High-Order Lohner-Type Algorithm for Rigorous Computation of Poincaré Maps in Systems of Delay Differential Equations with Several Delays
We present a Lohner-type algorithm for rigorous integration of systems of delay differential equations (DDEs) with multiple delays, and its application in computation of Poincaré maps, to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representati...
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| Published in | Foundations of computational mathematics Vol. 24; no. 4; pp. 1389 - 1454 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.08.2024
Springer Springer Nature B.V |
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| Online Access | Get full text |
| ISSN | 1615-3375 1615-3383 1615-3383 |
| DOI | 10.1007/s10208-023-09614-x |
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| Abstract | We present a Lohner-type algorithm for rigorous integration of systems of delay differential equations (DDEs) with multiple delays, and its application in computation of Poincaré maps, to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase space, and it exploits the smoothing of solutions occurring in DDEs to produce enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behaviour, for example, existence of (apparently) unstable periodic orbits in Mackey–Glass equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system). |
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| AbstractList | We present a Lohner-type algorithm for rigorous integration of systems of delay differential equations (DDEs) with multiple delays, and its application in computation of Poincaré maps, to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase space, and it exploits the smoothing of solutions occurring in DDEs to produce enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behaviour, for example, existence of (apparently) unstable periodic orbits in Mackey–Glass equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system). |
| Audience | Academic |
| Author | Zgliczyński, Piotr Szczelina, Robert |
| Author_xml | – sequence: 1 givenname: Robert surname: Szczelina fullname: Szczelina, Robert email: robert.szczelina@uj.edu.pl organization: Faculty of Mathematics and Computer Science, Jagiellonian University – sequence: 2 givenname: Piotr surname: Zgliczyński fullname: Zgliczyński, Piotr organization: Faculty of Mathematics and Computer Science, Jagiellonian University |
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| Snippet | We present a Lohner-type algorithm for rigorous integration of systems of delay differential equations (DDEs) with multiple delays, and its application in... |
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| StartPage | 1389 |
| SubjectTerms | Algorithms Applications of Mathematics Chaos theory Computation Computer Science Delay Differential equations Economics Linear and Multilinear Algebras Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Orbits Poincare maps |
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| Title | High-Order Lohner-Type Algorithm for Rigorous Computation of Poincaré Maps in Systems of Delay Differential Equations with Several Delays |
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