Search results for " 34C05"
showing 3 items of 3 documents
Multiplicity of fixed points and growth of ε-neighborhoods of orbits
2012
We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on epsilon of the length of epsilon-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered before (Elezovic, Zubrinic, Zupanovic) in the differentiable case, and related to the box dimension of the orbit. Here, we generalize these results to non-differentiable cases introducing a new notion of critical Minkowski order. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With the new definition, we recover the relationship between multiplicity o…
Infinite orbit depth and length of Melnikov functions
2019
Abstract In this paper we study polynomial Hamiltonian systems d F = 0 in the plane and their small perturbations: d F + ϵ ω = 0 . The first nonzero Melnikov function M μ = M μ ( F , γ , ω ) of the Poincare map along a loop γ of d F = 0 is given by an iterated integral [3] . In [7] , we bounded the length of the iterated integral M μ by a geometric number k = k ( F , γ ) which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system F and its orbit γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations d F + ϵ ω with arbitrary high length first nonzero Melnikov function M μ along…
Some Applications of the Poincaré-Bendixson Theorem
2021
We consider a C 1 vector field X defined on an open subset U of the plane, with compact closure. If X has no singular points and if U is simply connected, a weak version of the Poincaré-Bendixson Theorem says that the limit sets of X in U are empty but that one can defined non empty extended limit sets contained into the boundary of U. We give an elementary proof of this result, independent of the classical Poincaré-Bendixson Theorem. A trapping triangle T based at p, for a C 1 vector field X defined on an open subset U of the plane, is a topological triangle with a corner at a point p located on the boundary ∂U and a good control of the tranversality of X along the sides. The principal app…