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Research Papers

Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Delays

[+] Author and Article Information
Anwar Sadath

Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Ordnance Factory Estate,
Hyderabad, Telangana 502205, India

C. P. Vyasarayani

Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Ordnance Factory Estate,
Hyderabad, Telangana 502205, India
e-mail: vcprakash@iith.ac.in

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 13, 2014; final manuscript received September 22, 2014; published online April 9, 2015. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 10(6), 061008 (Nov 01, 2015) (7 pages) Paper No: CND-14-1124; doi: 10.1115/1.4028631 History: Received May 13, 2014; Revised September 22, 2014; Online April 09, 2015

In this paper, we develop Galerkin approximations for determining the stability of delay differential equations (DDEs) with time periodic coefficients and time periodic delays. Using a transformation, we convert the DDE into a partial differential equation (PDE) along with a boundary condition (BC). The PDE and BC we obtain have time periodic coefficients. The PDE is discretized into a system of ordinary differential equations (ODEs) using the Galerkin method with Legendre polynomials as the basis functions. The BC is imposed using the tau method. The resulting ODEs are time periodic in nature; thus, we resort to Floquet theory to determine the stability of the ODEs. We show through several numerical examples that the stability charts obtained from the Galerkin method agree closely with those obtained from direct numerical simulations.

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Figures

Grahic Jump Location
Fig. 1

(a) Time periodic coefficients a1,a2, and a3 obtained in the simulation of the first-order DDE with a single time periodic delay described in Eq. (33) and (b) stability diagram for the same. The red circles are stable regions obtained using the Galerkin method with N = 10, and the blue dots are stable regions obtained from the numerical integration of the DDE. The system parameters are ω1 = π/2, ω2 = π, and τ(t) = 0.6 − 0.4 sin(ω2t).

Grahic Jump Location
Fig. 2

Time response of Eq. (33) obtained using the Galerkin approximation and using numerical simulation. The system parameters are as follows: (a) k1 = 0.26 and k2 = −1.4, (b) k1 = 0.26 and k2 = −1.5, and (c) k1 = −0.5 and k2 = −1.

Grahic Jump Location
Fig. 3

(a) Time periodic coefficients a1, a2, a3, and a4 obtained in the simulation of the first-order DDE with two time periodic delays described in Eq. (36) and (b) stability diagram for the same. The red circles are stable regions obtained using the Galerkin method with N = 7, and the blue dots are stable regions obtained from the numerical integration of the DDE (Eq. (36)).

Grahic Jump Location
Fig. 4

Time response of Eq. (36) obtained using Galerkin approximation and numerical simulation. The parameters are (a) a = 1.5 and b = 1 and (b) a = −4.5 and b = 6.

Grahic Jump Location
Fig. 5

(a) Time periodic coefficients a1, a2, a3, a4, and a5 obtained in the simulation of the delayed damped Mathieu equation described in Eq. (37); and (b) stability diagram for the same. The red circles are stable regions obtained using the Galerkin method with N = 10, and the blue dots are stable regions obtained from numerical integration of the DDE.

Grahic Jump Location
Fig. 6

Time response of Eq. (37) obtained using the Galerkin approximation and numerical simulation. The parameters are (a) δ = 8 and  = 19.2, which is stable; and (b) δ = 0.4 and  = 6, which is unstable.

Grahic Jump Location
Fig. 7

The stability diagram of Eq. (38) with the parameters ζ = 0.02, RA = 0.1, RP = 2, and with (a) N = 8, (b) N = 10, and (c) N = 25

Grahic Jump Location
Fig. 8

The stability diagram of Eq. (38) with parameters ζ = 0.005, RA = 0.02, RP = 0.4, and with (a) N = 25, (b) N = 30, and (c) N = 35

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