Fractal sets attached to homogeneous quadratic maps in two variables
The fractal set attached to the iteration of a “generic” homogeneous quadratic map from the plane to itself is studied and depicted, by using a two-parametric family of normal forms obtained from the theory of invariants of symmetric bilinear maps F:R×R→R under the full linear group of the plane. Wh...
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| Published in: | Physica. D Vol. 245; no. 1; pp. 8 - 18 |
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Elsevier B.V
15-02-2013
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| Abstract | The fractal set attached to the iteration of a “generic” homogeneous quadratic map from the plane to itself is studied and depicted, by using a two-parametric family of normal forms obtained from the theory of invariants of symmetric bilinear maps F:R×R→R under the full linear group of the plane. While invariant theory classifies maps F:C×C→C on the complex plane, we confine ourselves to consider maps on the real plane, in order to include the results obtained from the theory of two-dimensional discrete dynamical systems. A discrete number of “topological types” for such fractals is conjectured to exist.
► Fractal sets attached to “generic” homogeneous quadratic maps are studied. ► Using invariant theory, two invariants give rise to only six “normal forms”. ► Normal forms partition the plane of invariants into eleven connected regions. ► Comparison of “topological types” for pairs of fractals, per region. ► Conjecture: only a finite and small number of distinct topological types exist. |
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| AbstractList | The fractal set attached to the iteration of a “generic” homogeneous quadratic map from the plane to itself is studied and depicted, by using a two-parametric family of normal forms obtained from the theory of invariants of symmetric bilinear maps F:R×R→R under the full linear group of the plane. While invariant theory classifies maps F:C×C→C on the complex plane, we confine ourselves to consider maps on the real plane, in order to include the results obtained from the theory of two-dimensional discrete dynamical systems. A discrete number of “topological types” for such fractals is conjectured to exist.
► Fractal sets attached to “generic” homogeneous quadratic maps are studied. ► Using invariant theory, two invariants give rise to only six “normal forms”. ► Normal forms partition the plane of invariants into eleven connected regions. ► Comparison of “topological types” for pairs of fractals, per region. ► Conjecture: only a finite and small number of distinct topological types exist. |
| Author | Muñoz Masqué, J. Hernández Encinas, L. Durán Díaz, R. |
| Author_xml | – sequence: 1 givenname: R. surname: Durán Díaz fullname: Durán Díaz, R. email: raul.duran@uah.es organization: Departamento de Automática, Universidad de Alcalá, Ctra. Madrid–Barcelona, km. 33,600, E-28871 Alcalá de Henares, Spain – sequence: 2 givenname: L. surname: Hernández Encinas fullname: Hernández Encinas, L. email: luis@iec.csic.es organization: Instituto de Seguridad de la Información, CSIC, C/Serrano 144, E-28006 Madrid, Spain – sequence: 3 givenname: J. surname: Muñoz Masqué fullname: Muñoz Masqué, J. email: jaime@iec.csic.es organization: Instituto de Seguridad de la Información, CSIC, C/Serrano 144, E-28006 Madrid, Spain |
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| Keywords | Fractal set Bilinear symmetric map Discrete quadratic dynamic systems Invariant functions |
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Graph. doi: 10.1016/0097-8493(89)90071-X contributor: fullname: Norton – start-page: 58 year: 2010 ident: 10.1016/j.physd.2012.11.002_br000095 article-title: Superior tricorns and multicorns contributor: fullname: Rani |
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| SubjectTerms | Bilinear symmetric map Discrete quadratic dynamic systems Fractal set Invariant functions |
| Title | Fractal sets attached to homogeneous quadratic maps in two variables |
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