A degenerate singularity generating geometric Lorenz attractors

A degenerate vector field singularity in R3 can generate a geometric Lorenz attractor in an arbitrarily small unfolding of it. This enables us to detect Lorenz-like chaos in some families of vector fields, merely by performing normal form calculations of order 3.

System Parameter Adaptive Control of a Spring Supported Truss Member

Structures that are described by bifurcating static and dynamic systems can deform drastically when parameters are varied so that the system passes through a degenerate singularity. An important nonlinear control problem for adaptive structures, in which the load is a bifurcation parameter, is to return the structure to its undeformed position by controlling a structure parameter other than the load. A simple structure whose behavior undergoes bifurcation as the load varies is a spring and pin supported rigid truss member. The critical point of its Hamiltonian pitchfork bifurcation corresponds to the cusp catastrophe degenerate singularity of its potential function. A diffeomorphic coordinate change reduces the potential to its four-jet, which determines the dynamic behavior. For this four-jet, under a fixed load greater than the critical value, two dynamic control schemes are proposed to return the system to its undeformed configuration. One strategy, which is robust and global and which requires damping, varies a structural parameter at a constant rate to shift the critical bifurcation value in state-control space. A slower and less robust method is a closed-loop feedback control, which stabilizes a center manifold approximation to the dynamic system at the critical point without using damping.