Related papers: One-dimensional Piecewise Smooth Rational Degree M…

We study the presence in the Lozi map of a type of abrupt order-to-order and order-to-chaos transitions which are mediated by an attractor made of a continuum of neutrally stable limit cycles, all with the same period.

We derive a normal form for a near-integrable, four-dimensional symplectic map with a fold or cusp singularity in its frequency mapping. The normal form is obtained for when the frequency is near a resonance and the mapping is approximately…

This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in circle maps and quasi-contractions and provide paths through this literature to prove sufficient conditions for the…

Piecewise smooth maps are known to exhibit a wide range of dynamical features including numerous types of periodic orbits. Predicting regions in parameter space where such periodic orbits might occur and determining their stability is…

Let $\Sigma$ be a compact oriented surface and $N$ a compact K\"ahler manifold with nonnegative holomorphic bisectional curvature. For a solution of harmonic map flow starting from an almost-holomorphic map $\Sigma \to N$ (in the energy…

The border-collision normal form is a piecewise-linear family of continuous maps that describe the dynamics near border-collision bifurcations. Most prior studies assume each piece of the normal form is invertible, as is generic from an…

We prove strong statistical stability of a large class of one-dimensional maps which may have an arbitrary finite number of discontinuities and of non-degenerate critical points and/or singular points with infinite derivative, and satisfy…

We study bifurcations of symmetric elliptic fixed points in the case of \emph{p}:\emph{q} resonances with odd $q\geq 3$. We consider the case where the initial area-preserving map $\bar z =\lambda z + Q(z,z^*)$ possesses the central…

We completely classify the orientable infinite-type surfaces $S$ such that $\operatorname{PMap}(S)$, the pure mapping class group, has automatic continuity. This classification includes surfaces with noncompact boundary. In the case of…

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…

We define two conformal structures on $S^1$ which give rise to a different view of the affine curvature flow and a new curvature flow, the ``$Q$-curvature flow". The steady state of these flows are studied. More specifically, we prove four…

Let M be a smooth connected compact surface, P be either the real line R^1 or the circle S^1, and f:M-->P be a smooth mapping. In a previous series of papers for the case when f is a Morse map the author calculated the homotopy types of…

We demonstrate with a minimal example that in Filippov systems (dynamical systems governed by discontinuous but piecewise smooth vector fields) stable periodic motion with sliding is not robust with respect to stable singular perturbations.…

We solve a problem on a construction of a separately continuous mapping with the given diagonal, which is the pointwise limit of a sequence of continuous mappings valued in an equiconnected space. We construct an example of a closed-valued…

We study a family of diffusively coupled chaotic maps on periodic d-dimensional square lattices. Even and odd sub-lattices are updated alternately, introducing an effective delay. As the coupling strength is increased, the system undergoes…

Bifurcation analysis collects techniques for characterizing the dependence of certain classes of solutions of a dynamical system on variations in problem parameters. Common solution classes of interest include equilibria and periodic…

This paper is concerned with the local bifurcation analysis around typical singularities of piecewise smooth planar dynamical systems. Three-parameter families of a class of non$-$smooth vector fields are studied and the tridimensional…

For a continuous map on a topological graph containing a loop $S$ it is possible to define the degree (with respect to the loop $S$) and, for a map of degree $1$, rotation numbers. We study the rotation set of these maps and the periods of…

The local zero structure of a smooth map may qualitatively change, when the map is subjected to small perturbations. The changes may include births and/or deaths of zeros. The qualitative properties are defined as the invariances of an…

For piecewise-smooth differential systems, in this paper we focus on crossing limit cycles and sliding loops bifurcating from a grazing loop connecting one high multiplicity tangent point. For the low multiplicity cases considered in…