Mike R. Jeffrey

The Two-Fold Singularity of Discontinuous Vector Fields ∗

When a vector field in

The Geometry of Generic Sliding Bifurcations

\mathbb{R}^3

Nondeterministic Chaos, and the Two-fold Singularity in Piecewise Smooth Flows ∗

is discontinuous on a smooth codimension one surface, it may be simultaneously tangent to both sides of the surface at generic isolated points (singularities). For a piecewise-smooth dynamical system governed by the vector field, we show that the local dynamics depends on a single quantity—the jump in direction of the vector field through the singularity. This quantity controls a bifurcation, in which the initially repelling singularity becomes the apex of a pair of parabolic invariant surfaces. The surfaces are smooth except where they intersect the discontinuity surface, and they divide local space into regions of attraction to, and repulsion from, the singularity.

Teixeira singularities in 3D switched feedback control systems

Using the singularity theory of scalar functions, we derive a classification of sliding bifurcations in piecewise-smooth flows. These are global bifurcations which occur when distinguished orbits become tangent to surfaces of discontinuity, called switching manifolds. The key idea of the paper is to attribute sliding bifurcations to singularities in the manifolds projection along the flow, namely, to points where the projection contains folds, cusps, and two-folds (saddles and bowls). From the possible local configurations of orbits we obtain sliding bifurcations. In this way we derive a complete classification of generic one-parameter sliding bifurcations at a smooth codimension one switching manifold in

Canards and curvature: nonsmooth approximation by pinching

n

Dynamics at a switching intersection: hierarchy, isonomy, and multiple-sliding

dimensions for

Canards and curvature: the ‘smallness of ε’ in slow–fast dynamics

n\ge3

Regularization of hidden dynamics in piecewise smooth flows

. We uncover previously unknown sliding bifurcations, all of which are catastrophic in nature. We also describe how the method can be extended to sliding bifurcations of codimension two or higher.

Structural stability of the two-fold singularity

A vector field is piecewise smooth if its value jumps across a hypersurface, and a two-fold singularity is a point where the flow is tangent to the hypersurface from both sides. Two-folds are generic in piecewise smooth systems of three or more dimensions. We derive the local dynamics of all possible two-folds in three dimensions, including nonlinear effects around certain bifurcations, finding that they admit a flow exhibiting chaotic but nondeterministic dynamics. In cases where the flow passes through the two-fold, upon reaching the singularity it is unique in neither forward nor backward time, meaning the causal link between inward and outward dynamics is severed. In one scenario this occurs recurrently. The resulting flow makes repeated, but nonperiodic, excursions from the singularity, whose path and amplitude is not determined by previous excursions. We show that this behavior is robust and has many of the properties associated with chaos. Local geometry reveals that the chaotic behavior can be eliminated by varying a single parameter: the angular jump of the vector field across the two-fold.

Two-folds in nonsmooth dynamical systems

This paper is concerned with the analysis of a singularity that can occur in three-dimensional discontinuous feedback control systems. The singularity is the two-fold — a tangency of orbits to both sides of a switching manifold. Particular attention is placed on the Teixeira singularity, which previous literature suggests as a mechanism for dynamical transitions in this class of systems. We show that such a singularity cannot occur in classical single-input single-output systems in the Lur’e form. It is, however, a potentially dangerous phenomenon in multiple-input multiple-output switched control systems. The theoretical derivation is illustrated by means of a representative example.