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A linear condition determining local or global existence for nonlinear problems

John Neuberger, John Neuberger, James Swift (2013)

Open Mathematics

Given a nonlinear autonomous system of ordinary or partial differential equations that has at least local existence and uniqueness, we offer a linear condition which is necessary and sufficient for existence to be global. This paper is largely concerned with numerically testing this condition. For larger systems, principals of computations are clear but actual implementation poses considerable challenges. We give examples for smaller systems and discuss challenges related to larger systems. This...

A note on the structure of quadratic Julia sets

Karsten Keller (1997)

Commentationes Mathematicae Universitatis Carolinae

In a series of papers, Bandt and the author have given a symbolic and topological description of locally connected quadratic Julia sets by use of special closed equivalence relations on the circle called Julia equivalences. These equivalence relations reflect the landing behaviour of external rays in the case of local connectivity, and do not apply completely if a Julia set is connected but fails to be locally connected. However, rational external rays land also in the general case. The present...

A topological invariant for pairs of maps

Marcelo Polezzi, Claudemir Aniz (2006)

Open Mathematics

In this paper we develop the notion of contact orders for pairs of continuous self-maps (f, g) from ℝn, showing that the set Con(f, g) of all possible contact orders between f and g is a topological invariant (we remark that Con(f, id) = Per(f)). As an interesting application of this concept, we give sufficient conditions for the graphs of two continuous self-maps from ℝ intersect each other. We also determine the ordering of the sets Con(f, 0) and Con(f, h), for h ∈ Hom(ℝ) such that f ∘ h = h ∘...

Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps

Feliks Przytycki (1994)

Fundamenta Mathematicae

We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if A is completely invariant (i.e. f - 1 ( A ) = A ), and if μ is an arbitrary f-invariant measure with positive Lyapunov exponents on ∂A, then μ-almost every point q ∈ ∂A is accessible along a curve from A. In fact, we prove the accessibility of every “good” q, i.e. one for which “small neigh bourhoods arrive at large scale” under iteration of f. This generalizes the...

An exotic flow on a compact surface

N. Markley, M. Vanderschoot (2000)

Colloquium Mathematicae

In 1988 Anosov [1] published the construction of an example of a flow (continuous real action) on a cylinder or annulus with a phase portrait strikingly different from our normal experience. It contains orbits whose ο m e g a -limit sets contain a non-periodic orbit along with a simple closed curve of fixed points, but these orbits do not wrap down on this simple closed curve in the usual way. In this paper we modify some of Anosov’s methods to construct a flow on a surface of genus 2 with equally striking...

Bounce trajectories in plane tubular domains

Roberto Peirone (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We state that in opportune tubular domains any two points are connected by a bounce trajectory and that there exist non-trivial periodic bounce trajectories.

Chaos made visual.

M. de Guzmán (1996)

Revista Matemática de la Universidad Complutense de Madrid

In this paper we show how the main properties of chaos can be fully visualized at the light of a very easy to handle object, the tent function. Although very concrete, this case is representative of a very large number of examples, with more or less the same properties.

Chaotic behaviour of the map x ↦ ω(x, f)

Emma D’Aniello, Timothy Steele (2014)

Open Mathematics

Let K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ω f: 2ℕ → K(2ℕ) defined as ω f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω f is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity of ω f and...

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