Stability of the Damped Mathieu Equation With Time Delay

2003 ◽  
Vol 125 (2) ◽  
pp. 166-171 ◽  
Author(s):  
T. Insperger ◽  
G. Ste´pa´n
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Delay Differential Equation ◽  
Mathieu Equation ◽  
Important Class ◽  
System Parameters ◽  
Oscillatory Systems ◽  
Stability Chart ◽  
The Stability ◽  
Delay Differential

In the space of the system parameters, the stability charts are determined for the delayed and damped Mathieu equation defined as x¨t+κx˙t+δ+ε cos txt=bxt−2π. This stability chart makes the connection between the Strutt-Ince chart of the damped Mathieu equation and the Hsu-Bhatt-Vyshnegradskii chart of the autonomous second order delay-differential equation. The combined charts describe the intriguing stability properties of an important class of delayed oscillatory systems subjected to parametric excitation.

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 .

MULTISCROLL CHAOTIC ATTRACTORS FROM A HYSTERESIS BASED TIME-DELAY DIFFERENTIAL EQUATION

2010 ◽  
Vol 20 (10) ◽  
pp. 3275-3281 ◽  
Author(s):  
SELÇUK KILINÇ ◽  
MÜŞTAK E. YALÇIN ◽  
SERDAR ÖZOGUZ
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Delay Differential Equation ◽  
Order Differential Equation ◽  
Chaotic Attractors ◽  
Nonlinear Characteristic ◽  
Circuit Implementation ◽  
First Order ◽  
State Variable ◽  
Delay Differential

In this paper, the generation of multiscroll chaotic attractors derived from a time-delay differential equation is presented. The proposed system is represented by only one first-order differential equation including time-delayed state variable, and employs hysteresis function as the nonlinear characteristic. The generalization of the introduced system is based on adding multihysteresis nonlinear characteristic which leads to n-scroll chaotic attractors. The circuit implementation of the proposed system and some experimental results referring to two-, three-, four-, and five-scroll chaotic attractors are reported.

Machine Tool Chatter and Surface Quality in Milling Processes

2004 ◽  
Author(s):  
Tama´s Insperger ◽  
Janez Gradisek ◽  
Martin Kalveram ◽  
Ga´bor Ste´pa´n ◽  
Klaus Weinert ◽  
...  
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Periodic Motion ◽  
Milling Process ◽  
Location Error ◽  
Surface Location Error ◽  
Stability Chart ◽  
Stable Periodic ◽  
Surface Location ◽  
Delay Differential

Two degree of freedom model of milling process is investigated. The governing equation of motion is decomposed into two parts: an ordinary differential equation describing the stable periodic motion of the tool and a delay-differential equation describing chatter. Stability chart is derived by using semi-discretization method for the delay-differential equation corresponding to the chatter motion. The stable periodic motion of the tool and the associated surface location error are obtained by a conventional solution technique of ordinary differential equations. Stability chart and surface location error are determined for milling process. It is shown that at spindle speeds, where high depths of cut are available through stable machining, the surface location error is large. The phase portrait of the tool is also analyzed for different spindle speeds. Theoretical predictions are qualitatively confirmed by experiments.

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.

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