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.

References

References
1.
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
.
2.
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
.
3.
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
.
4.
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
.
5.
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
.
6.
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
.
7.
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
.
8.
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
.
9.
Yakubovich
,
V. A.
, and
Starzhinski
,
V. M.
,
1975
,
Linear Differential Equations With Periodic Coefficients
, Parts I and II,
Wiley
,
New York
.
10.
Lukes
,
D. L.
,
1982
,
Differential Equations: Classical to Controlled
,
Academic Press
,
New York
.
11.
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
.
12.
Helton
,
B. W.
,
1968
, “
Logarithms of Matrices
,”
Proc. Am. Math. Soc.
,
19
(
3
), pp.
733
738
.
13.
Wouk
,
A.
,
1965
, “
Integral Representation of the Logarithm of Matrices and Operators
,”
J. Math. Anal. Appl.
,
11
, pp.
131
138
.
You do not currently have access to this content.