相关论文: Jacobson's Theorem near saddle-node bifurcations

A periodic perturbation generates a complicated dynamics close to separatrices and saddle points. We construct an asymptotic solution which is close to the separatrix for the unperturbed Duffing's oscillator over a long time. This solution…

We show a Wolff-Denjoy type theorem in the case of a one-parameter continuous semigroups of nonexpansive mappings in which there is a compact mapping. Using the notion of attractor we are also able to prove some specific properties directly…

A quadratic H\'enon map is an automorphism of $\C^2$ of the form $h:(x,y)\mapsto (\l^{1/2} (x^2+c)-\l y,x)$. It has a constant Jacobian equal to $\l$ and has two fixed points. If $\lambda$ is on the unit circle (one says $h$ is…

We revisit here the dynamics of an engineered dimer granular crystal under an external periodic drive in the presence of dissipation. Earlier findings included a saddle-node bifurcation, whose terminal point initiated the observation of…

We show that a one-dimensional differential equation depending on a parameter $\mu$ with a saddle-node bifurcation at $\mu =0$ can be modelled by an extended normal form $\dot y = \nu (\mu )-y^2+a(\mu )y^3$, where the functions $\nu$ and…

Let $I_1=[a_0,a_1),\ldots,I_{k}= [a_{k-1},a_k)$ be a partition of the interval $I=[0,1)$ into $k$ subintervals. Let $f:I\to I$ be a map such that each restriction $f|_{I_i}$ is an increasing Lipschitz contraction. We prove that any $f$…

We consider a dynamical system undergoing a saddle-node bifurcation with an explicitly time dependent parameter~$p(t)$. The combined dynamics can be considered as a dynamical systems where $p$ is a slowly evolving parameter. Here, we…

Non-smooth saddle-node bifurcations give rise to minimal sets of interesting geometry built of so-called strange non-chaotic attractors. We show that certain families of quasiperiodically driven logistic differential equations undergo a…

We investigate the evolution of families of periodic orbits in a bisymmetrical potential made up of a two-dimensional harmonic oscillator with only one quartic perturbing term, in a number of resonant cases. Our main objective is to compute…

One- and two-parameter families of flows in $R^3$ near an Andronov-Hopf bifurcation (AHB) are investigated in this work. We identify conditions on the global vector field, which yield a rich family of multimodal orbits passing close to a…

All possible orbital trajectories and their analytical expressions in the Schwarzschild metric are presented in a single complete map characterized by two dimensionless parameters. While three possible pairs of parameters with different…

Junctions and interfaces consisting of unconventional superconductors provide an excellent experimental playground to study exotic phenomena related to the phase of the order parameter. Not only the complex structure of unconventional order…

In the class of nonlinear one-parameter real maps we study those with bifurcation that exhibits period doubling cascade. The fixed points of such a map form a finite discrete real set with dimension (2^n)m, where m is the (odd) number of…

In this paper we study measurable dynamics for the widest reasonable class of smooth one dimensional maps. Three principle decompositions are described in this class : decomposition of the global measure-theoretical attractor into primitive…

We investigate the dynamics of maps of the real line whose behavior on an invariant interval is close to a rational rotation on the circle. We concentrate on a specific two-parameter family, describing the dynamics arising from models in…

In order to model the processes taking place in systems with Josephson contacts, a differential equation on a torus with three parameters is used. One of the parameters of the system can be considered small and the methods of the fast--slow…

We study the dynamics of generic unfoldings of saddle-node circle local diffeomorphisms from the measure theoretical point of view, obtaining statistical stability results for deterministic and random perturbations in these kind of…

We study a parametric family of piecewise rotations of the torus, in the limit in which the rotation number approaches the rational value 1/4. There is a region of positive measure where the discontinuity set becomes dense in the limit; we…

We study the dynamics of strongly dissipative H\'enon-like maps, around the first bifurcation parameter $a^*$ at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We prove that $a^*$ is a full…

Properties of spatially dependent relaxation oscillations near a SNIPER bifurcation are described. A SNIPER bifurcation creates a large-amplitude long-period periodic orbit via the annihilation of a pair of fixed points in a saddle-node…