ANALYSIS OF TIME-PERIODIC NONLINEAR DYNAMICAL-SYSTEMS UNDERGOING BIFURCATIONS

In this study a new procedure for analysis of nonlinear dynamical syst ems with periodically varying parameters under critical conditions is presented through an application of the Liapunov-Floquet (GF) transfor mation. The LF transformation is obtained by computing the state trans ition matrix associated with the linear part of the problem. The eleme nts of the state transition matrix are expressed in terms of Chebyshev polynomials in time t which is suitable for algebraic manipulations. Application of Floquet theory and the eigen-analysis of the state tran sition matrix at the end of one principal period provides the LF trans formation matrix in terms of the Chebyshev polynomials. Since this is a periodic matrix, the GF transformation matrix has a Fourier represen tation. It is well known that such a transformation converts a linear periodic system into a linear time-invariant one. When applied to quas i-linear equations with periodic coefficients, a dynamically similar s ystem is obtained whose linear part is time-invariant and the nonlinea r part consists of coefficients which are periodic. Due to this proper ty of the GF transformation, a periodic orbit in original coordinates will have a fixed point representation in the transformed coordinates. In this study, the bifurcation analysis of the transformed equations, obtained after the application of the LF transformation, is conducted by employing time-dependent center manifold reduction and time-depend ent normal form theory. The above procedures are analogous to existing methods that are employed in the study of bifurcations of autonomous systems. For the two physical examples considered, the three generic c odimension one bifurcations namely, Hopf, flip and fold bifurcations a re analyzed. In the first example, the primary bifurcations of a param etrically excited single degree of freedom pendulum is studied. As a s econd example, a double inverted pendulum subjected to a periodic load ing which undergoes Hopf or flip bifurcation is analyzed. The methodol ogy is semi-analytic in nature and provides quantitative measure of st ability when compared to point mappings method. Furthermore, the techn ique is applicable also to those systems where the periodic term of th e linear part does not contain a small parameter which is certainly no t the case with perturbation or averaging methods. The conclusions of the study are substantiated by numerical simulations. It is believed t hat analysis of this nature has been reported for the first time for t his class of systems.