Complex dynamics in a quasi-periodic plasma perturbations model

Xin Zhang; Shuangling Yang

doi:10.3934/dcdsb.2020272

Article Contents

2021, Volume 26, Issue 8: 4013-4043. Doi: 10.3934/dcdsb.2020272

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Complex dynamics in a quasi-periodic plasma perturbations model

Xin Zhang^1, and
Shuangling Yang^2, ,

1.
School of Applied Mathematics, Nanjing University of Finance & Economics, 210023 Nanjing, China
2.
School of Mathematics, Jilin Univesity, 130012 Changchun, China

^* Corresponding author: Shuangling Yang
^* Corresponding author: Shuangling Yang

Received: December 31, 2019

Revised: June 30, 2020

Early access: September 2020

Published: August 2021

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Abstract

In this paper, the complex dynamics of a quasi-periodic plasma perturbations (QPP) model, which governs the interplay between a driver associated with pressure gradient and relaxation of instability due to magnetic field perturbations in Tokamaks, are studied. The model consists of three coupled ordinary differential equations (ODEs) and contains three parameters. This paper consists of three parts: (1) We study the stability and bifurcations of the QPP model, which gives the theoretical interpretation of various types of oscillations observed in [Phys. Plasmas, 18(2011):1-7]. In particular, assuming that there exists a finite time lag $ \tau $ between the plasma pressure gradient and the speed of the magnetic field, we also study the delay effect in the QPP model from the point of view of Hopf bifurcation. (2) We provide some numerical indices for identifying chaotic properties of the QPP system, which shows that the QPP model has chaotic behaviors for a wide range of parameters. Then we prove that the QPP model is not rationally integrable in an extended Liouville sense for almost all parameter values, which may help us distinguish values of parameters for which the QPP model is integrable. (3) To understand the asymptotic behavior of the orbits for the QPP model, we also provide a complete description of its dynamical behavior at infinity by the Poincaré compactification method. Our results show that the input power $ h $ and the relaxation of the instability $ \delta $ do not affect the global dynamics at infinity of the QPP model and the heat diffusion coefficient $ \eta $ just yield quantitative, but not qualitative changes for the global dynamics at infinity of the QPP model.

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Mathematics Subject Classification: Primary:34C15, 34C23;Secondary:37G15, 37N25.

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Figure 1. The equilibrium $ E_{+} = (0,1,1) $ is asymptotically for system (7) with $ \tau = 0.9<\tau_{0} $

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Figure 2. Bifurcating periodic solution for system (7) at $ E_{+} = (0,1,1) $ with $ \tau = 1.1>\tau_{0} $

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Figure 3. Bifurcation diagram at $ E_{+} $ in $ (\tau,x) $, $ (\tau,y) $ and $ (\tau,z) $ space, respectively

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Figure 4. The two and three dimensional phase portraits of system (6), illustrating its chaotic behavior, are shown for values of parameters $ \delta = 0.6 $, $ \eta = 0.1 $ and $ h = 3 $

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Figure 5. The corresponding graphs of Lyapunov exponents for (a) $ \delta = 0.5, \eta = 0.1 $ and $ 0<h\leq50 $; (b) $ \delta = 0.5, h = 2 $ and $ 0<\eta\leq50 $ and (c) $ h = 2, \eta = 0.1 $ and $ 0<\delta\leq50 $

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Figure 6. Orientation of the local charts $ U_{i} $ and $ V_{i} $($ i = 1,2,3 $) in the positive endpoints of the $ x $, $ y $ and $ z $ axis

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Figure 7. Dynamics of system (6) on the Poincaré sphere at infinity in the local charts $ U_{i} $ and $ V_{i}(i = 1,2,3) $: the solutions tend toward the equilibria in the $ z_{1}- $axis for $ \eta>0 $ and outward of this line for $ \eta<0 $ as $ t\rightarrow+\infty $

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Figure 8. Dynamics of system (6) on the sphere at infinity

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