Stability of Real Parametric Polynomial Discrete Dynamical Systems
We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameter λ and generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to real mth degre...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2015/680970 |
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| author | Fermin Franco-Medrano Francisco J. Solis |
| author_facet | Fermin Franco-Medrano Francisco J. Solis |
| author_sort | Fermin Franco-Medrano |
| collection | DOAJ |
| description | We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameter λ and generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to real mth degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept of canonical polynomial maps which are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termed Product Position Function for a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates and even when chaos arises, as it passes through what we have termed stability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three. |
| format | Article |
| id | doaj-art-8206b4daa4ea48ee81030e03fab82065 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-8206b4daa4ea48ee81030e03fab820652025-02-03T01:31:39ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2015-01-01201510.1155/2015/680970680970Stability of Real Parametric Polynomial Discrete Dynamical SystemsFermin Franco-Medrano0Francisco J. Solis1Applied Mathematics, CIMAT, 36240 Guanajuato, GTO, MexicoApplied Mathematics, CIMAT, 36240 Guanajuato, GTO, MexicoWe extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameter λ and generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to real mth degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept of canonical polynomial maps which are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termed Product Position Function for a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates and even when chaos arises, as it passes through what we have termed stability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.http://dx.doi.org/10.1155/2015/680970 |
| spellingShingle | Fermin Franco-Medrano Francisco J. Solis Stability of Real Parametric Polynomial Discrete Dynamical Systems Discrete Dynamics in Nature and Society |
| title | Stability of Real Parametric Polynomial Discrete Dynamical Systems |
| title_full | Stability of Real Parametric Polynomial Discrete Dynamical Systems |
| title_fullStr | Stability of Real Parametric Polynomial Discrete Dynamical Systems |
| title_full_unstemmed | Stability of Real Parametric Polynomial Discrete Dynamical Systems |
| title_short | Stability of Real Parametric Polynomial Discrete Dynamical Systems |
| title_sort | stability of real parametric polynomial discrete dynamical systems |
| url | http://dx.doi.org/10.1155/2015/680970 |
| work_keys_str_mv | AT ferminfrancomedrano stabilityofrealparametricpolynomialdiscretedynamicalsystems AT franciscojsolis stabilityofrealparametricpolynomialdiscretedynamicalsystems |