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Research Papers

Response and Stability Analysis of Periodic Delayed Systems With Discontinuous Distributed Delay

[+] Author and Article Information
Oleg A. Bobrenkov1

Department of Mechanical and Aerospace Engineering,  New Mexico State University, Las Cruces, NM 88003chaalis@nmsu.edu

Morad Nazari

Department of Mechanical and Aerospace Engineering,  New Mexico State University, Las Cruces, NM 88003morad@nmsu.edu

Eric A. Butcher

Department of Mechanical and Aerospace Engineering,  New Mexico State University, Las Cruces, NM 88003eab@nmsu.edu

1

Corresponding author. Current address: Department of Mechanical Engineering, The University of Texas at Dallas, Richardson, Texas 75080, e-mail: oleg.bobrenkov@utdallas.edu.

J. Comput. Nonlinear Dynam 7(3), 031010 (Apr 03, 2012) (12 pages) doi:10.1115/1.4005925 History: Received July 27, 2011; Accepted November 30, 2011; Published April 03, 2012

In this paper, the analysis of delay differential equations with periodic coefficients and discontinuous distributed delay is carried out through discretization by the Chebyshev spectral continuous time approximation (ChSCTA). These features are introduced in the delayed Mathieu equation with discontinuous distributed delay which is used as an illustrative example. The efficiency of stability analysis is improved by using shifted Chebyshev polynomials for computing the monodromy matrix, as well as the adaptive meshing of the parameter plane. An idea for a method for numerical integration of periodic DDEs with discontinuous distributed delay based on existing MATLAB functions is proposed.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 14

Stability results for Eq. 26 in the (a, c1 ) plane for different values of c while b = 0.1, c2  = 0.1 (first row) and c2  = 4 (second row) are kept constant. In the respective columns from 1 to 5, the variable parameter values are 0, 0.5, 1, 1.5, and 2. The shaded regions are stable.

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Figure 13

Stability charts for Eq. 26 in the (a, c) plane for different values of c1 while c2  = 0.1 is kept constant (first row) and different values of c2 while c1  = 0.1 is kept constant (second row). In the respective columns from 1 to 5, the variable parameter values are 0, 0.2, 0.4, 0.6, and 0.8. The shaded regions are stable.

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Figure 12

Stability charts for Eq. 26 in the (a, c) plane for different values of b and c1  = c2 . The parameter values are b = 0.1, 1, and 2 in the first, second, and third row, respectively, and c1  = c2  = 0, 0.1, 0.2, 0.3, and 0.4 in the respective columns from 1 to 5. The shaded regions are stable.

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Figure 11

Time series obtained by integrating Eq. 15 using dde23 (solid), Euler method (dashed), and by integrating Eq. 20 using ode45 (dotted). Ten points per distributed delay integral were used (Np  = 37 for Eq. 20) along with the parameter set (a, b, c, c1 , c2 ) = (5, 1, 0.1, 0.1, 0.1).

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Figure 10

The results of convergence analysis for Eq. 15 performed for the point on the stability chart in Fig. 9 The parameter values are (a, b, c, c1 , c2 ) = (2.7112, 0.1, 0.1, 0.1, 0.1). Spectral radii, error, and computation time versus the number of collocation points Np are plotted in the first, second, and third columns, respectively. The first two rows are produced for 20 and 30 polynomials, while numerical integration is used for the third row. The fluctuations in the error charts show that the spectral convergence has been achieved in the system.

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Figure 9

Stability chart with the point (a, c) = (2.7112, 0.1) marked by the asterisk for which the convergence tests were performed. The values of other parameters are (b, c1 , c2 ) = (0.1, 0.1, 0.1).

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Figure 8

Stability charts for Eq. 15 in the (a, c1 ) plane for different values of c while b = 0.1, c2  = 0.1 (first row) and c2  = 2 (second row) are kept constant. In the respective columns from 1 to 5, the variable parameter values are 0, 0.5, 1, 1.5, and 2. The shaded regions are stable.

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Figure 7

Stability charts for Eq. 15 in the (a, c) plane for different values of c1 while c2  = 0.1 is kept constant (first row) and different values of c2 while c1  = 0.1 is kept constant (second row). In the respective columns from 1 to 5, the variable parameter values are 0, 0.2, 0.3, 0.4, and 0.5. The shaded regions are stable.

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Figure 6

(a) Stability chart for Eq. 15 in the three-dimensional (a, c, b) plane for c1  = c2  = 0.1. (b) Three-dimensional stability diagram for the delayed Mathieu equation obtained in [42]. The points within the volumetric regions are stable.

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Figure 5

Stability results for Eq. 15 in the (a, c) plane for different values of b and c1  = c2 . The shaded regions are stable. The parameter values are: b = 0.1, 1, and 2 in the first, second, and third row, respectively, and c1  = c2  = 0, 0.05, 0.1, 0.2, and 0.3 in the respective columns from 1 to 5. The shaded regions are stable.

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Figure 4

Chebyshev collocation points within each time delay interval of discretization of Eq. 15 (above) plotted versus the discontinuous distributed delay weighting function (below)

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Figure 3

Adaptive meshing scheme of the parameter plane

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Figure 2

Chebyshev collocation points as defined in Eq. 4

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Figure 1

The qualitative difference between the DDE systems with discontinuous coefficients whose stability was analyzed in [38] by multiple-interval Chebyshev collocation method and those with discontinuous distributed delay is illustrated. (a) Discontinuous coefficient variation in the period-to-period mapping and (b) variation of distributed delay weight functions ηi (θ, t) periodic in t and discontinuous in θ (see Eq. 1) and locations of discontinuities in the discontinuous distributed delay integration parameter θ depending on the current time t* of the system evolution. Instead of using the surface representation of the two-parametric function ηi (θ, t), it is plotted versus θ for some fixed values of t = t* . The following notation is also used: ρ is a portion of the period T that identifies the discontinuity location, T is the period of the discontinuous periodic coefficient g(t) equal to the discrete time delay, and t0 , …, tN are the Chebyshev collocation points ordered right to left.

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