Two-fold singularities in nonsmooth dynamics-Higher dimensional analogs.

When a system of ordinary differential equations is discontinuous along some threshold, its flow may become tangent to that threshold from one side or the other, creating a fold singularity, or from both sides simultaneously, creating a two-fold singularity. The classic two-fold exhibits intricate local dynamics and accumulating sequences of local bifurcations and is by now rather well understood, but it is just the simplest of an infinite hierarchy of two-folds and multi-folds in higher dimensions. These arise when a system is discontinuous along multiple intersecting thresholds, and the induced sliding flows on those thresholds become tangent to their intersections. We show here, surprisingly, that these higher dimensional analogs of the two-fold reduce to the equations of the classic two-fold, providing the first step into their study and a new tool to understand higher dimensional systems with discontinuities.

Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking.

The collapse of flows onto hypersurfaces where their vector fields are discontinuous creates highly robust states called sliding modes. The way flows exit from such sliding modes can lead to complex and interesting behaviour about which little is currently known. Here, we examine the basic mechanisms by which a flow exits from sliding, either along a switching surface or along the intersection of two switching surfaces, with a view to understanding sliding and exit when many switches are involved. On a single switching surface, exit occurs via tangency of the flow to the switching surface. Along an intersection of switches, exit can occur at a tangency with a lower codimension sliding flow, or by a spiralling of the flow that exhibits geometric divergence (infinite steps in finite time). Determinacy-breaking can occur where a singularity creates a set-valued flow in an otherwise deterministic system, and we resolve such dynamics as far as possible by blowing up the switching surface into a switching layer. We show preliminary simulations exploring the role of determinacy-breaking events as organizing centres of local and global dynamics.

Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems.

Switches in real systems take many forms, such as impacts, electronic relays, mitosis, and the implementation of decisions or control strategies. To understand what is lost, and what can be retained, when we model a switch as an instantaneous event, requires a consideration of so-called hidden terms. These are asymptotically vanishing outside the switch, but can be encoded in the form of nonlinear switching terms. A general expression for the switch can be developed in the form of a series of sigmoid functions. We review the key steps in extending Filippov's method of sliding modes to such systems. We show how even slight nonlinear effects can hugely alter the behaviour of an electronic control circuit, and lead to "hidden" attractors inside the switching surface.

Mathematical analysis of a model for the growth of the bovine corpus luteum.

The corpus luteum (CL) is an ovarian tissue that grows in the wound space created by follicular rupture. It produces the progesterone needed in the uterus to maintain pregnancy. Rapid growth of the CL and progesterone transport to the uterus require angiogenesis, the creation of new blood vessels from pre-existing ones, a process which is regulated by proteins that include fibroblast growth factor 2 (FGF2). In this paper we develop a system of time-dependent ordinary differential equations to model CL growth. The dependent variables represent FGF2, endothelial cells (ECs), luteal cells, and stromal cells (like pericytes), by assuming that the CL volume is a continuum of the three cell types. We assume that if the CL volume exceeds that of the ovulated follicle, then growth is inhibited. This threshold volume partitions the system dynamics into two regimes, so that the model may be classified as a Filippov (piecewise smooth) system. We show that normal CL growth requires an appropriate balance between the growth rates of luteal and stromal cells. We investigate how angiogenesis influences CL growth by considering how the system dynamics depend on the dimensionless EC proliferation rate, ρ5. We find that weak (low ρ5) or strong (high ρ5) angiogenesis leads to 'pathological' CL growth, since the loss of CL constituents compromises progesterone production or delivery. However, for intermediate values of ρ5, normal CL growth is predicted. The implications of these results for cow fertility are also discussed. For example, inadequate angiogenesis has been linked to infertility in dairy cows.

Nondeterminism in the limit of nonsmooth dynamics.

Discontinuous time derivatives are used to model threshold-dependent switching in such diverse applications as dry friction, electronic control, and biological growth. In a continuous flow, a discontinuous derivative can generate multiple outcomes from a single initial state. Here we show that well-defined solution sets exist for flows that become multivalued due to grazing a discontinuity. Loss of determinism is used to quantify dynamics in the limit of infinite sensitivity to initial conditions, then applied to the dynamics of a superconducting resonator and a negatively damped oscillator.

Nondeterministic dynamics of a mechanical system.

A mechanical system is presented exhibiting a nondeterministic singularity, that is, a point in an otherwise deterministic system where forward time trajectories become nonunique. A Coulomb friction force applies linear and angular forces to a wheel mounted on a turntable. In certain configurations, the friction force is not uniquely determined. When the dynamics evolves past the singularity and the mechanism slips, the future state becomes uncertain up to a set of possible values. For certain parameters, the system repeatedly returns to the singularity, giving recurrent yet unpredictable behavior that constitutes nondeterministic chaotic dynamics. The robustness of the phenomenon is such that we expect it to persist with more sophisticated friction models, manifesting as extreme sensitivity to initial conditions, and complex global dynamics attributable to a local loss of determinism in the limit of discontinuous friction.