Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems

In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study

Bifurcation problems with octahedral symmetry

We analyse local bifurcation problems with octahedral symmetry using results from singularity theory. The thesis is split up into three sections. §1 comprises the bifurcation theory, and §3 contains a full singularity theory classification up to topological codimension one. The classification relies heavily upon new results about the recognition problem. These results are presented in §2 together with several examples drawn from equivariant bifurcation theory. These examples illustrate the new methods more clearly than the work in §3. In §1 we look at nondegenerate bifurcation problems equivariant with respect to the standard action of the octahedral group on Rᶟ. We find three branches of symmetry-breaking bifurcation corresponding to the three maximal isotropy subgroups of the symmetry group with one-dimensional fixed-point subspaces. Locally, one of these branches is never stable, but precisely one of the other branches is stable if and only if all three branches bifurcate supercritically. In §2 we simplify the recognition problem by decomposing the group of equivalences into a unipotent group and a group of matrices. Building upon results of Bruce, du Plessis & Wall, we show that in many cases the unipotent problem can be solved by just using linear algebra. We give a necessary and sufficient condition for this, namely that the tangent space be invariant under unipotent equivalence. In addition we develop methods for checking whether the tangent space is invariant. The classification theorem in §3 gives a list of seven normal forms together with recognition problem solutions and universal unfoldings. Certain anomalies arise when comparing these results with those in Si. We reconcile the anomalies by giving a qualitative classification in addition to the standard classification. An application to barium titanate crystals is considered briefly

Structural stability of the two-fold singularity

At a two-fold singularity, the velocity vector of a flow switches discontinuously across a codimension one switching manifold, between two directions that both lie tangent to the manifold. Particularly intricate dynamics arises when the local flow curves toward the switching manifold from both sides, a case referred to as the Teixeira singularity. The flow locally performs two different actions: it winds around the singularity by crossing repeatedly through, and passes through the singularity by sliding along, the switching manifold. The case when the number of rotations around the singularity is infinite has been analyzed in detail. Here we study the case when the flow makes a finite, but previously unknown, number of rotations around the singularity between incidents of sliding. We show that the solution is remarkably simple: the maximum and minimum numbers of rotations made anywhere in the flow differ only by one and increase incrementally with a single parameter -the angular jump in the flow direction across the switching manifold at the singularity

On Matching, and Even Rectifying, Dynamical Systems through Koopman Operator Eigenfunctions

Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of nonlinear dynamic behavior (e.g. through normal forms). In this paper we will argue that the use of the Koopman operator and its spectrum is particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven algorithm developments. We believe, and document through illustrative examples, that this can nontrivially extend the use and applicability of the Koopman spectral theoretical and computational machinery beyond modeling and prediction, towards what can be considered as a systematic discovery of "Cole-Hopf-type" transformations for dynamics.Comment: 34 pages, 10 figure

A NORMALLY ELLIPTIC HAMILTONIAN BIFURCATION

A universal local bifurcation analysis is presented of an autonomous Hamiltonian system around a certain equilibrium point. This central equilibrium has a double zero eigenvalue, the other eigenvalues being in general position. Main emphasis is given to the 2 degrees of freedom case where these other eigenvalues are purely imaginary. By normal form techniques and Singularity Theory unfoldings are obtained. having 'integrable' approximations related to the Elliptic and Hyperbolic Umbilic Catastrophes

A NORMALLY ELLIPTIC HAMILTONIAN BIFURCATION

A universal local bifurcation analysis is presented of an autonomous Hamiltonian system around a certain equilibrium point. This central equilibrium has a double zero eigenvalue, the other eigenvalues being in general position. Main emphasis is given to the 2 degrees of freedom case where these other eigenvalues are purely imaginary. By normal form techniques and Singularity Theory unfoldings are obtained. having 'integrable' approximations related to the Elliptic and Hyperbolic Umbilic Catastrophes