keyword(s):  
Time Delay ◽  
Population Density ◽  
Positive Equilibrium ◽  
Delay Model ◽  
Stability Criteria ◽  
State Dependent ◽  
State Dependent Delay ◽  
Cooperation Model ◽  
Increasing Function ◽  
Positivity And Boundedness

AbstractIn this paper, we propose and analyze a cooperation model with harvesting and state-dependent delay, which is assumed to be an increasing function of the population density with lower and upper bound. The main purpose of this article is to obtain the dynamics of our model analytically by controlling the harvesting. We present results on positivity and boundedness of all populations. Criteria for the existence of all equilibria and uniqueness of a positive equilibrium are given by controlling the harvesting. Finally, the global exponentially asymptotical stability criteria of model is obtained by the improved Hanalay inequality.

scholarly journals Singular Hopf bifurcation in a differential equation with large state-dependent delay

2014 ◽  
Vol 470 (2162) ◽  
pp. 20130596 ◽  
Author(s):  
G. Kozyreff ◽  
T. Erneux
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Gradual Change ◽  
Classical State ◽  
Sustained Oscillations ◽  
State Dependent ◽  
Analytical Construction ◽  
State Dependent Delay ◽  
Delay Differential ◽  
Sawtooth Oscillations

We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations. The dependence of the delay on the solution can be either linear or nonlinear, with at least quadratic dependence. In the former case, an asymptotic connection is made with the Rayleigh oscillator. In the latter, van der Pol’s equation is derived for the small-amplitude oscillations. SDD differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant-delay differential equation, but explicit analytical construction of solutions are rare. This paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for SDD problems with large delays.

State Dependent Regenerative Effect in Milling Processes

2011 ◽  
Vol 6 (4) ◽  
Author(s):  
Dániel Bachrathy ◽  
Gábor Stépán ◽  
János Turi
Keyword(s):  
Periodic Solutions ◽  
Large Amplitude ◽  
Chip Thickness ◽  
Hysteresis Phenomenon ◽  
Delay Model ◽  
Periodic Motions ◽  
Stability Calculation ◽  
State Dependent ◽  
Fold Bifurcation ◽  
State Dependent Delay

The governing equation of milling processes is generalized with the help of accurate chip thickness derivation resulting in a state dependent delay model. This model is valid for large amplitude machine tool vibrations and uses accurate nonlinear screen functions describing the entrance and exit positions of the cutting edges of the milling tool relative to the workpiece. The periodic motions of this nonlinear system are calculated by a shooting method. The stability calculation is based on the linearization around these periodic solutions by means of the semidiscretization method applied for the corresponding time-periodic delay system. Predictor-corrector method is developed to continue the periodic solutions as the bifurcation parameter, that is, the axial immersion is varied. It is observed that the influence of the state dependent delay on linear stability can be significant close to resonance where large amplitude forced vibrations occur. The existence of an unusual fold bifurcation is shown where a kind of hysteresis phenomenon appears between two different stable periodic motions.

scholarly journals Stability and Hopf Bifurcation for a Cell Population Model with State-Dependent Delay

2010 ◽  
Vol 70 (5) ◽  
pp. 1611-1633 ◽  
Author(s):  
Mostafa Adimy ◽  
Fabien Crauste ◽  
My Lhassan Hbid ◽  
Redouane Qesmi
Keyword(s):  
Hopf Bifurcation ◽  
Cell Population ◽  
Population Model ◽  
State Dependent ◽  
State Dependent Delay ◽  
A Cell
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