### Abstract

A new technique is presented for symbolic computation of local stability boundaries and bifurcation surfaces for nonlinear multidimensional time-periodic dynamical systems as an explicit function of the system parameters. This is made possible by the recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. 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 surfaces in the parameter space as polynomials of the system parameters. Further, the method may be used in conjunction with a series expansion to obtain perturbation-like expressions for the bifurcation boundaries. Because this method is not based on expansion in terms of a small parameter, it can be successfully applied to periodic systems whose internal excitation is strong. Also, the proposed method appears to be more efficient in terms of cpu time than the truncated point mapping method. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.

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

Pages (from-to) | 1-21 |

Number of pages | 21 |

Journal | Nonlinear Dynamics |

Volume | 17 |

Issue number | 1 |

State | Published - 1998 |

Externally published | Yes |

### Fingerprint

### Keywords

- Bifurcation
- Nonlinear
- Stability
- Symbolic computation
- Time-periodic

### ASJC Scopus subject areas

- Computational Mechanics
- Mechanical Engineering
- Mechanics of Materials

### Cite this

*Nonlinear Dynamics*,

*17*(1), 1-21.

**Symbolic Computation of Local Stability and Bifurcation Surfaces for Nonlinear Time-Periodic Systems.** / Butcher, Eric; Sinha, S. C.

Research output: Contribution to journal › Article

*Nonlinear Dynamics*, vol. 17, no. 1, pp. 1-21.

}

TY - JOUR

T1 - Symbolic Computation of Local Stability and Bifurcation Surfaces for Nonlinear Time-Periodic Systems

AU - Butcher, Eric

AU - Sinha, S. C.

PY - 1998

Y1 - 1998

N2 - A new technique is presented for symbolic computation of local stability boundaries and bifurcation surfaces for nonlinear multidimensional time-periodic dynamical systems as an explicit function of the system parameters. This is made possible by the recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. 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 surfaces in the parameter space as polynomials of the system parameters. Further, the method may be used in conjunction with a series expansion to obtain perturbation-like expressions for the bifurcation boundaries. Because this method is not based on expansion in terms of a small parameter, it can be successfully applied to periodic systems whose internal excitation is strong. Also, the proposed method appears to be more efficient in terms of cpu time than the truncated point mapping method. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.

AB - A new technique is presented for symbolic computation of local stability boundaries and bifurcation surfaces for nonlinear multidimensional time-periodic dynamical systems as an explicit function of the system parameters. This is made possible by the recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. 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 surfaces in the parameter space as polynomials of the system parameters. Further, the method may be used in conjunction with a series expansion to obtain perturbation-like expressions for the bifurcation boundaries. Because this method is not based on expansion in terms of a small parameter, it can be successfully applied to periodic systems whose internal excitation is strong. Also, the proposed method appears to be more efficient in terms of cpu time than the truncated point mapping method. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.

KW - Bifurcation

KW - Nonlinear

KW - Stability

KW - Symbolic computation

KW - Time-periodic

UR - http://www.scopus.com/inward/record.url?scp=0032157864&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032157864&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032157864

VL - 17

SP - 1

EP - 21

JO - Nonlinear Dynamics

JF - Nonlinear Dynamics

SN - 0924-090X

IS - 1

ER -