Bifurcation Analysis of Nonlinear Periodic Systems via Lyapunov - FloquetTechnique

摘要

A technique has been presented for computing the Lyapunov-Floquet transformationmatrix for multidimensional nonlinear systems with periodically varying parameters by first computing the state transition matrix (STM) of the linearized system in a symbolic form via Chebyshev polynomials over the principal period. This STM is then factored to evaluate the Lyapunov-Floquet transformation matrix which is used to transform the nonlinear periodic system to one with time invariant linear coefficients. The subsequent application of center manifold and normal form theories may then be used to determine the stability and obtain an accurate analytical solution which is suitable for algebraic manipulations. Unlike the averaging and perturbation techniques, the proposed technique does not require the existence of a small parameter multiplying the time varying terms. Application of this method to externally excited systems such as periodically loaded columns and rotordynamic systems has given an accurate representation of the resonance conditions which arise in the system. Also, a method for obtaining the STM of the linearized system with parametric dependence has been shown. When combined with the stability and bifurcation theory of discrete maps, accurate local bifurcation surfaces may be obtained in closed form in the parameter space. Results indicate that this technique is well converged and is computationally more efficient than comparable schemes such as point mapping.