Planar embeddings with a globally attracting fixed point

Alarcón, Begoña; Guíñez, Víctor; Gutiérrez, Carlos

doi:10.1016/j.na.2007.05.005

Cited by 14 publications

(29 citation statements)

References 11 publications

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“…If h(x) = Ax then h = r 2 with r(x) = Bx, det B < 0, as soon as the matrix A has two positive eigenvalues λ 1 = λ 2 . Condition (iii) is inspired by the work of Alarcón, Guíñez and Gutiérrez [1]. They introduced a similar assumption to get results on global asymptotic stability.…”

Section: Computing the Rotation Numbermentioning

confidence: 99%

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Attractors with vanishing rotation number

Ortega¹,

Portal²

2011

J. Eur. Math. Soc.

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Abstract. Given an orientation-preserving homeomorphism of the plane, a rotation number can be associated with each locally attracting fixed point. Assuming that the homeomorphism is dissipative and the rotation number vanishes we prove the existence of a second fixed point. The main tools in the proof are Carathéodory prime ends and fixed point index. The result is applicable to some concrete problems in the theory of periodic differential equations.

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Section: Computing the Rotation Numbermentioning

confidence: 99%

“…The proof in [1] is based on Brouwer's theory of fixed point free homeomorphisms of the plane. Next we present an alternative proof based on the theory of prime ends.…”

Section: Invariant Rays and Population Dynamicsmentioning

confidence: 99%

Attractors with vanishing rotation number

Ortega¹,

Portal²

2011

J. Eur. Math. Soc.

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“…(b) Fix( h) = {p} and there exists an arc I C S 2 with end points at p and oo such that h( I ) = r · The proof in [2] is based on Brouwer's theory of fixed point free homeomorphisms of the plane. Ortega and Ruiz del Portal give in [30] an alternative proof based on the theory of prime ends.…”

Section: Dynamics Of Plane Continuamentioning

confidence: 99%

Unstable manifold, Conley index and fixed points of flows

Barge

Sanjurjo

2014

Journal of Mathematical Analysis and Applications

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We study dynamical and topological properties of the unstable manifold of isolated invariant compacta of flows. We show that some parts of the unstable manifold admit sections carrying a considerable amount of information. These sections enable the construction of parallelizable structures which facilitate the study of the flow. From this fact, many nice consequences are derived, specially in the case of plane continua. For instance, we give an easy method of calculation of the Conley index provided we have some knowledge of the unstable manifold and, as a consequence, a relation between the Brouwer degree and the unstable manifold is established for smooth vector fields. We study the dynamics of non-saddle sets, properties of existence or non-existence of fixed points of flows and conditions under which attractors are fixed points, Morse decompositions, preservation of topological properties by continuation and classify the bifurcations taking place at a critical point.

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“…Later, other conditions have been added to condition I, like the one of having the infinite as a repeller, for trying to obtain GAS. Nevertheless, assuming also these additional conditions it turns out that it is possible to obtain dynamical systems for which the origin is not GAS, see [2]. In [3,6] there are examples of polynomial maps defined in R n , n ≥ 3, satisfying the condition and having unbounded orbits.…”

mentioning

confidence: 99%

“…It is already known that even for n = 2, condition I does not imply GAS of the fixed point for maps of the form (2). The counterexample presented in [7], based on a family introduced in [1], is given by the map…”

mentioning

confidence: 99%