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Research Papers

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

[+] Author and Article Information
Anwar Sadath

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

C. P. Vyasarayani

Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Ordnance Factory Estate,
Andhra Pradesh 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 November 25, 2013; final manuscript received March 3, 2014; published online January 12, 2015. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 10(2), 021011 (Mar 01, 2015) (7 pages) Paper No: CND-13-1298; doi: 10.1115/1.4026989 History: Received November 25, 2013; Revised March 03, 2014; Online January 12, 2015

A numerical method to determine the stability of delay differential equations (DDEs) with time periodic coefficients is proposed. The DDE is converted into an equivalent partial differential equation (PDE) with a time periodic boundary condition (BC). The PDE, along with its BC, is then converted into a system of ordinary differential equations (ODEs) with time periodic coefficients using the Galerkin least squares approach. In the Galerkin approach, shifted Legendre polynomials are used as basis functions, allowing us to obtain explicit expressions for the approximate system of ODEs. We analyze the stability of the discretized ODEs, which represent an approximate model of the DDEs, using Floquet theory. We use numerical examples to show that the stability charts obtained with our method are in excellent agreement with those existing in the literature and those obtained from direct numerical simulation.

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Figures

Grahic Jump Location
Fig. 1

Stability diagram obtained using the Galerkin method with (a) N = 3, (b) N = 5, and (c) N = 7. The thick circled lines are the stability boundaries reported in Ref. [4]. The system parameters are c = 0.1, k1 = -0.04, ω = 1, and τ = 2π.

Grahic Jump Location
Fig. 2

Comparison between solutions obtained using the Galerkin approximation [y(0,t)] and through the direct numerical simulation of Eq. (28). We use the following parameters: N = 7, c = 0.1, k1 = -0.04, ω = 1, τ = 2π, and δ = 0.4; (a) ε = 0.4 and (b) ɛ = 0.5.

Grahic Jump Location
Fig. 3

(a) Eigenvalues of the state transition matrix when the parameter ɛ is varied from 0 to 0.7 with δ = 0.4. Stability is lost through a flip bifurcation. (b) Eigenvalues of the state transition matrix when the parameter ε is varied from 0 to 1.6 with δ = 1.5. In this case, stability is lost through a transcritical bifurcation. The system parameters are c = 0.1, k1 = -0.04, ω = 1, τ = 2π, and N = 7.

Grahic Jump Location
Fig. 4

Stability diagram obtained using the Galerkin method with N = 5. The black dots are stable regions obtained using the Galerkin method, and the red circles are stable regions obtained from numerical simulation of the DDE (Eq. (29)). The system parameters are c = 0.2, k1 = -0.01, ω = 2π, and τ = 2π.

Grahic Jump Location
Fig. 5

Comparison between solutions obtained using the Galerkin approximation [y(0,t)] and through the direct numerical simulation of Eq. (29). We use the following parameters: N = 7, c = 0.2, k1 = -0.01, ω = 2π, and τ = 2π; (a) δ = 6 and ε = 2 and (b) δ = 10 and ɛ = 8.

Grahic Jump Location
Fig. 6

Stability diagram obtained using the Galerkin method with N = 17. The black dots are stable regions obtained using the Galerkin method, and the red circles are stable regions obtained from numerical simulation of the DDE (Eq. (30)). The system parameters are c = 0.05, k1 = -0.2, k2 = -0.1, ω = π, τ1 = 2π, and τ2 = π.

Grahic Jump Location
Fig. 7

Stability diagrams obtained using the Galerkin method with N = 11. The thick circled lines are stability boundaries from [4]. The system parameters are τ = 2π; (a) c = 0.1 and (b) c = 0.2.

Grahic Jump Location
Fig. 8

Stability diagram obtained using the Galerkin method with N = 20. The black dots are stable regions obtained using the Galerkin method and the red circles are stable regions obtained from numerical simulation of the DDE (Eq. (30)). The system parameters are ζ = 0.001, ɛ = 0.2, and τ1 = 2π.

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