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This paper consists in discussing some issues on generic local classification of typical singularities of $2D$ piecewise smooth vector fields when the switching set is an algebraic variety. The main focus is to obtain classification results…

Finding a non-sofic hyperbolic group will resolve two major problems in geometric group theory: Are there non sofic groups? Are there non residually finite hyperbolic groups? In this paper, we propose a new probabilistic approach to this…

For piecewise-linear maps the stable and unstable manifolds of hyperbolic periodic solutions are themselves piecewise-linear. Hence compact subsets of these manifolds can be represented using polytopes (i.e. polygons, in the case of…

Let G be a graph. It was proved that if G is a planar graph without {4, 6, 7}-cycles and without two 5-cycles sharing exactly one edge, then G 3-colorable. We observed that the proof of this result is not correct.

We investigate the standard stable manifold theorem in the context of a partially hyperbolic singu-larity of a vector field depending on a parameter. We prove some estimates on the size of the neighbourhood where the local stable manifold…

We consider $C^2$ vector fields in the three dimensional sphere with an attracting heteroclinic cycle between two periodic hyperbolic solutions with real Floquet multipliers. The proper basin of this attracting set exhibits historic…

One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular…

We study $C^1$-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices and prove that they exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets with…

We study planar piecewise quadratic differential systems of Kolmogorov type. Specifically, we consider systems with both coordinate axes invariant and with a separation line that is straight and distinct from the invariant axes. The main…

We present a necessary and sufficient condition for existence of a contractible, non-separating and noncontractible separating Hamiltonian cycle in the edge graph of polyhedral maps on surfaces. In particular, we show the existence of…

Using a quasi-linear version of Hodge theory, holomorphic vector bundles in a neighbourhood of a given polystable bundle on a compact Kaehler manifold are shown to be (poly)stable if and only if their corresponding classes are (poly)stable…

Heterodimensional cycles are heteroclinic cycles that connect periodic orbits whose unstable manifolds have different dimensions. This is a source of nonhyperbolic dynamics and unstable dimension variability. For smooth invertible maps…

Experiments observing the liquid surface in a vertically oscillating container have indicated that modeling the dynamics of such systems require maps that admit states at infinity. In this paper we investigate the bifurcations in such a…

We conjecture that the balanced complete bipartite graph $K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil}$ contains more cycles than any other $n$-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent…

We present a new solution for fundamental problems in nonlinear dynamical systems: finding, verifying, and stabilizing cycles. The solution we propose consists of a new control method based on mixing previous states of the system (or the…

In this paper, we apply the averaging method via Brouwer degree in a class of planar systems given by a linear center perturbed by a sum of continuous homogeneous vector fields, to study lower bounds for their number of limit cycles. Our…

Consider the family of graphs without $ k $ node-disjoint odd cycles, where $ k $ is a constant. Determining the complexity of the stable set problem for such graphs $ G $ is a long-standing problem. We give a polynomial-time algorithm for…

Consider two conditions on a graph: (1) each 5-cycle is not a subgraph of 5-wheel and does not share exactly one edge with 3-cycle, and (2) each 5-cycle is not adjacent to two 3-cycles and is not adjacent to a 4-cycle with chord. We show…

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 $\epsilon\rightarrow 0$. In slow-fast systems, the slow…

This paper investigates the multiplicity and the number of limit cycles for planar piecewise linear system divided into two regions by a straight line and each linear subsystem has a node. Through constructing Poincare half maps and a…