## Computation, Continuation And Bifurcation Analysis Of Periodic Solutions Of Delay Differential Equations (1997)

Citations: | 9 - 4 self |

### BibTeX

@MISC{Luzyanina97computation,continuation,

author = {Tatyana Luzyanina and Koen Engelborghs and Kurt Lust and Dirk Roose},

title = {Computation, Continuation And Bifurcation Analysis Of Periodic Solutions Of Delay Differential Equations},

year = {1997}

}

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### Abstract

this paper we present a new numerical method for the e#cient computation of periodic solutions of DDEs and the determination of their stability. This approach exploits the fact that zero is a cluster point for the Floquet multipliers [Hale, 1977]. Usually only a few Floquet multipliers have large modulus. Therefore we can use the ideas behind the Newton--Picard method developed for the computation of periodic solutions of dissipative systems of PDEs [Roose et al., 1995; Lust et al., 1996]. This algorithm can compute branches of periodic solutions and at little extra cost also the dominant, stability-determining Floquet multipliers. We now extend this method to compute periodic solutions of a DDE or a system of DDEs. We avoid an approximation of a DDE by a high order system of ODEs, which has been widespread in the mathematical and engineering literature [Bank & Burns, 1978]. The disadvantage of this approach is that an extremely large system of ODEs is necessary to obtain a good approximation for a periodic solution of a DDE, which leads to a very expensive method. In Sec. 2 we formulate the periodicity problem for DDEs in an infinite dimensional space and suggest a finite dimensional approximation of this problem through the discretization of an initial function on the delay interval. Using the discrete version of the Poincare operator, we construct a nonlinear system which allows to compute the segment of the periodic solution on the delay interval and the value of its period. This leads to a single-shooting approach which is the starting point of the Newton--Picard algorithm. In Sec. 3 we briefly describe the main ideas of the Newton--Picard algorithm and consider its application to DDEs. Results of numerical experiments are presented in Sec. 4. In Sec. 5 we summariz...