Chaotic numerical instabilities arising in the transition from differential to difference nonlinear equations
For computational purposes, a numerical algorithm maps a differential equation into an often complex difference equation whose structure and stability depends on the scheme used. When considering nonlinear models, standard and nonstandard integration routines can act invasively and numerical chaotic...
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| Format: | Article |
| Language: | English |
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Wiley
2000-01-01
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| Series: | Discrete Dynamics in Nature and Society |
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| Online Access: | http://dx.doi.org/10.1155/S1026022600000029 |
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| _version_ | 1832545235155025920 |
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| author | Alicia Serfaty de Markus |
| author_facet | Alicia Serfaty de Markus |
| author_sort | Alicia Serfaty de Markus |
| collection | DOAJ |
| description | For computational purposes, a numerical algorithm maps a differential equation into an often complex difference equation whose structure and stability depends on the scheme used. When considering nonlinear models, standard and nonstandard integration routines can act invasively and numerical chaotic instabilities may arise. However, because nonstandard schemes offer a direct and generally simpler finite-difference representations, in this work nonstandard constructions were tested over three different systems: a photoconductor model, the Lorenz equations and the Van der Pol equations. Results showed that although some nonstandard constructions created a chaotic dynamics of their own, there was found a construction in every case that greatly reduced or successfully removed numerical chaotic instabilities. These improvements represent a valuable development to incorporate into more sophisticated algorithms. |
| format | Article |
| id | doaj-art-6393175e37de46bb97882d2aad590116 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2000-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-6393175e37de46bb97882d2aad5901162025-02-03T07:26:20ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2000-01-0141212810.1155/S1026022600000029Chaotic numerical instabilities arising in the transition from differential to difference nonlinear equationsAlicia Serfaty de Markus0Centro de Estudios Avanzados en Optica and Centro de Astrofísica Teórica, Fac. de Ciencias, La Hechicera, Universidad de los Andes, AP 26, Mérida 5101, VenezuelaFor computational purposes, a numerical algorithm maps a differential equation into an often complex difference equation whose structure and stability depends on the scheme used. When considering nonlinear models, standard and nonstandard integration routines can act invasively and numerical chaotic instabilities may arise. However, because nonstandard schemes offer a direct and generally simpler finite-difference representations, in this work nonstandard constructions were tested over three different systems: a photoconductor model, the Lorenz equations and the Van der Pol equations. Results showed that although some nonstandard constructions created a chaotic dynamics of their own, there was found a construction in every case that greatly reduced or successfully removed numerical chaotic instabilities. These improvements represent a valuable development to incorporate into more sophisticated algorithms.http://dx.doi.org/10.1155/S1026022600000029ChaosNumerical chaotic instabilitiesNonstandard schemesDifference equations. |
| spellingShingle | Alicia Serfaty de Markus Chaotic numerical instabilities arising in the transition from differential to difference nonlinear equations Discrete Dynamics in Nature and Society Chaos Numerical chaotic instabilities Nonstandard schemes Difference equations. |
| title | Chaotic numerical instabilities arising in the transition from differential to difference nonlinear equations |
| title_full | Chaotic numerical instabilities arising in the transition from differential to difference nonlinear equations |
| title_fullStr | Chaotic numerical instabilities arising in the transition from differential to difference nonlinear equations |
| title_full_unstemmed | Chaotic numerical instabilities arising in the transition from differential to difference nonlinear equations |
| title_short | Chaotic numerical instabilities arising in the transition from differential to difference nonlinear equations |
| title_sort | chaotic numerical instabilities arising in the transition from differential to difference nonlinear equations |
| topic | Chaos Numerical chaotic instabilities Nonstandard schemes Difference equations. |
| url | http://dx.doi.org/10.1155/S1026022600000029 |
| work_keys_str_mv | AT aliciaserfatydemarkus chaoticnumericalinstabilitiesarisinginthetransitionfromdifferentialtodifferencenonlinearequations |