Planar maps whose second iterate has a unique fixed point

Alarcón, Begoña; Gutiérrez, Carlos; Martínez-Alfaro, José

doi:10.1080/10236190701698155

Cited by 6 publications

(5 citation statements)

References 17 publications

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“…In the appendix of [7] it is shown that the eigenvalues of DF 4 (x, y) lie in the open unit disk, establishing 5). Statement 3) follows as a direct consequence of Corollary 2 in [2] and the same estimates on the eigenvalues.…”

Section: Example With An Orbit Of Periodmentioning

confidence: 67%

A local but not global attractor for a Z_n-symmetric map

Alarcón¹,

Castro²,

Labouriau³

2012

jsing

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Abstract. There are many tools for studying local dynamics. An important problem is how this information can be used to obtain global information. We present examples for which local stability does not carry on globally. To this purpose we construct, for any natural n ≥ 2, planar maps whose symmetry group is Z n having a local attractor that is not a global attractor. The construction starts from an example with symmetry group Z 4 . We show that although this example has codimension 3 as a Z 4 -symmetric map-germ, its relevant dynamic properties are shared by two 1-parameter families in its universal unfolding. The same construction can be applied to obtain examples that are also dissipative. The symmetry of these maps forces them to have rational rotation numbers.

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Section: Example With An Orbit Of Periodmentioning

confidence: 67%

A local but not global attractor for a Z_n-symmetric map

Alarcón¹,

Castro²,

Labouriau³

2012

jsing

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“…In many contexts, this condition is simply known as f is dissipative, see for instance [3]. The fourth is not really a condition, it is a consequence of the previous ones, as shown by Corollary 2 of [2].…”

Section: Point At Infinity Is a Repeller;mentioning

confidence: 91%

Horseshoes for a Generalized Markus–Yamabe Example

Addas-Zanata

Gomes

2011

Qual. Theory Dyn. Syst.

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“…The issue of uniqueness of a fixed point has been addressed by Alarcón et al [4] who gave simple conditions for planar maps under which the origin is the unique fixed point.…”

Section: Introductionmentioning

confidence: 99%

Global dynamics for symmetric planar maps

Alarcón¹,

Castro²,

Labouriau³

2013

Discrete &Amp; Continuous Dynamical Systems - A

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We consider sufficient conditions to determine the global dynamics for equivariant maps of the plane with a unique fixed point which is also hyperbolic. When the map is equivariant under the action of a compact Lie group, it is possible to describe the local dynamics. In particular, if the group contains a reflection, there is a line invariant by the map. This allows us to use results based on the theory of free homeomorphisms to describe the global dynamical behaviour. We briefly discuss the case when reflections are absent, for which global dynamics may not follow from local dynamics near the unique fixed point.

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Planar maps whose second iterate has a unique fixed point

Abstract: Let ǫ > 0, F : R 2 → R 2 be a differentiable (not necessarily C 1 ) map and Spec(F ) be the set of (complex) eigenvalues of the derivative DF p when p varies in R 2 .

Cited by 6 publications

References 17 publications

A local but not global attractor for a Z_n-symmetric map

A local but not global attractor for a Z_n-symmetric map

Horseshoes for a Generalized Markus–Yamabe Example

Global dynamics for symmetric planar maps

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