scholarly journals On the Stability and Boundedness of Solutions of Certain Non-Autonomous Delay Differential Equation of Third Order

2016 ◽  
Vol 07 (06) ◽  
pp. 457-467
Author(s):  
Akinwale L. Olutimo ◽  
Daniel O. Adams
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Boundedness Of Solutions ◽  
Third Order ◽  
The Stability ◽  
Delay Differential

scholarly journals Uncharted Stable Peninsula for Multivariable Milling Tools by High-Order Homotopy Perturbation Method

2020 ◽  
Vol 10 (21) ◽  
pp. 7869 ◽  
Author(s):  
Jose de la Luz Sosa ◽  
Daniel Olvera-Trejo ◽  
Gorka Urbikain ◽  
Oscar Martinez-Romero ◽  
Alex Elías-Zúñiga ◽  
...  
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Delay Differential Equation ◽  
Homotopy Perturbation Method ◽  
Multiple Delays ◽  
Third Order ◽  
Method Performance ◽  
Homotopy Perturbation ◽  
The Stability ◽  
Delay Differential

In this work, a new method for solving a delay differential equation (DDE) with multiple delays is presented by using second- and third-order polynomials to approximate the delayed terms using the enhanced homotopy perturbation method (EMHPM). To study the proposed method performance in terms of convergency and computational cost in comparison with the first-order EMHPM, semi-discretization and full-discretization methods, a delay differential equation that model the cutting milling operation process was used. To further assess the accuracy of the proposed method, a milling process with a multivariable cutter is examined in order to find the stability boundaries. Then, theoretical predictions are computed from the corresponding DDE finding uncharted stable zones at high axial depths of cut. Time-domain simulations based on continuous wavelet transform (CWT) scalograms, power spectral density (PSD) charts and Poincaré maps (PM) were employed to validate the stability lobes found by using the third-order EMHPM for the multivariable tool.

scholarly journals Non-Spiking Laser Controlled by a Delayed Feedback

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2069
Author(s):  
Anton V. Kovalev ◽  
Evgeny A. Viktorov ◽  
Thomas Erneux
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Optical Feedback ◽  
Laser Pulses ◽  
Equation Model ◽  
Rate Equations ◽  
Differential Equation Model ◽  
The Stability ◽  
Delay Differential ◽  
Intensity Laser

In 1965, Statz et al. (J. Appl. Phys. 30, 1510 (1965)) investigated theoretically and experimentally the conditions under which spiking in the laser output can be completely suppressed by using a delayed optical feedback. In order to explore its effects, they formulate a delay differential equation model within the framework of laser rate equations. From their numerical simulations, they concluded that the feedback is effective in controlling the intensity laser pulses provided the delay is short enough. Ten years later, Krivoshchekov et al. (Sov. J. Quant. Electron. 5394 (1975)) reconsidered the Statz et al. delay differential equation and analyzed the limit of small delays. The stability conditions for arbitrary delays, however, were not determined. In this paper, we revisit Statz et al.’s delay differential equation model by using modern mathematical tools. We determine an asymptotic approximation of both the domains of stable steady states as well as a sub-domain of purely exponential transients.

τ-D Decomposition to the One Dimensional Delay Differential Equation by Fixed the Coefficient b<0

2013 ◽  
Vol 785-786 ◽  
pp. 1418-1422
Author(s):  
Ai Gao
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Delay Differential Equation ◽  
Stability Domain ◽  
Transcendental Equation ◽  
One Dimension ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Delay Differential

In this paper, we provide a partition of the roots of a class of transcendental equation by using τ-D decomposition ,where τ>0,a>0,b<0 and the coefficient b is fixed.According to the partition, one can determine the stability domain of the equilibrium and get a Hopf bifurcation diagram that can provide the Hopf bifurcation curves in the-parameter space, for one dimension delay differential equation .

scholarly journals On the stability and boundedness of solutions of nonlinear third order differential equations with delay

Filomat ◽  
2010 ◽  
Vol 24 (3) ◽  
pp. 1-10 ◽  
Author(s):  
Cemil Tunç
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Relevant Literature ◽  
Stability Result ◽  
Boundedness Of Solutions ◽  
Third Order ◽  
Functional Differential ◽  
The Stability ◽  
Equations With Delay

By defining a Lyapunov functional, we investigate the stability and boundedness of solutions to nonlinear third order differential equation with constant delay, r : x'''(t) + g(x(t), x'(t))x''(t) + f (x(t - r), x'(t - r)) + h(x(t - r)) = p(t, x(t), x'(t), x(t - r), x'(t - r), x''(t)), when p(t, x(t), x'(t), x(t - r), x'(t - r), x''(t)) = 0 and ? 0, respectively. Our results achieve a stability result which exists in the relevant literature of ordinary nonlinear third order differential equations without delay to the above functional differential equation for the stability and boundedness of solutions. An example is introduced to illustrate the importance of the results obtained.

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