Numerical bifurcation analysis of delay differential equations arising from physiological modeling

2001 ◽  
Vol 42 (4) ◽  
pp. 361-385 ◽  
Author(s):  
K. Engelborghs ◽  
V. Lemaire ◽  
J. Bélair ◽  
D. Roose
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Physiological Modeling ◽  
Numerical Bifurcation ◽  
Delay Differential ◽  
Numerical Bifurcation Analysis

NUMERICAL BIFURCATION ANALYSIS OF DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

2001 ◽  
Vol 11 (03) ◽  
pp. 737-753 ◽  
Author(s):  
TATYANA LUZYANINA ◽  
KOEN ENGELBORGHS ◽  
DIRK ROOSE
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Constant Delay ◽  
State Dependent ◽  
State Dependent Delay ◽  
Numerical Bifurcation ◽  
Delay Differential ◽  
Steady State Solutions ◽  
Numerical Bifurcation Analysis

In this paper we apply existing numerical methods for bifurcation analysis of delay differential equations with constant delay to equations with state-dependent delay. In particular, we study the computation, continuation and stability analysis of steady state solutions and periodic solutions. We collect the relevant theory and describe open theoretical problems in the context of bifurcation analysis. We present computational results for two examples and compare with analytical results whenever possible.

Chatter Stability Analysis Using the Matrix Lambert Function and Bifurcation Analysis

2006 ◽  
Author(s):  
Sun Yi ◽  
Patrick W. Nelson ◽  
A. Galip Ulsoy
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Lambert Function ◽  
Chatter Stability ◽  
Approximate Methods ◽  
Linear Delay ◽  
The Matrix ◽  
The Stability ◽  
Delay Differential

We investigate the stability of the regenerative machine tool chatter problem, in a turning process modeled using delay differential equations (DDEs). An approach using the matrix Lambert function for the analytical solution to systems to delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert function, known to be useful for solving scalar first order DDEs, has recently been extended to a matrix Lambert function approach to solve systems of DDEs. The essential advantage of the matrix Lambert approach is not only the similarity to the concept of the state transition matrix in linear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy, and certain other advantages, when compared to traditional graphical, computational and approximate methods.

scholarly journals Computation of double Hopf points for delay differential equations

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Yingxiang Xu ◽  
Tingting Shi
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Delay Systems ◽  
System Of Equations ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Defining System ◽  
Numerical Bifurcation ◽  
Delay Differential ◽  
Branch Switching

AbstractRelating to the crucial problem of branch switching, the calculation of codimension 2 bifurcation points is one of the major issues in numerical bifurcation analysis. In this paper, we focus on the double Hopf points for delay differential equations and analyze in detail the corresponding eigenspace, which enable us to obtain the finite dimensional defining system of equations of such points, instead of an infinite dimensional one that happens naturally for delay systems. We show that the double Hopf point, together with the corresponding eigenvalues, eigenvectors and the critical values of the bifurcation parameters, is a regular solution of the finite dimensional defining system of equations, and thus can be obtained numerically through applying the classical iterative methods. We show our theoretical findings by a numerical example.

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