Related papers: The isolated saddle-node bifurcation occurring ins…
This paper presents results concerning bifurcations of 2D piecewise-smooth dynamical systems governed by vector fields. Generic three-parameter families of a class of Non-Smooth Vector Fields are studied and the bifurcation diagrams are…
In this paper we study conditions for the existence of the close orbit in piecewise continuous Matsumoto-Chua System. Our interest are in a pseudo saddle-node bifurcation in the first return application. In this bifurcation the close orbit…
In this paper, we are concerned with studying the existence of invariant complex manifolds of two-dimensional holomorphic systems. From the geometric singular perturbation theory we know that if a slow-fast system has associated a normally…
We study the renormalization operator of circle homeomorphisms with a break point and show that it possesses a hyperbolic horseshoe attractor.
We analyse three codimension-two bifurcations occurring in nonsmooth systems, when a non-hyperbolic cycle (fold, flip, and Neimark-Sacker cases, both in continuous- and discrete-time) interacts with one of the discontinuity boundaries…
A family of periodic perturbations of an attracting robust heteroclinic cycle defined on the two-sphere is studied by reducing the analysis to that of a one-parameter family of maps on a circle. The set of zeros of the family forms a…
This work is devoted to the study of global connections between typical generic singularities, named $T$-singularities, in piecewise smooth dynamical systems. Such a singularity presents the so-called nonsmooth diabolo, which consists on a…
A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is 3-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the…
In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddle-node bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we…
A saddle-node bifurcation cascade is studied in the logistic equation, whose bifurcation points follow an expression formally identical to the one given by Feigenbaum for period doubling cascade. The Feigenbaum equation is generalized…
In this paper are provided some sufficient conditions for a non autonomous scalar differential equation to have saddle node, transcritical and pitchfork bifurcations using higher order derivatives.
We describe all possible topological structures of typical one-parameter bifurcations of gradient flows on the 2-sphere with holes in the case that the number of singular point of flows is at most six. To describe structures, we separatrix…
We consider diffeomorphisms $f$ with heterodimensional cycles of co-index two, associated with saddles $P$ and $Q$ having unstable indices $\ell$ and $\ell+2$, respectively. In a partially hyperbolic setting, where a two-dimensional center…
Random diffeomorphisms with bounded absolutely continuous noise are known to possess a finite number of stationary measures. We discuss dependence of stationary measures on an auxiliary parameter, thus describing bifurcations of families of…
The saddle-node bifurcation on an invariant circle (SNIC) is one of the codimension-one routes to creation or destruction of a periodic orbit in a continuous-time dynamical system. It governs the transition from resting behaviour to…
In dimension three and under certain regularity assumptions, we construct a renormalisation scheme at the heterodimensional tangency of a non-transverse heterodimensional cycle associated with a pair of saddle-foci whose limit dynamic is a…
Singular Hopf bifurcation occurs in generic families of vector-fields with two slow variables and one fast variable. Normal forms for this bifurcation depend upon several parameters, and the dynamics displayed by the normal forms is…
We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodal maps (with arbitrary combinatorics), in any even degree $d$. We then conclude that orbits of renormalization are asymptotic to…
Taking advantage of the recently developed L-ALE framework [Sierra-Ausin \textit{et al.}, Phys. Rev. Fluids {\bf{7}}, 113603 (2022)], we characterize the linear dynamics of an incompressible gas bubble immersed in a biaxial straining flow.…
We study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of…