### Abstract

A second-order accurate numerical method has been proposed for the solution of a coupled non-linear oscillator featuring in chemical kinetics. Although implicit by construction, the method enables the solution of the model initial-value problem (IVP) to be computed explicitly. The second-order method is constructed by taking a linear combination of first-order methods. The stability analysis of the system suggests the existence of a Hopf bifurcation, which is confirmed by the numerical method. Both the critical point of the continuous system and the fixed point of the numerical method will be seen to have the same stability properties. The second-order method is more competitive in terms of numerical stability than some well-known standard methods (such as the Runge-Kutta methods of order two and four).

Original language | English (US) |
---|---|

Pages (from-to) | 325-340 |

Number of pages | 16 |

Journal | Journal of Mathematical Chemistry |

Volume | 28 |

Issue number | 4 |

State | Published - Dec 2000 |

Externally published | Yes |

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### Keywords

- Coupled oscillator
- Hopf bifurcation
- Numerical method
- Stability

### ASJC Scopus subject areas

- Chemistry(all)
- Applied Mathematics

### Cite this

*Journal of Mathematical Chemistry*,

*28*(4), 325-340.

**Numerical solutions for a coupled non-linear oscillator.** / Gumel, Abba; Langford, W. F.; Twizell, E. H.; Wu, J.

Research output: Contribution to journal › Article

*Journal of Mathematical Chemistry*, vol. 28, no. 4, pp. 325-340.

}

TY - JOUR

T1 - Numerical solutions for a coupled non-linear oscillator

AU - Gumel, Abba

AU - Langford, W. F.

AU - Twizell, E. H.

AU - Wu, J.

PY - 2000/12

Y1 - 2000/12

N2 - A second-order accurate numerical method has been proposed for the solution of a coupled non-linear oscillator featuring in chemical kinetics. Although implicit by construction, the method enables the solution of the model initial-value problem (IVP) to be computed explicitly. The second-order method is constructed by taking a linear combination of first-order methods. The stability analysis of the system suggests the existence of a Hopf bifurcation, which is confirmed by the numerical method. Both the critical point of the continuous system and the fixed point of the numerical method will be seen to have the same stability properties. The second-order method is more competitive in terms of numerical stability than some well-known standard methods (such as the Runge-Kutta methods of order two and four).

AB - A second-order accurate numerical method has been proposed for the solution of a coupled non-linear oscillator featuring in chemical kinetics. Although implicit by construction, the method enables the solution of the model initial-value problem (IVP) to be computed explicitly. The second-order method is constructed by taking a linear combination of first-order methods. The stability analysis of the system suggests the existence of a Hopf bifurcation, which is confirmed by the numerical method. Both the critical point of the continuous system and the fixed point of the numerical method will be seen to have the same stability properties. The second-order method is more competitive in terms of numerical stability than some well-known standard methods (such as the Runge-Kutta methods of order two and four).

KW - Coupled oscillator

KW - Hopf bifurcation

KW - Numerical method

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=0034340246&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034340246&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034340246

VL - 28

SP - 325

EP - 340

JO - Journal of Mathematical Chemistry

JF - Journal of Mathematical Chemistry

SN - 0259-9791

IS - 4

ER -