Periodic solutions of nonlinear ordinary differential equations computed by a boundary shape function method and a generalized derivative-free Newton method
In the paper, the period of an n-dimensional nonlinear dynamical system is computed by a formula derived in an (n+1)-dimensional augmented state space. The periodic conditions and nonlinear first-order ordinary differential equations constitute a specific periodic boundary value problem within a tim...
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| Vydané v: | Mechanical systems and signal processing Ročník 184; s. 109712 |
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| Médium: | Journal Article |
| Jazyk: | English |
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Elsevier Ltd
01.02.2023
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| ISSN: | 0888-3270, 1096-1216 |
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| Abstract | In the paper, the period of an n-dimensional nonlinear dynamical system is computed by a formula derived in an (n+1)-dimensional augmented state space. The periodic conditions and nonlinear first-order ordinary differential equations constitute a specific periodic boundary value problem within a time interval, whose length is an unknown finite constant. Two periodic problems are considered: (I) boundary values are given and (II) boundary values are unknown. A boundary shape function method (BSFM), using the derived shape functions, is devised to an initial value problem with the initial values of new variables given, whereas the terminal values and period are determined by iterative algorithms. The periodic solutions obtained by the BSFM satisfy the periodic conditions automatically. For the sake of comparison, the iterative algorithms based on the shooting method are developed, directly implementing the Poincaré map with the fictitious time integration method to determine the periodic solutions, where the periodic conditions are transformed to a mathematically equivalent scalar equation. Owing to the implicit, non-differentiable and nonlinear property of the scalar equation, we develop a generalized derivative-free Newton method (GDFNM) to solve the periodic problem of case (I), which can pick up very accurate period through a few iterations. In numerical examples the computed order of convergence displays the merit of the proposed iterative algorithms. The BSFM and GDFNM are better than the shooting method from the aspects of convergence speed, accuracy and stability. A conventional Poincaré mapping method is introduced to solve the periodic problems with the same parameters. The BSFM converges faster and more accurate than the Poincaré mapping method and is less sensitive to the initial guesses of initial values and period.
•The novel schemes are different from the conventional methods.•The computed order of convergence displays the merit of the proposed algorithms.•The BSFM and GDFNM are better than the shooting method. |
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| AbstractList | In the paper, the period of an n-dimensional nonlinear dynamical system is computed by a formula derived in an (n+1)-dimensional augmented state space. The periodic conditions and nonlinear first-order ordinary differential equations constitute a specific periodic boundary value problem within a time interval, whose length is an unknown finite constant. Two periodic problems are considered: (I) boundary values are given and (II) boundary values are unknown. A boundary shape function method (BSFM), using the derived shape functions, is devised to an initial value problem with the initial values of new variables given, whereas the terminal values and period are determined by iterative algorithms. The periodic solutions obtained by the BSFM satisfy the periodic conditions automatically. For the sake of comparison, the iterative algorithms based on the shooting method are developed, directly implementing the Poincaré map with the fictitious time integration method to determine the periodic solutions, where the periodic conditions are transformed to a mathematically equivalent scalar equation. Owing to the implicit, non-differentiable and nonlinear property of the scalar equation, we develop a generalized derivative-free Newton method (GDFNM) to solve the periodic problem of case (I), which can pick up very accurate period through a few iterations. In numerical examples the computed order of convergence displays the merit of the proposed iterative algorithms. The BSFM and GDFNM are better than the shooting method from the aspects of convergence speed, accuracy and stability. A conventional Poincaré mapping method is introduced to solve the periodic problems with the same parameters. The BSFM converges faster and more accurate than the Poincaré mapping method and is less sensitive to the initial guesses of initial values and period.
•The novel schemes are different from the conventional methods.•The computed order of convergence displays the merit of the proposed algorithms.•The BSFM and GDFNM are better than the shooting method. |
| ArticleNumber | 109712 |
| Author | Chang, Chih-Wen Liu, Chein-Shan |
| Author_xml | – sequence: 1 givenname: Chein-Shan surname: Liu fullname: Liu, Chein-Shan organization: Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan – sequence: 2 givenname: Chih-Wen orcidid: 0000-0001-9846-0694 surname: Chang fullname: Chang, Chih-Wen email: cwchang@nuu.edu.tw organization: Department of Mechanical Engineering, National United University, Miaoli 36063, Taiwan |
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| Cites_doi | 10.1016/j.chaos.2006.05.072 10.3390/math9233070 10.5539/jmr.v13n6p10 10.1016/0020-7462(71)90010-2 10.1007/s11071-014-1391-4 10.1016/j.matcom.2021.06.019 10.1016/j.ymssp.2022.109261 10.3390/sym14071313 10.1142/S0218127499001024 10.1007/s11071-013-0813-z 10.1016/j.aml.2019.106151 10.1016/j.matcom.2011.05.007 10.1016/j.jcp.2005.10.026 10.1016/j.ymssp.2020.107157 10.3390/sym14040778 10.1016/j.ijnonlinmec.2006.01.006 10.1016/S0020-7462(98)00048-1 10.1515/ijnsns-2019-0209 10.1016/j.jsv.2006.03.047 10.1016/S0020-7462(00)00069-X 10.1016/j.camwa.2009.03.034 10.1137/S0036144500375292 10.1504/IJANS.2016.085806 10.2307/2373181 10.1006/jsvi.1996.0228 10.1080/10407790.2020.1713623 10.1016/j.jsv.2007.05.021 10.1016/0022-247X(86)90076-4 10.1016/S0893-9659(00)00100-2 10.1016/S0020-7462(98)00085-7 10.1016/S0167-2789(02)00603-6 10.1186/s13661-020-01436-y |
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| Keywords | Boundary shape function method Iterative algorithm Nonlinear dynamical system Periodic solution Generalized derivative-free Newton method |
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| SubjectTerms | Boundary shape function method Generalized derivative-free Newton method Iterative algorithm Nonlinear dynamical system Periodic solution |
| Title | Periodic solutions of nonlinear ordinary differential equations computed by a boundary shape function method and a generalized derivative-free Newton method |
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