A double saddle-node bifurcation theorem
نویسندگان
چکیده
We consider an abstract equation F (λ, u) = 0 with one parameter λ, where F ∈ C(R × X,Y ), p ≥ 2 is a nonlinear differentiable mapping, and X,Y are Banach spaces. We apply Lyapunov-Schmidt procedure and Morse Lemma to obtain a “double” saddle-node bifurcation theorem near a degenerate point with a two-dimensional kernel. It is shown that the solution set of the equation is the union of two parabola-like curves with same vertex, and it is interesting that the two curves can be on the different sides of bifurcation point.
منابع مشابه
Jakobson’s Theorem near saddle-node bifurcations
We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. It was previously shown that for a parameter set of positive Lebesgue density at the bifurcation, the maps possess attracting periodic orbits of high period. We show that there is also a parameter set of positive density at the bifurcation, for which the maps exhibit abs...
متن کاملIntermittency and Jakobson’s theorem near saddle-node bifurcations
We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. We show that there is a parameter set of positive but not full Lebesgue density at the bifurcation, for which the maps exhibit absolutely continuous invariant measures which are supported on the largest possible interval. We prove that these measures converge weakly to a...
متن کاملPredicting Non-Stationary and Stochastic Activation of Saddle-Node Bifurcation
Accurately predicting the onset of large behavioral deviations associated with saddlenode bifurcations is imperative in a broad range of sciences and for a wide variety of purposes, including ecological assessment, signal amplification, and microscale mass sensing. In many such practices, noise and non-stationarity are unavoidable and everpresent influences. As a result, it is critical to simul...
متن کاملBifurcation and Stability of Two-dimensional Double-diffusive Convection
In this article, we present a bifurcation and stability analysis on the double-diffusive convection. The main objective is to study 1) the mechanism of the saddle-node bifurcation and hysteresis for the problem, 2) the formation, stability and transitions of the typical convection structures, and 3) the stability of solutions. It is proved in particular that there are two different types of tra...
متن کاملThe Geometry of Quadratic Polynomial Differential Systems with a Finite and an Infinite Saddle-Node (A, B)
Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert’s 16th problem [Hilbert, 1900, Hilbert, 1902], are still open for this family. Our goal is to make a global study of the family QsnSN o...
متن کامل