### Abstract

The normal forms for time-periodic nonlinear variational equations with arbitrary linear Jordan form undergoing bifurcations of high co-dimension are found. First, the equations are transformed via the Lyapunov-Floquet (L-F) transformation into an equivalent form in which the linear matrix is constant with degenerate nonsemisimple linear eigenvalues while the nonlinear monomials have periodic coefficients. By considering the resulting coupling of the bases of the near identity transformation, the solvability condition for an arbitrary Jordan matrix is then derived. It is shown that time-independent and/or time-dependent nonlinear resonance terms remain in the normal form for various Jordan matrices. Specifically, the normal forms for quadratic and cubic nonlinearities with the following linear Jordan forms are explicitly derived: double zero eigenvalues (co-dimension two bifurcation), triple zero eigenvalues (co-dimension three bifurcation), and two repeated pairs of purely imaginary eigenvalues (co-dimension two bifurcation). A commutative system with cubic nonlinearities and a double inverted pendulum with a periodic follower force are used as illustrative examples.

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

Pages (from-to) | 29-53 |

Number of pages | 25 |

Journal | Nonlinear Dynamics |

Volume | 30 |

Issue number | 1 |

DOIs | |

State | Published - Oct 2002 |

Externally published | Yes |

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### Keywords

- Lyapunov-Floquet transformation
- Nonsemisimple eigenvalues
- Normal forms
- Time-periodic systems

### ASJC Scopus subject areas

- Mechanical Engineering
- Mechanics of Materials
- Computational Mechanics

### Cite this

**Normal forms for high co-dimension bifurcations of nonlinear time-periodic systems with nonsemisimple eigenvalues.** / Butcher, Eric.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Normal forms for high co-dimension bifurcations of nonlinear time-periodic systems with nonsemisimple eigenvalues

AU - Butcher, Eric

PY - 2002/10

Y1 - 2002/10

N2 - The normal forms for time-periodic nonlinear variational equations with arbitrary linear Jordan form undergoing bifurcations of high co-dimension are found. First, the equations are transformed via the Lyapunov-Floquet (L-F) transformation into an equivalent form in which the linear matrix is constant with degenerate nonsemisimple linear eigenvalues while the nonlinear monomials have periodic coefficients. By considering the resulting coupling of the bases of the near identity transformation, the solvability condition for an arbitrary Jordan matrix is then derived. It is shown that time-independent and/or time-dependent nonlinear resonance terms remain in the normal form for various Jordan matrices. Specifically, the normal forms for quadratic and cubic nonlinearities with the following linear Jordan forms are explicitly derived: double zero eigenvalues (co-dimension two bifurcation), triple zero eigenvalues (co-dimension three bifurcation), and two repeated pairs of purely imaginary eigenvalues (co-dimension two bifurcation). A commutative system with cubic nonlinearities and a double inverted pendulum with a periodic follower force are used as illustrative examples.

AB - The normal forms for time-periodic nonlinear variational equations with arbitrary linear Jordan form undergoing bifurcations of high co-dimension are found. First, the equations are transformed via the Lyapunov-Floquet (L-F) transformation into an equivalent form in which the linear matrix is constant with degenerate nonsemisimple linear eigenvalues while the nonlinear monomials have periodic coefficients. By considering the resulting coupling of the bases of the near identity transformation, the solvability condition for an arbitrary Jordan matrix is then derived. It is shown that time-independent and/or time-dependent nonlinear resonance terms remain in the normal form for various Jordan matrices. Specifically, the normal forms for quadratic and cubic nonlinearities with the following linear Jordan forms are explicitly derived: double zero eigenvalues (co-dimension two bifurcation), triple zero eigenvalues (co-dimension three bifurcation), and two repeated pairs of purely imaginary eigenvalues (co-dimension two bifurcation). A commutative system with cubic nonlinearities and a double inverted pendulum with a periodic follower force are used as illustrative examples.

KW - Lyapunov-Floquet transformation

KW - Nonsemisimple eigenvalues

KW - Normal forms

KW - Time-periodic systems

UR - http://www.scopus.com/inward/record.url?scp=0036772341&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036772341&partnerID=8YFLogxK

U2 - 10.1023/A:1020340116695

DO - 10.1023/A:1020340116695

M3 - Article

AN - SCOPUS:0036772341

VL - 30

SP - 29

EP - 53

JO - Nonlinear Dynamics

JF - Nonlinear Dynamics

SN - 0924-090X

IS - 1

ER -