On a deformed version of the two-disk dynamo system

Cristian Lăzureanu; Camelia Petrişor; Ciprian Hedrea

Applications of Mathematics (2021)

  • Volume: 66, Issue: 3, page 345-372
  • ISSN: 0862-7940

Abstract

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We give some deformations of the Rikitake two-disk dynamo system. Particularly, we consider an integrable deformation of an integrable version of the Rikitake system. The deformed system is a three-dimensional Hamilton-Poisson system. We present two Lie-Poisson structures and also symplectic realizations. Furthermore, we give a prequantization result of one of the Poisson manifold. We study the stability of the equilibrium states and we prove the existence of periodic orbits. We analyze some properties of the energy-Casimir mapping ℰ𝒞 associated to our system. In many cases the dynamical behavior of such systems is related with some geometric properties of the image of the energy-Casimir mapping. These connections were observed in the cases when the image of ℰ𝒞 is a convex proper subset of ℝ 2 . In order to point out new connections, we choose deformation functions such that Im ( ℰ𝒞 ) = ℝ 2 . Using the images of the equilibrium states through the energy-Casimir mapping we give parametric equations of some special orbits, namely heteroclinic orbits, split-heteroclinic orbits, and split-homoclinic orbits. Finally, we implement the mid-point rule to perform some numerical integrations of the considered system.

How to cite

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Lăzureanu, Cristian, Petrişor, Camelia, and Hedrea, Ciprian. "On a deformed version of the two-disk dynamo system." Applications of Mathematics 66.3 (2021): 345-372. <http://eudml.org/doc/297592>.

@article{Lăzureanu2021,
abstract = {We give some deformations of the Rikitake two-disk dynamo system. Particularly, we consider an integrable deformation of an integrable version of the Rikitake system. The deformed system is a three-dimensional Hamilton-Poisson system. We present two Lie-Poisson structures and also symplectic realizations. Furthermore, we give a prequantization result of one of the Poisson manifold. We study the stability of the equilibrium states and we prove the existence of periodic orbits. We analyze some properties of the energy-Casimir mapping $\mathcal \{EC\}$ associated to our system. In many cases the dynamical behavior of such systems is related with some geometric properties of the image of the energy-Casimir mapping. These connections were observed in the cases when the image of $\mathcal \{EC\}$ is a convex proper subset of $\mathbb \{R\}^2$. In order to point out new connections, we choose deformation functions such that Im$(\mathcal \{EC\})=\mathbb \{R\}^2.$ Using the images of the equilibrium states through the energy-Casimir mapping we give parametric equations of some special orbits, namely heteroclinic orbits, split-heteroclinic orbits, and split-homoclinic orbits. Finally, we implement the mid-point rule to perform some numerical integrations of the considered system.},
author = {Lăzureanu, Cristian, Petrişor, Camelia, Hedrea, Ciprian},
journal = {Applications of Mathematics},
keywords = {integrable deformation; Hamilton-Poisson system; stability; energy-Casimir mapping; periodic orbit; heteroclinic orbit; mid-point rule},
language = {eng},
number = {3},
pages = {345-372},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a deformed version of the two-disk dynamo system},
url = {http://eudml.org/doc/297592},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Lăzureanu, Cristian
AU - Petrişor, Camelia
AU - Hedrea, Ciprian
TI - On a deformed version of the two-disk dynamo system
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 345
EP - 372
AB - We give some deformations of the Rikitake two-disk dynamo system. Particularly, we consider an integrable deformation of an integrable version of the Rikitake system. The deformed system is a three-dimensional Hamilton-Poisson system. We present two Lie-Poisson structures and also symplectic realizations. Furthermore, we give a prequantization result of one of the Poisson manifold. We study the stability of the equilibrium states and we prove the existence of periodic orbits. We analyze some properties of the energy-Casimir mapping $\mathcal {EC}$ associated to our system. In many cases the dynamical behavior of such systems is related with some geometric properties of the image of the energy-Casimir mapping. These connections were observed in the cases when the image of $\mathcal {EC}$ is a convex proper subset of $\mathbb {R}^2$. In order to point out new connections, we choose deformation functions such that Im$(\mathcal {EC})=\mathbb {R}^2.$ Using the images of the equilibrium states through the energy-Casimir mapping we give parametric equations of some special orbits, namely heteroclinic orbits, split-heteroclinic orbits, and split-homoclinic orbits. Finally, we implement the mid-point rule to perform some numerical integrations of the considered system.
LA - eng
KW - integrable deformation; Hamilton-Poisson system; stability; energy-Casimir mapping; periodic orbit; heteroclinic orbit; mid-point rule
UR - http://eudml.org/doc/297592
ER -

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