Orbital dynamics on invariant sets of contact Hamiltonian systems

Qihuai Liu; Pedro J. Torres

doi:10.3934/dcdsb.2021297

Article Contents

2022, Volume 27, Issue 10: 5821-5844. Doi: 10.3934/dcdsb.2021297

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Orbital dynamics on invariant sets of contact Hamiltonian systems

Qihuai Liu^1, and
Pedro J. Torres^2, ,

1.
School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin 541002, China
2.
Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, Granada 18071, Spain

^*Corresponding author: Pedro J. Torres
^*Corresponding author: Pedro J. Torres

Received: May 31, 2021

Revised: September 30, 2021

Early access: December 2021

Published: October 2022

The first author is supported by National Natural Science Foundation of China (Grant No. 11771105), Guangxi Natural Science Foundation (Grants No. 2017GXNSFFA198012 and 2018GXNSFAA138177)

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Abstract

In this paper, we shall give new insights on dynamics of contact Hamiltonian flows, which are gaining importance in several branches of physics as they model a dissipative behaviour. We divide the contact phase space into three parts, which are corresponding to three differential invariant sets $ \Omega_\pm, \Omega_0 $. On the invariant sets $ \Omega_\pm $, under some geometric conditions, the contact Hamiltonian system is equivalent to a Hamiltonian system via the Hölder transformation. The invariant set $ \Omega_0 $ may be composed of several equilibrium points and heteroclinic orbits connecting them, on which contact Hamiltonian system is conservative. Moreover, we have shown that, under general conditions, the zero energy level domain is a domain of attraction. In some cases, such a domain of attraction does not have nontrivial periodic orbits. Some interesting examples are presented.

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Mathematics Subject Classification: Primary: 34C25; Secondary: 34C14, 37J55.

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Figure 1. In $ \mathbb{R}^3 $, the invariant set $ \Omega_0 $ of system (10) consists of two equilibrium points $ (0,0,\pm\sqrt{2}) $ and infinitely many heteroclinic orbits, where we take $ h = 1 $

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Figure 2. The invariant set $ \Omega_0 $ of system (11) consists of four equilibrium points $ (0,0,\pm(R\pm r)) $ and infinitely many heteroclinic orbits, where we take $ R = 2, r = 1 $. Three different curves with red, blue and green colors correspond to three heteroclinic orbits of system (11) starting from the initial values $ (-0.01,0.01,-2.999933331759) $, $ (0.01,-0.01,-1.0000000025) $, $ (-0.01,0.01,1.0000000025) $, respectively

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Figure 3. The periodic orbits are located on the planes perpendicular to the $ xOp $-plane

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