The regularized visible fold revisited

Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems

In this paper we study the Sotomayor-Teixeira regularization of a general visible fold singularity of a Filippov system. Extending Geometric Fenichel Theory beyond the fold with asymptotic methods, we determine there the deviation of the orbits of the regularized system from the generalized solutions of the Filippov one. This result is applied to the regularization of some global sliding bifurcations as the Grazing-Sliding of periodic orbits or the Sliding Homoclinic to a Saddle, as well as to some classical problems in dry friction. Roughly speaking, we see that locally, and also globally, th…

The number of limit cycles for regularized piecewise polynomial systems is unbounded

In this paper, we extend the slow divergence-integral from slow-fast systems, due to De Maesschalck, Dumortier and Roussarie, to smooth systems that limit onto piecewise smooth ones as $ε\rightarrow 0$. In slow-fast systems, the slow divergence-integral is an integral of the divergence along a canard cycle with respect to the slow time and it has proven very useful in obtaining good lower and upper bounds of limit cycles in planar polynomial systems. In this paper, our slow divergence-integral is based upon integration along a generalized canard cycle for a piecewise smooth two-fold bifurcatio…

正则分段光滑系统中的慢发散积分

Slow divergence integral in regularized piecewise smooth systems

该研究定义了正则平面分段光滑系统中沿滑动段的慢发散积分，证明其光滑等价不变性，并将其应用于VI₃型可见-不可见二重折叠模型，关联Minkowski维数与滑动极限环数目。

Fold singularities of nonsmooth and slow-fast dynamical systems -- equivalence through regularization

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 …

On the use of blow up to study regularizations of singularities of piecewise smooth dynamical systems in $\mathbb{R}^3$

In this paper we use the blow up method of Dumortier and Roussarie \cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to Krupa and Szmolyan \cite{krupa_extending_2001}, to study the regularization of singularities of piecewise smooth dynamical systems \cite{filippov1988differential} in $\mathbb R^3$. Using the regularization method of Sotomayor and Teixeira \cite{Sotomayor96}, first we demonstrate the power of our approach by considering the case of a fold line. We quickly recover a main result of Bonet and Seara \cite{reves_regularization_2014} in a simple manner. The…

Slow-fast normal forms arising from piecewise smooth vector fields

We studied piecewise smooth differential systems of the form $$\dot{z} = Z(z) = \dfrac{1 + \operatorname{sgn}(F)}{2}X(z) + \dfrac{1 - \operatorname{sgn}(F)}{2}Y(z),$$ where $F: \mathbb{R}^{n}\rightarrow \mathbb{R}$ is a smooth map having 0 as a regular value. We consider linear regularizations of the vector field $Z$ given by $$\dot{z}= Z_{\varepsilon}(z) = \dfrac{1 + φ(F/\varepsilon)}{2}X(z) +\dfrac{1 - φ(F /\varepsilon)}{2}Y(z),$$where $φ$ is a transition function (not necessarily monotonic) and nonlinear regularizations of the vector field $Z$ whose transition function is monotonic. It is a…