Versal Deformation and Local Bifurcation Analysis of Time-Periodic Nonlinear Systems

Abstract

In this study a local semi-analytical method of quantitativebifurcation analysis for time-periodic nonlinear systems is presented.In the neighborhood of a local bifurcation point the system equationsare simplified via Lyapunov–Floquet transformation whichtransforms the linear part of the equation into a dynamically equivalenttime-invariant form. Then the time-periodic center manifoldreduction is used to separate the `critical' states and reduce thedimension of the system to a possible minimum. The center manifoldequations can be simplified further via time-dependent normal formtheory. For most codimension one cases these nonlinear normal forms arecompletely time-invariant. Versal deformation of thesetime-invariant normal forms can be found and the bifurcation phenomenoncan be studied in the neighborhood of the critical point. However, ingeneral, it is not a trivial task to find a quantitatively correctversal deformation for time-periodic systems. In order to do so, onemust find a relationship between the bifurcation parameter of theoriginal time-periodic system and the versal deformation parameter ofthe time-invariant normal form. Essentially one needs to find theeigenvalues of the fundamental solution matrix of the time-periodicproblem in terms of the system parameters, which, in general, cannot bedone due to computational difficulties. In this work two ideas areproposed to achieve this goal. The eigenvalues of the fundamentalsolution matrix can be related to the versal deformation parameter bysensitivity analysis and an approximation of any desired order can beobtained. This idea requires a symbolic computational procedure whichcan be very time consuming in some cases. An alternative method issuggested for faster results in which a second or higher order curvefitting technique is used to find the relationship. Once thisrelationship is established, closed form post-bifurcation steady-statesolutions can be obtained for flip, symmetry breaking, transcritical andsecondary Hopf bifurcations. Unlike averaging and perturbation methods,the proposed technique is applicable at any bifurcation point in theparameter space. As physical examples, a simple and a double pendulumsubjected to periodic parametric excitation are considered. A simple twodegrees of freedom model is also studied and the results are comparedwith those obtained from the traditional averaging method. All resultsare verified by numerical integration. It is observed that the proposedtechnique yields results which are very close to the numericalsolutions, unlike the averaging method.