In this work, we study the dynamics of piecewise smooth systems on a codimension-2 transverse intersection of two codimension-1 discontinuity sets. The Filippov convention can be extended to such intersections, but this approach does not provide a unique sliding vector and, as opposed to the classical sliding vector-field on codimension-1 discontinuity manifolds, there is no agreed notion of stability in the codimension-2 context. In this paper, we perform a regularization of the piecewise smooth system, introducing two regularization functions and a small perturbation parameter. Then, based on singular perturbation theory, we define sliding and stability of sliding through a critical manifold of the singularly perturbed, regularized system. We show that this notion of sliding vector-field coincides with the Filippov one. The regularized system gives a parameterized surface (the canopy) independent of the regularization functions. This surface serves as our natural basis to derive new and simple geometric criteria on the existence, multiplicity and stability of the sliding flow, depending only on the smooth vector fields around the intersection. Interestingly, we are able to show that if there exist two sliding vector-fields then one is a saddle and the other is of focus/node/center type. This means that there is at most one stable sliding vector-field. We then investigate the effect of the choice of the regularization functions, and, using a blowup approach, we demonstrate the mechanisms through which sliding behavior can appear or disappear on the intersection and describe what consequences this has on the dynamics on the adjacent codimension-1 discontinuity sets. Finally, we show the existence of canard explosions of regularizations of PWS systems in $\mathbb R^3$ that depend on a single unfolding parameter.
Mathematical models involving switches --- in the form of differential equations with discontinuities --- can accomodate real-world non-idealities through perturbations by hysteresis, time-delay, discretization, and noise. These are used to model the processes associated with switching in electronic control, mechanical contact, predator-prey preferences, and genetic or cellular regulation. The effect of such perturbations on rapid switching dynamics about a single switch are somewhat benign: in the limit that the size of the perturbation goes to zero the dynamics is given by Filippov's sliding…
The second part of the Hilbert's sixteenth problem consists in determining the upper bound $\mathcal{H}(n)$ for the number of limit cycles that planar polynomial vector fields of degree $n$ can have. For $n\geq2$, it is still unknown whether $\mathcal{H}(n)$ is finite or not. The main achievements obtained so far establish lower bounds for $\mathcal{H}(n)$. Regarding asymptotic behavior, the best result says that $\mathcal{H}(n)$ grows as fast as $n^2\log(n)$. Better lower bounds for small values of $n$ are known in the research literature. In the recent paper "Some open problems in low dimens…
We consider a piecewise smooth system in the neighborhood of a co-dimension 2 discontinuity manifold $Σ$. Within the class of Filippov solutions, if $Σ$ is attractive, one should expect solution trajectories to slide on $Σ$. It is well known, however, that the classical Filippov convexification methodology is ambiguous on $Σ$. The situation is further complicated by the possibility that, regardless of how sliding on $Σ$ is taking place, during sliding motion a trajectory encounters so-called generic first order exit points, where $Σ$ ceases to be attractive. In this work, we attempt to underst…
Work on standard piecewise-smooth (PWS) dynamical systems, with codimension-1 discontinuity sets, relies on the Filippov framework, which does not always readily generalise to systems with higher codimension discontinuities. These higher order degeneracies occur in applications, with spatial Coulomb friction being the prominent example. In this work, we consider PWS systems with isolated codimension-2 discontinuity sets using regularization and blowup to study the dynamics. We present a general framework and give specific results for the local dynamics in a class of problems, and generalising …
In this paper we study the maximum number $N$ of limit cycles that can exhibit a planar piecewise linear differential system formed by two pieces separated by a straight line. More precisely, we prove that this maximum number satisfies $2\leq N \leq 3$ if one of the two linear differential systems has its equilibrium point on the straight line of discontinuity.
We study self-similarity problem for two classes of flows: (1) special flows over circle rotations and under roof functions with symmetric logarithmic singularities (2) special flows over interval exchange transformations and under roof functions which are of two types * piecewise constant with one additional discontinuity which is not a discontinuity of the IET; * piecewise linear over exchanged intervals with non-zero slope. We show that if $\{T^f_t\}_{t\in\mathbb R}$ is as in (1) then for a full measure set of rotations, and for every two distinct natural numbers $K$ and $L$, we have that $…
In this article we study the existence of limit cycles in families of piecewise smooth differential equations having the unit circle as discontinuity region. We consider families presenting singularities of center or saddle type, visible or invisible, as well as the case without singularities. We establish an upper bound for the number of limit cycles and give examples showing that the maximum number of limit cycles can be reached. We also discuss the existence of homoclinic cycles for such differential equations in the saddle-center case.
The two-fold singularity has played a significant role in our understanding of uniqueness and stability in piecewise smooth dynamical systems. When a vector field is discontinuous at some hypersurface, it can become tangent to that surface from one side or the other, and tangency from both sides creates a two-fold singularity. The local flow bears a superficial resemblance to so-called {\it folded singularities} in (smooth) slow-fast systems, which arise at the intersection of attractive and repelling branches of slow invariant manifolds, important in the local study of canards and mixed mode …
Switch-like behaviour in dynamical systems may be modelled by highly nonlinear functions, such as Hill functions or sigmoid functions, or alternatively by piecewise-smooth functions, such as step functions. Consistent modelling requires that piecewise-smooth and smooth dynamical systems have similar dynamics, but the conditions for such similarity are not well understood. Here we show that by smoothing out a piecewise-smooth system one may obtain dynamics that is inconsistent with the accepted wisdom --- so-called Filippov dynamics --- at a discontinuity, even in the piecewise-smooth limit. By…
Piecewise smooth dynamical systems make use of discontinuities to model switching between regions of smooth evolution. This introduces an ambiguity in prescribing dynamics at the discontinuity: should it be given by a limiting value on one side or other of the discontinuity, or a member of some set containing those values? One way to remove the ambiguity is to regularize the discontinuity, the most common being either to smooth out the discontinuity, or to introduce a hysteresis between switching in one direction or the other across the discontinuity. Here we show that the two can in general l…
The Filippov convention is widely used in the literature to define vector fields on a discontinuity set of piecewise-continuous vector fields. The aim of this paper is to give a sufficient and necessary condition for an interpolation scheme of piecewise-continuous vector fields to coincide with the Filippov convention. That is, we show that a map from a space of piecewise-continuous vector fields with two components to the space of vector fields coincides with the Filippov convention where the latter can be applied, if it is sufficiently well-behaved as a generalization of continuous vector fi…
In this paper, by using Picard-Fuchs equations and Chebyshev criterion, we study the bifurcate of limit cycles for quadratic Hamilton system $S^{(2)}$ and $S^{(3)}$: $\dot{x}= y+2axy+by^2$, $\dot{y}=-x+x^2-ay^2$ with $a\in(-\frac{1}{2},1)$, $b=(1-a)(1+2a)^{1/2}$ and $a=1$, $b=0$ respectively, under perturbations of piecewise smooth polynomials with degree $n$. The discontinuity is the line $y=0$. We bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles fo…
A two-fold singularity is a point on a discontinuity surface of a piecewise-smooth vector field at which the vector field is tangent to the surface on both sides. Due to the double tangency, forward evolution from a two-fold is typically ambiguous. This is an especially serious issue for two-folds that are reached by the forward orbits of a non-zero measure set of initial points. The purpose of this paper is to explore the concept of perturbing the vector field so that forward evolution is well-defined, and characterising the perturbed dynamics in the limit that the size of the perturbation te…
This paper studies the equivalence between differentiable and non-differentiable dynamics in Rn. Filippov's theory of discontinuous differential equations allows us to find flow solutions of dynamical systems whose vector fields undergo switches at thresholds in phase space. The canonical convex combination at the discontinuity is only the linear part of a nonlinear combination that more fully explores Filippov's most general problem: the differential inclusion. Here we show how recent work relating discontinuous systems to singular limits of continuous (or regularized) systems extends to nonl…
We study a class of discontinuous vector fields brought to our attention by multi-legged animal locomotion. Such vector fields arise not only in biomechanics, but also in robotics, neuroscience, and electrical engineering, to name a few domains of application. Under the conditions that (i) the vector field's discontinuities are locally confined to a finite number of smooth submanifolds and (ii) the vector field is transverse to these surfaces in an appropriate sense, we show that the vector field yields a well-defined flow that is Lipschitz continuous and piecewise-differentiable. This implies…
The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter $ε\rightarrow 0$. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this …
The {\it two-fold singularity} has played a significant role in our understanding of uniqueness and stability in piecewise smooth dynamical systems. When a vector field is discontinuous at some hypersurface, it can become tangent to that surface from one side or the other, and tangency from both sides creates a two-fold singularity. The flow this creates bears a superficial resemblance to so-called {\it folded singularities} in (smooth) slow-fast systems, which arise at the intersection of attractive and repelling branches of slow invariant manifolds, important in the local study of canards an…
In multiple time-scale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: they are slowly varying with respect to a control parameter, except for an exponentially small range of values where they grow extremely rapidly. This sudden growth, called a canard explosion, has been encountered in many applications ranging from chemistry to neuronal dynamics, aerospace engineering and eco…
A two-fold is a singular point on the discontinuity surface of a piecewise-smooth vector field, at which the vector field is tangent to the discontinuity surface on both sides. If an orbit passes through an invisible two-fold (also known as a Teixeira singularity) before settling to regular periodic motion, then the phase of that motion cannot be determined from initial conditions, and in the presence of small noise the asymptotic phase of a large number of sample solutions is highly random. In this paper we show how the probability distribution of the asymptotic phase depends on the global no…
In this work we consider piecewise smooth vector fields $X$ defined in $\R^n\setminus Σ$, where $Σ$ is a self-intersecting switching manifold. A double regularization of $X$ is a 2-parameter family of smooth vector fields $X_{\e.η}$, $\e,η>0,$ satisfying that $X_{\e,η}$ converges pointwise to $X$ on $\R^n\setminusΣ$, when $\e,η\rightarrow 0$. We define the sliding region on the non regular part of $Σ$ as a limit of invariant manifolds of $X_{\e.η}$. Since the double regularization provides a slow--fast system, the GSP-theory (geometric singular perturbation theory) is our main tool.
Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions
Mannheim Institute for Intelligent Systems in Medicine, Heidelberg University（海德堡大学曼海姆智能医学研究所）；Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University（海德堡大学跨学科科学计算中心）；Heidelberg Institute for Theoretical Studies (HITS), Heidelberg University（海德堡大学理论研究 institute）；Central Institute for Computer Engineering (ZITI), Heidelberg University（海德堡大学计算机工程中心）；CZS Heidelberg Initiative for Model-Based AI (MBAI), Heidelberg University（海德堡模型驱动人工智能倡议）
