### Abstract

In this study dynamically equivalent time-invariant forms are obtained for linear and non-linear systems with periodically varying coefficients via Lyapunov-Floquet (L-F) transformation. These forms are equivalent in the sense that the local stability and bifurcation characteristics are identical for both systems in the entire parameter space. It is well known that the L-F transformation converts a linear periodic first order system into a time-invariant one. In the first part of this study a set of linear second order periodic equations is converted into an equivalent set of time-independent second order equations through a sequence of linear transformations. Then the transformations are applied to a time-periodic quadratic Hamiltonian to obtain its equivalent time-invariant form. In the second part, time-invariant forms of nonlinear equations are studied. The application of L-F transformation to a quasi-linear periodic equation converts the linear part to a time-invariant form and leaves the non-linear part with time-periodic coefficients. Dynamically equivalent time-invariant forms are obtained via time-periodic center manifold reduction and time-dependent normal form theory. Such forms are constructed for general hyperbolic systems and for some simple critical cases, including that of one zero eigenvalue and a purely imaginary pair. As a physical example of these techniques, a single and a double inverted pendulum subjected to periodic parametric excitation are considered. The results thus obtained are verified by numerical simulation.

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

Pages (from-to) | 203-221 |

Number of pages | 19 |

Journal | Nonlinear Dynamics |

Volume | 16 |

Issue number | 3 |

State | Published - 1998 |

Externally published | Yes |

### Fingerprint

### Keywords

- Critical systems
- Nonlinear
- Time-invariant forms
- Time-periodic systems

### ASJC Scopus subject areas

- Computational Mechanics
- Mechanical Engineering
- Mechanics of Materials

### Cite this

*Nonlinear Dynamics*,

*16*(3), 203-221.

**Construction of Dynamically Equivalent Time-Invariant Forms for Time-Periodic Systems.** / Sinha, S. C.; Butcher, Eric; Dávid, A.

Research output: Contribution to journal › Article

*Nonlinear Dynamics*, vol. 16, no. 3, pp. 203-221.

}

TY - JOUR

T1 - Construction of Dynamically Equivalent Time-Invariant Forms for Time-Periodic Systems

AU - Sinha, S. C.

AU - Butcher, Eric

AU - Dávid, A.

PY - 1998

Y1 - 1998

N2 - In this study dynamically equivalent time-invariant forms are obtained for linear and non-linear systems with periodically varying coefficients via Lyapunov-Floquet (L-F) transformation. These forms are equivalent in the sense that the local stability and bifurcation characteristics are identical for both systems in the entire parameter space. It is well known that the L-F transformation converts a linear periodic first order system into a time-invariant one. In the first part of this study a set of linear second order periodic equations is converted into an equivalent set of time-independent second order equations through a sequence of linear transformations. Then the transformations are applied to a time-periodic quadratic Hamiltonian to obtain its equivalent time-invariant form. In the second part, time-invariant forms of nonlinear equations are studied. The application of L-F transformation to a quasi-linear periodic equation converts the linear part to a time-invariant form and leaves the non-linear part with time-periodic coefficients. Dynamically equivalent time-invariant forms are obtained via time-periodic center manifold reduction and time-dependent normal form theory. Such forms are constructed for general hyperbolic systems and for some simple critical cases, including that of one zero eigenvalue and a purely imaginary pair. As a physical example of these techniques, a single and a double inverted pendulum subjected to periodic parametric excitation are considered. The results thus obtained are verified by numerical simulation.

AB - In this study dynamically equivalent time-invariant forms are obtained for linear and non-linear systems with periodically varying coefficients via Lyapunov-Floquet (L-F) transformation. These forms are equivalent in the sense that the local stability and bifurcation characteristics are identical for both systems in the entire parameter space. It is well known that the L-F transformation converts a linear periodic first order system into a time-invariant one. In the first part of this study a set of linear second order periodic equations is converted into an equivalent set of time-independent second order equations through a sequence of linear transformations. Then the transformations are applied to a time-periodic quadratic Hamiltonian to obtain its equivalent time-invariant form. In the second part, time-invariant forms of nonlinear equations are studied. The application of L-F transformation to a quasi-linear periodic equation converts the linear part to a time-invariant form and leaves the non-linear part with time-periodic coefficients. Dynamically equivalent time-invariant forms are obtained via time-periodic center manifold reduction and time-dependent normal form theory. Such forms are constructed for general hyperbolic systems and for some simple critical cases, including that of one zero eigenvalue and a purely imaginary pair. As a physical example of these techniques, a single and a double inverted pendulum subjected to periodic parametric excitation are considered. The results thus obtained are verified by numerical simulation.

KW - Critical systems

KW - Nonlinear

KW - Time-invariant forms

KW - Time-periodic systems

UR - http://www.scopus.com/inward/record.url?scp=0032115332&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032115332&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032115332

VL - 16

SP - 203

EP - 221

JO - Nonlinear Dynamics

JF - Nonlinear Dynamics

SN - 0924-090X

IS - 3

ER -