Dynamics at a Switching Intersection: Hierarchy, Isonomy, and Multiple Sliding

Abstract

If a set of ordinary differential equations is discontinuous along some threshold, solutions can be found that are continuous, if sometimes multi-valued. We show the extent to which unique solutions can be found in general cases when the threshold takes the form of finitely many intersecting manifolds. If the intersections are transversal, finitely many solutions can be found that slide along the threshold. They are obtained by a hierarchical application of convex combinations to form a differential inclusion. The system chooses between these solutions by means of an instantaneous dummy system. No assumptions on attractivity are required and all switches are treated equally, so the standard 'Filippov' method is extended to intersections of discontinuity manifolds in the most natural way possible. The corresponding result in the setting of equivalent control is also given, allowing more general systems than typical linear control forms to be solved.

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

“…The set-valued vector field in (5) contains vector field values that are dynamically irrelevant in the sense that the flow cannot follow them for any non-vanishing interval of time. Those values the flow can follow may be found by re-writing the vector as a canopy combination [20] of the values of f in the neighbourhood of a point on the switching surface,…”

“…These are sliding modes (an extension of Filippov's sliding modes [14,20] for r = 1). The values of the λ i 's corresponding to sliding modes are then given by…”

Hidden Bifurcations and Attractors in Nonsmooth Dynamical Systems

“…We shall also apply them to systems with multiple switches, for which they are easily extended as follows. At a point where r switching surfaces intersect, given by h 1 = h 2 = ... = h r = 0, the combination (3) generalises (see [11]) tȯ…”

“…This example is taken from a model of protein product concentrations x i in a two gene regulatory system [4,24], where the switches come in the form of Hill functions When x 1 = 0 and x 2 = 0, the system (33) is smooth and easily solved. We can apply the transition dynamics (11) to each of the switching thresholds x 1 = 0 and x 2 = 0 independently. We can work in terms of either λ i or Z i , let us choose the former.…”

Solution Continuation on a Discontinuity Set

“…Доопределение скорости скольжения в усредненной интерпретации нетрудно рас-пространить на случай x ∈ R n , s ∈ R m , m < n. Множество V u (x, t) включает конеч-ное либо бесконечное число значений вектора u, из которых можно выбрать K ≤ 2 m значений, обеспечивающих выполнение условий притяжения (4).…”