### Abstract

A symbolic computational technique is used to study the secondary bifurcations of a parametrically excited simple pendulum as an explicit function of the periodic parameter. This is made possible by the recent development of an algorithm which approximates the fundamental solution matrix of linear time-periodic systems in terms of system parameters in symbolic form. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet transition matrix (FTM), or the linear part of the Poincaré map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation boundaries in the parameter space. Since this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. By repeating the linearization and computational procedure after each bifurcation of an equilibrium or periodic solution, it is shown how the bifurcation locations as well as the new linearized equations may be obtained in closed form as a function of the periodic parameter. Bifurcation diagrams are constructed and the results are compared with those obtained elsewhere using the point mapping method.

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

Pages (from-to) | 627-637 |

Number of pages | 11 |

Journal | International Journal of Bifurcation and Chaos in Applied Sciences and Engineering |

Volume | 8 |

Issue number | 3 |

State | Published - Mar 1998 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- General
- Applied Mathematics

### Cite this

**Symbolic computation of secondary bifurcations in a parametrically excited simple pendulum.** / Butcher, Eric; Sinha, S. C.

Research output: Contribution to journal › Article

*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 8, no. 3, pp. 627-637.

}

TY - JOUR

T1 - Symbolic computation of secondary bifurcations in a parametrically excited simple pendulum

AU - Butcher, Eric

AU - Sinha, S. C.

PY - 1998/3

Y1 - 1998/3

N2 - A symbolic computational technique is used to study the secondary bifurcations of a parametrically excited simple pendulum as an explicit function of the periodic parameter. This is made possible by the recent development of an algorithm which approximates the fundamental solution matrix of linear time-periodic systems in terms of system parameters in symbolic form. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet transition matrix (FTM), or the linear part of the Poincaré map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation boundaries in the parameter space. Since this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. By repeating the linearization and computational procedure after each bifurcation of an equilibrium or periodic solution, it is shown how the bifurcation locations as well as the new linearized equations may be obtained in closed form as a function of the periodic parameter. Bifurcation diagrams are constructed and the results are compared with those obtained elsewhere using the point mapping method.

AB - A symbolic computational technique is used to study the secondary bifurcations of a parametrically excited simple pendulum as an explicit function of the periodic parameter. This is made possible by the recent development of an algorithm which approximates the fundamental solution matrix of linear time-periodic systems in terms of system parameters in symbolic form. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet transition matrix (FTM), or the linear part of the Poincaré map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation boundaries in the parameter space. Since this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. By repeating the linearization and computational procedure after each bifurcation of an equilibrium or periodic solution, it is shown how the bifurcation locations as well as the new linearized equations may be obtained in closed form as a function of the periodic parameter. Bifurcation diagrams are constructed and the results are compared with those obtained elsewhere using the point mapping method.

UR - http://www.scopus.com/inward/record.url?scp=0032018648&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032018648&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032018648

VL - 8

SP - 627

EP - 637

JO - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

JF - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

SN - 0218-1274

IS - 3

ER -