Detection of the onset of numerical chaotic instabilities by lyapunov exponents
It is commonly found in the fixed-step numerical integration of nonlinear differential equations that the size of the integration step is opposite related to the numerical stability of the scheme and to the speed of computation. We present a procedure that establishes a criterion to select the larg...
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| Format: | Article |
| Language: | English |
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Wiley
2001-01-01
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| Series: | Discrete Dynamics in Nature and Society |
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| Online Access: | http://dx.doi.org/10.1155/S1026022601000127 |
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| _version_ | 1832556133929189376 |
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| author | Alicia Serfaty De Markus |
| author_facet | Alicia Serfaty De Markus |
| author_sort | Alicia Serfaty De Markus |
| collection | DOAJ |
| description | It is commonly found in the fixed-step numerical integration of nonlinear differential equations that the size of the integration step is opposite related to the numerical stability of the scheme and to the speed of computation. We present a procedure that
establishes a criterion to select the largest possible step size before the onset of chaotic numerical instabilities, based upon the observation that computational chaos does not occur in a smooth, continuous way, but rather abruptly, as detected by examining the largest Lyapunov exponent as a function of the step size. For completeness, examination of the bifurcation diagrams with the step reveals the complexity imposed by the algorithmic discretization, showing the robustness of a scheme to numerical instabilities, illustrated here for explicit and implicit Euler schemes. An example of numerical suppression of chaos is also provided. |
| format | Article |
| id | doaj-art-296a8413da08478faa5a1a9ef5be5918 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2001-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-296a8413da08478faa5a1a9ef5be59182025-02-03T05:46:13ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2001-01-016212112810.1155/S1026022601000127Detection of the onset of numerical chaotic instabilities by lyapunov exponentsAlicia Serfaty De Markus0Centro de Estudios Avanzados en Optica and Centro de Astrofisica Teórica, Facultad de Ciencias, La Hechicera, Universidad de Los Andes, Mérida 5051, VenezuelaIt is commonly found in the fixed-step numerical integration of nonlinear differential equations that the size of the integration step is opposite related to the numerical stability of the scheme and to the speed of computation. We present a procedure that establishes a criterion to select the largest possible step size before the onset of chaotic numerical instabilities, based upon the observation that computational chaos does not occur in a smooth, continuous way, but rather abruptly, as detected by examining the largest Lyapunov exponent as a function of the step size. For completeness, examination of the bifurcation diagrams with the step reveals the complexity imposed by the algorithmic discretization, showing the robustness of a scheme to numerical instabilities, illustrated here for explicit and implicit Euler schemes. An example of numerical suppression of chaos is also provided.http://dx.doi.org/10.1155/S1026022601000127Numerical instabilities; Difference equations; Lyapunov exponents; Fixed-step schemes; Chaos. |
| spellingShingle | Alicia Serfaty De Markus Detection of the onset of numerical chaotic instabilities by lyapunov exponents Discrete Dynamics in Nature and Society Numerical instabilities; Difference equations; Lyapunov exponents; Fixed-step schemes; Chaos. |
| title | Detection of the onset of numerical chaotic instabilities by lyapunov exponents |
| title_full | Detection of the onset of numerical chaotic instabilities by lyapunov exponents |
| title_fullStr | Detection of the onset of numerical chaotic instabilities by lyapunov exponents |
| title_full_unstemmed | Detection of the onset of numerical chaotic instabilities by lyapunov exponents |
| title_short | Detection of the onset of numerical chaotic instabilities by lyapunov exponents |
| title_sort | detection of the onset of numerical chaotic instabilities by lyapunov exponents |
| topic | Numerical instabilities; Difference equations; Lyapunov exponents; Fixed-step schemes; Chaos. |
| url | http://dx.doi.org/10.1155/S1026022601000127 |
| work_keys_str_mv | AT aliciaserfatydemarkus detectionoftheonsetofnumericalchaoticinstabilitiesbylyapunovexponents |