Local (Classical) And Global Bifurcations In Non-linear, Non-gradient Autonomous Dissipative Structural Systems

Abstract

A general analysis for non-linear, non-gradient, multiple-parameter structural systems with or without damping described by autonomous ordinary differential equations is comprehensively presented. Attention is focused on bifurcational systems under follower loading exhibiting a trivial stable fundamental path from which a post-divergence or post-oscillatory instability may occur. Conditions for establishing regions of existence of adjacent equilibria, critical and stability conditions as well as different types of bifurcations (with zero, double zero and pure imaginary eigenvalues), for a smooth variation of the control parameter, are thoroughly explored and discussed. Global (dynamic) bifurcations (associated with stable limit cycles), irrelevant to any characteristic properties of the Jacobian eigenvalues, are discovered in a certain small neighborhood of a compound branching. It seems that such global bifurcations with trajectories passing through the saddles of the trivial path are due to the interaction of the first and second buckling modes which occur in the aforementioned small region of adjacent equilibria. Loading discontinuity phenomena for values of the control parameter defining the foregoing compound branching are detected. The effect of damping and other findings based on local (classical) analyses are compared with the results of this nonlinear analysis, and serious discrepancies are observed showing that a precise modelling must be generic; including both non-linearities and damping.The analysis is illustrated by using as a model Ziegler's classical model, the response of which is fully assessed.