DCDS-B
Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations
Antonio Pumariño José Ángel Rodríguez Joan Carles Tatjer Enrique Vigil
Discrete & Continuous Dynamical Systems - B 2014, 19(2): 523-541 doi: 10.3934/dcdsb.2014.19.523
For certain 3D-homoclinic tangencies where the unstable manifold of the saddle point involved in the homoclinic tangency has dimension two, many different strange attractors have been numerically observed for the corresponding family of limit return maps. Moreover, for some special value of the parameter, the respective limit return map is conjugate to what was called bidimensional tent map. This piecewise affine map is an example of what we call now Expanding Baker Map, and the main objective of this paper is to show how many of the different attractors exhibited for the limit return maps resemble the ones observed for Expanding Baker Maps.
keywords: Expanding Baker Maps. strange attractors Limit return maps
DCDS
On the double pendulum: An example of double resonant situations
Antonio Pumariño Claudia Valls
Discrete & Continuous Dynamical Systems - A 2004, 11(2&3): 413-448 doi: 10.3934/dcds.2004.11.413
As a model of double resonant situations, we study fast periodic perturbations of a double pendulum. The associated dynamical system presents periodic orbits whose invariant manifolds split under the perturbation. The main purpose of this paper is to analytically show that this splitting is given, in first order, by the Melnikov function and give a lower bound for such splitting in terms of the perturbative parameter. Many results used in "simple pendulum cases" have to be adapted in order to give a description of the intricate dynamics exhibited by these periodic perturbations.
keywords: A double pendulum invariant manifolds by the Melnikov function.
DCDS-B
Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphisms
Antonio Pumariño Joan Carles Tatjer
Discrete & Continuous Dynamical Systems - B 2007, 8(4): 971-1005 doi: 10.3934/dcdsb.2007.8.971
We numerically analyse different kinds of one-dimensional and two-dimensional attractors for the limit return map associated to the unfolding of homoclinic tangencies for a large class of three-dimensional dissipative diffeomorphisms. Besides describing the topological properties of these attractors, we often numerically compute their Lyapunov exponents in order to clarify where two-dimensional strange attractors can show up in the parameter space. Hence, we are specially interested in the case in which the unstable manifold of the periodic saddle taking part in the homoclinic tangency has dimension two.
keywords: Lyapunov exponents. Invariant curves strange attractors
DCDS
Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors
Antonio Pumariño José Ángel Rodríguez Enrique Vigil
Discrete & Continuous Dynamical Systems - A 2018, 38(2): 941-966 doi: 10.3934/dcds.2018040

For a two parameter family of two-dimensional piecewise linear maps and for every natural number $n$, we prove not only the existence of intervals of parameters for which the respective maps are n times renormalizable but also we show the existence of intervals of parameters where the coexistence of at least $2^n$ strange attractors takes place. This family of maps contains the two-dimensional extension of the classical one-dimensional family of tent maps.

keywords: Piecewise linear maps renormalization strange attractor tent maps
DCDS
Renormalizable Expanding Baker Maps: Coexistence of strange attractors
Antonio Pumariño José Ángel Rodríguez Enrique Vigil
Discrete & Continuous Dynamical Systems - A 2017, 37(3): 1651-1678 doi: 10.3934/dcds.2017068

We introduce the concept of Expanding Baker Maps and renormalizable Expanding Baker Maps in a two-dimensional scenario. For a one-parameter family of Expanding Baker Maps we prove the existence of an interval of parameters for which the respective transformation is renormalizable. Moreover, we show the existence of intervals of parameters for which coexistence of strange attractors takes place.

keywords: Piecewise linear maps renormalization strange attractor
DCDS-B
Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps
Antonio Pumariño José Ángel Rodríguez Enrique Vigil
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-14 doi: 10.3934/dcdsb.2018201

We characterize the attractors for a two-parameter class of two-dimensional piecewise affine maps. These attractors are strange attractors, probably having finitely many pieces, and coincide with the support of an ergodic absolutely invariant probability measure. Moreover, we demonstrate that every compact invariant set with non-empty interior contains one of these attractors. We also prove the existence, for each natural number $ n, $ of an open set of parameters in which the respective transformation exhibits at least $ 2^n $ non connected two-dimensional strange attractors each one of them formed by $ 4^n $ pieces.

keywords: Piecewise affine maps strange attractors invariant measures

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