In this study dynamically equivalent time-invariant forms are obtained for linear and non-linear systems with periodically varying coefficients via Lyapunov–Floquet (L–F) transformation. These forms are equivalent in the sense that the local stability and bifurcation characteristics are identical for both systems in the entire parameter space. It is well known that the L–F transformation converts a linear periodic first order system into a time-invariant one. In the first part of this study a set of linear second order periodic equations is converted into an equivalent set of time-independent second order equations through a sequence of linear transformations. Then the transformations are applied to a time-periodic quadratic Hamiltonian to obtain its equivalent time-invariant form. In the second part, time-invariant forms of nonlinear equations are studied. The application of L–F transformation to a quasi-linear periodic equation converts the linear part to a time-invariant form and leaves the non-linear part with time-periodic coefficients. Dynamically equivalent time-invariant forms are obtained via time-periodic center manifold reduction and time-dependent normal form theory. Such forms are constructed for general hyperbolic systems and for some simple critical cases, including that of one zero eigenvalue and a purely imaginary pair. As a physical example of these techniques, a single and a double inverted pendulum subjected to periodic parametric excitation are considered. The results thus obtained are verified by numerical simulation.
[1]
Y. Kuznetsov.
Elements of Applied Bifurcation Theory
,
2023,
Applied Mathematical Sciences.
[2]
Vladimir Igorevich Arnold,et al.
Geometrical Methods in the Theory of Ordinary Differential Equations
,
1983
.
[3]
On averaged and normal form equations
,
1995
.
[4]
E. Butcher,et al.
Canonical Perturbation of a Fast Time-Periodic Hamiltonian via Liapunov-Floquet Transformation
,
1998
.
[5]
A. Nayfeh,et al.
Applied nonlinear dynamics : analytical, computational, and experimental methods
,
1995
.
[6]
S. C. Sinha,et al.
Analysis of time-periodic nonlinear dynamical systems undergoing bifurcations
,
1995,
Nonlinear Dynamics.
[7]
V. A. I︠A︡kubovich,et al.
Linear differential equations with periodic coefficients
,
1975
.
[8]
H. Broer,et al.
Normal forms and bifurcations of planar vector fields
,
1995
.
[9]
S. Sinha,et al.
Liapunov-Floquet Transformation: Computation and Applications to Periodic Systems
,
1996
.
[10]
S. Chow,et al.
Normal Forms and Bifurcation of Planar Vector Fields
,
1994
.
[11]
S. Sinha,et al.
An efficient computational scheme for the analysis of periodic systems
,
1991
.
[12]
S. Sinha,et al.
Optimal control of mechanical systems subjected to periodic loading via chebyshev polynomials
,
1993
.