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
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U2 - 10.1016/j.physd.2018.08.003
DO - 10.1016/j.physd.2018.08.003
M3 - Article
AN - SCOPUS:85052813371
SN - 0167-2789
VL - 386-387
SP - 14
EP - 22
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
ER -