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An attraction result and an index theorem for continuous flows on ℝ n × [ 0 , ∞ )

Klaudiusz Wójcik (1997)

Annales Polonici Mathematici

We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on E = ℝ n + 1 for which ∂ E = ℝ n × 0 is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on ℝ n × [ 0 , ∞ ) .

Analysis of singularities and of integrability of ODE's by algorithms of Power Geometry

Alexander D. Bruno (2011)

Banach Center Publications

Here we present basic ideas and algorithms of Power Geometry and give a survey of some of its applications. In Section 2, we consider one generic ordinary differential equation and demonstrate how to find asymptotic forms and asymptotic expansions of its solutions. In Section 3, we demonstrate how to find expansions of solutions to Painlevé equations by this method, and we analyze singularities of plane oscillations of a satellite on an elliptic orbit. In Section 4, we consider the problem of local...

Applications of the Euler characteristic in bifurcation theory.

Slawomir Rybicki (1991)

Publicacions Matemàtiques

Let f: Rn x Rn → Rn be a continuous map such that f(0,λ) = 0 for all λ ∈ Rk. In this article we formulate, in terms of the Euler characteristic of algebraic sets, sufficient conditions for the existence of bifurcation points of the equation f(x,λ) = 0. Moreover we apply these results in bifurcation theory to ordinary differential equations. It is worth to point out that in the last paragraph we show how to verify, by computer, the assumptions of the theorems of this paper.

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