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Research Papers

# Symbolic Computation of Quantities Associated With Time-Periodic Dynamical Systems1

[+] Author and Article Information
W. Grant Kirkland

Nonlinear Systems Research Laboratory,
Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: gkirkland@auburn.edu

S. C. Sinha

Life Fellow ASME
Professor
Nonlinear Systems Research Laboratory,
Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: ssinha@eng.auburn.edu

2Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 1, 2015; final manuscript received March 30, 2016; published online May 13, 2016. Assoc. Editor: Bogdan I. Epureanu.

J. Comput. Nonlinear Dynam 11(4), 041022 (May 13, 2016) (10 pages) Paper No: CND-15-1197; doi: 10.1115/1.4033382 History: Received July 01, 2015; Revised March 30, 2016

## Abstract

Many dynamical systems can be modeled by a set of linear/nonlinear ordinary differential equations with periodic time-varying coefficients. The state transition matrix (STM) $Φ(t,α)$, associated with the linear part of the equation, can be expressed in terms of the periodic Lyapunov–Floquét (L-F) transformation matrix $Q(t,α)$ and a time-invariant matrix $R(α)$ containing a set of symbolic system parameters $α.$ Computation of $Q(t,α)$ and $R(α)$ in symbolic form as a function of $α$ is of paramount importance in stability, bifurcation analysis, and control system design. In earlier studies, since $Q(t,α)$ and $R(α)$ were available only in numerical forms, general results for parameter unfolding and control system design could not be obtained in the entire parameter space. In 2009, an attempt was made by Butcher et al. (2009, “Magnus' Expansion for Time-Periodic Systems: Parameter Dependent Approximations,” Commun. Nonlinear Sci. Numer. Simul., 14(12), pp. 4226–4245) to compute the $Q(t,α)$ matrix in a symbolic form using the Magnus expansions with some success. In this work, an efficient technique for symbolic computation of $Q(t,α)$ and $R(α)$ matrices is presented. First, $Φ(t,α)$ is computed symbolically using the shifted Chebyshev polynomials and Picard iteration method as suggested in the literature. Then, $R(α)$ is computed using a Gaussian quadrature integral formula. Finally, $Q(t,α)$ is computed using the matrix exponential summation method. Using mathematica, this approach has successfully been applied to the well-known Mathieu equation and a four-dimensional time-periodic system in order to demonstrate the applications of the proposed method to linear as well as nonlinear problems.

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## References

Sinha, S. C. , and Wu, D. H. , 1991, “ An Efficient Computational Schemed for the Analysis of Periodic Systems,” J. Sound Vib., 151(1), pp. 91–117.
Sinha, S. C. , Pandiyan, R. , and Bibb, J. S. , 1996, “ Liapunov-Floquet Transformation: Computation and Applications to Periodic Systems,” ASME J. Vib. Acoust., 118(2), pp. 209–219.
Sinha, S. C. , and Joseph, P. , 1994, “ Control of General Dynamics Systems With Periodically Varying Parameters Via Liapunov-Floquet Transformation,” ASME J. Dyn. Syst., Meas., Control, 116(4), pp. 650–658.
Sinha, S. C. , and Pandiyan, R. , 1994, “ Analysis of Quasilinear Dynamical Systems With Periodic Coefficients Via Liapunov-Floquet Transformation,” Int. J. Non-Linear Mech., 29(5), pp. 687–702.
Sinha, S. C. , and Butcher, E. A. , 1997, “ Symbolic Computation of Fundamental Solution Matrices for Time Periodic Dynamical Systems,” J. Sound Vib., 206(1), pp. 61–85.
Dávid, A. , and Sinha, S. C. , 2000, “ Versal Deformation and Local Bifurcation Analysis of Time-Periodic Nonlinear Systems,” J. Nonlinear Dyn., 21(4), pp. 317–336.
Dávid, A. , and Sinha, S. C. , 2003, “ Bifurcation Control of Nonlinear Systems With Time-Periodic Coefficients,” ASME J. Dyn. Syst., Meas., Control, 125(5), pp. 541–548.
Butcher, E. A. , Sari, M. , Bueler, E. , and Carlson, T. , 2009, “ Magnus' Expansion for Time-Periodic Systems: Parameter Dependent Approximations,” Commun. Nonlinear Sci. Numer. Simul., 14(12), pp. 4226–4245.
Yakubovich, V. A. , and Starzhinski, V. M. , 1975, Linear Differential Equations With Periodic Coefficients, Parts I and II, Wiley, New York.
Lukes, D. L. , 1982, Differential Equations: Classical to Controlled, Academic Press, New York.
Dieci, L. , Morini, B. , and Papini, A. , 1996, “ Computational Techniques for Real Logarithms of Matrices,” SIAM J. Matrix Anal. Appl., 17(3), pp. 570–593.
Helton, B. W. , 1968, “ Logarithms of Matrices,” Proc. Am. Math. Soc., 19(3), pp. 733–738.
Wouk, A. , 1965, “ Integral Representation of the Logarithm of Matrices and Operators,” J. Math. Anal. Appl., 11, pp. 131–138.

## Figures

Fig. 1

CPU time to compute R̂(α) versus number of Gaussian nodes for the DME

Fig. 2

CPU time to compute Q̂(t,α) versus number of summation terms for the DME

Fig. 3

Relative error versus number of Gaussian nodes of the R̂ matrix (stable DME)

Fig. 4

R̂ matrix over two periods of the DME (stable)

Fig. 5

Relative error versus number of summation terms of the Q̂ matrix (stable DME)

Fig. 6

T-periodic Q̂(t) matrix of the DME (stable)

Fig. 7

2T-periodic Q̂(t) matrix of the DME (unstable)

Fig. 8

Inverted double pendulum (IDP) [8]

Fig. 9

CPU time to compute Q̂(t,α) versus number of summation terms for the IDP

Fig. 10

Relative error versus number of Gaussian nodes of the R̂ matrix for the IDP

Fig. 11

R̂ matrix over two periods of the IDP

Fig. 12

Relative error versus number of expansion terms of the Q̂ matrix for the IDP

Fig. 13

T-periodic Q̂11 and Q̂12 of the IDP

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