AN ABSTRACT DIFFERENTIAL EQUATION ARISING FROM CELL POPULATION DYNAMICS

We provide an analysis of the stability and bifurcation properties of the solutions of an abstract differential nonlinear equation arising from cell population dynamics. The work surveyed here stems from a remark we made with respect to these equations: that it is possible to associate to any of them a delay differential equation on an infinite dimensional vector space. Perturbation theory for nonlinear equations similar to the one known for delay differential equations on finite dimensional spaces could possibly yield the same results as for those equations.

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

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.785-786.1418 ◽

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 .

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Stability of Up-milling and Down-milling Operations with Variable Spindle Speed

Journal of Vibration and Control ◽

10.1177/1077546309341131 ◽

2010 ◽

Vol 16 (7-8) ◽

pp. 1151-1168 ◽

Cited By ~ 42

Author(s):

Xinhua Long ◽

B. Balachandran

Keyword(s):

Spindle Speed ◽

Depth Of Cut ◽

Control Parameters ◽

Infinite Dimensional ◽

Dimensional Matrix ◽

Finite Dimensional ◽

Milling Operations ◽

The Stability ◽

Delay Differential ◽

Milling Dynamics

In this article, a stability treatment is presented for up-milling and down-milling processes with a variable spindle speed (VSS). This speed variation is introduced by superimposing a sinusoidal modulation on a nominal spindle speed. The VSS milling dynamics is described by a set of delay differential equations with time varying periodic coefficients and a time delay. A semi-discretization scheme is used to discretize the system over one period, and the infinite-dimensional transition matrix is reduced to a finite-dimensional matrix over this period. The eigenvalues of this finite-dimensional matrix provide information on VSS milling stability with respect to control parameters, such as the axial depth of cut and the nominal spindle speed. The stability charts obtained for VSS milling operations are compared with those obtained for constant spindle speed milling operations, and the benefits of VSS milling operations are discussed.

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Stability Analysis of a Variable Spindle Speed Milling Process

Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

10.1115/detc2005-85065 ◽

2005 ◽

Cited By ~ 2

Author(s):

X.-H. Long ◽

B. Balachandran

Keyword(s):

Milling Process ◽

Spindle Speed ◽

Depth Of Cut ◽

Infinite Dimensional ◽

Dimensional Matrix ◽

Finite Dimensional ◽

Milling Operations ◽

Milling Operation ◽

The Stability ◽

Delay Differential

In this effort, a stability treatment is presented for a milling process with a variable spindle speed (VSS). This variation is caused by superimposing a sinusoidal modulation on a nominal spindle speed. The dynamics of the VSS milling process is described by a set of delay differential equations (DDEs) with time varying periodic coefficients and a time delay. A semi-discretization scheme is used to discretize the system over one period, and the infinite dimensional transition matrix is converted to a finite dimensional matrix over this period. The eigenvalues of this finite dimensional matrix are used to determine the stability of the VSS milling operation with respect to selected control parameters, such as the axis depth of cut and the nominal spindle speed. The benefits of VSS milling operations are discussed by comparing the stability charts obtained for VSS milling operations with those obtained for constant spindle speed (CSS) milling operations.

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