### Abstract

In this paper, we deal with the solenoidal conservative Lie algebra associated to the classical normal form of Hopf-zero singular system. We concentrate on the study of some representations and [Formula presented]-equivariant normal form for such singular differential equations. First, we list some of the representations that this Lie algebra admits. The vector fields from this Lie algebra could be expressed by the set of ordinary differential equations where the first two of them are in the canonical form of a one-degree of freedom Hamiltonian system and the third one depends upon the first two variables. This representation is governed by the associated Poisson algebra to one sub-family of this Lie algebra. Euler's form, vector potential, and Clebsch representation are other representations of this Lie algebra that we list here. We also study the non-potential property of vector fields with Hopf-zero singularity from this Lie algebra. Finally, we examine the unique normal form with non-zero cubic terms of this family in the presence of the symmetry group [Formula presented]. The theoretical results of normal form theory are illustrated with the modified Chua's oscillator.

Original language | English |
---|---|

Pages (from-to) | 14-22 |

Number of pages | 9 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 386-387 |

Early online date | 18 Aug 2018 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

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### Keywords

- Clebsch representation
- Conservative and solenoidal vector field
- Euler's form
- Hopf-pitchfork singularities
- Normal form
- Vector potential

### Cite this

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**On the representations and [Formula presented]-equivariant normal form for solenoidal Hopf-zero singularities.** / Mokhtari, Fahimeh.

Research output: Contribution to Journal › Article › Academic › peer-review

TY - JOUR

T1 - On the representations and [Formula presented]-equivariant normal form for solenoidal Hopf-zero singularities

AU - Mokhtari, Fahimeh

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In this paper, we deal with the solenoidal conservative Lie algebra associated to the classical normal form of Hopf-zero singular system. We concentrate on the study of some representations and [Formula presented]-equivariant normal form for such singular differential equations. First, we list some of the representations that this Lie algebra admits. The vector fields from this Lie algebra could be expressed by the set of ordinary differential equations where the first two of them are in the canonical form of a one-degree of freedom Hamiltonian system and the third one depends upon the first two variables. This representation is governed by the associated Poisson algebra to one sub-family of this Lie algebra. Euler's form, vector potential, and Clebsch representation are other representations of this Lie algebra that we list here. We also study the non-potential property of vector fields with Hopf-zero singularity from this Lie algebra. Finally, we examine the unique normal form with non-zero cubic terms of this family in the presence of the symmetry group [Formula presented]. The theoretical results of normal form theory are illustrated with the modified Chua's oscillator.

AB - In this paper, we deal with the solenoidal conservative Lie algebra associated to the classical normal form of Hopf-zero singular system. We concentrate on the study of some representations and [Formula presented]-equivariant normal form for such singular differential equations. First, we list some of the representations that this Lie algebra admits. The vector fields from this Lie algebra could be expressed by the set of ordinary differential equations where the first two of them are in the canonical form of a one-degree of freedom Hamiltonian system and the third one depends upon the first two variables. This representation is governed by the associated Poisson algebra to one sub-family of this Lie algebra. Euler's form, vector potential, and Clebsch representation are other representations of this Lie algebra that we list here. We also study the non-potential property of vector fields with Hopf-zero singularity from this Lie algebra. Finally, we examine the unique normal form with non-zero cubic terms of this family in the presence of the symmetry group [Formula presented]. The theoretical results of normal form theory are illustrated with the modified Chua's oscillator.

KW - Clebsch representation

KW - Conservative and solenoidal vector field

KW - Euler's form

KW - Hopf-pitchfork singularities

KW - Normal form

KW - Vector potential

UR - http://www.scopus.com/inward/record.url?scp=85052813371&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052813371&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2018.08.003

DO - 10.1016/j.physd.2018.08.003

M3 - Article

VL - 386-387

SP - 14

EP - 22

JO - Physica D. Nonlinear Phenomena

JF - Physica D. Nonlinear Phenomena

SN - 0167-2789

ER -