August  2021, 26(8): 4013-4043. doi: 10.3934/dcdsb.2020272

Complex dynamics in a quasi-periodic plasma perturbations model

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

Received  January 2020 Revised  July 2020 Published  August 2021 Early access  September 2020

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.

Citation: Xin Zhang, Shuangling Yang. Complex dynamics in a quasi-periodic plasma perturbations model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4013-4043. doi: 10.3934/dcdsb.2020272
References:
[1]

R. Balescu, M. Vlad and F. Spineanu, Tokamap: A model of a partially stochastic toroidal magnetic field, In Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasmas (Carry-Le Rouet, 1997), volume 511 of Lecture Notes in Phys., pages 243-261. Springer, Berlin, 1998.

[2]

P. J. Morrison, Magnetic field lines, Hamiltonian dynamics, and nontwist systems, Phys. Plasmas, 7 (2000), 2279-2289.  doi: 10.1063/1.874062.

[3]

B. Shi, Magnetic Confinement Fusion: Principles and Practices, Beijing, Atomic Energy Press (in Chinese), 1999.

[4]

Zohm and Hartmut, The physics of edge localized modes (elms) and their role in power and particle exhaust, Plasma Physics & Controlled Fusion, 38 (1996), 1213-1223. 

[5]

H. NatiqS. BanerjeeA. P. Misra and M. R. M. Said, Degenerating the butterfly attractor in a plasma perturbation model using nonlinear controllers, Chaos Solitons Fractals, 122 (2019), 58-68.  doi: 10.1016/j.chaos.2019.03.009.

[6]

C. Kieu, Q. Wang and D. Yan, Dynamical transitions of the quasi-periodic plasma model, Nonlinear Dyn, 96 (2019), 323-338. doi: 10.1007/s11071-019-04792-2.

[7]

D. ConstantinescuO. DumbrajsV. IgochineK. LacknerH. Zohm and A. U. Team, Bifurcations and fast-slow dynamics in a low-dimensional model for quasi-periodic plasma perturbations, Romanian Reports in Physics, 67 (2015), 1049-1060. 

[8]

D. Constantinescu, O. Dumbrajs, V. Igochine, K. Lackner, R. Meyer-Spasche and H. Zohm, A low-dimensional model system for quasi-periodic plasma perturbations, Physics of Plasmas, 18 (2011), 062307. doi: 10.1063/1.3600209.

[9]

A. A. ElsadanyAm r Elsonbaty and H. N. Agiza, Qualitative dynamical analysis of chaotic plasma perturbations model, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 409-423.  doi: 10.1016/j.cnsns.2017.11.020.

[10]

E. A. González Velasco, Generic properties of polynomial vector fields at infinity, Transactions of the American Mathematical Society, 143 (1969), 201-222.  doi: 10.1090/S0002-9947-1969-0252788-8.

[11]

A. Cima and J. Llibre, Bounded polynomial vector fields, Transactions of the American Mathematical Society, 318 (1990), 557-579.  doi: 10.1090/S0002-9947-1990-0998352-5.

[12]

M. R. A. GouveiaM. Messias and C. Pessoa, Bifurcations at infinity, invariant algebraic surfaces, homoclinic and heteroclinic orbits and centers of a new Lorenz-like chaotic system, Nonlinear Dynamics, 84 (2016), 703-713.  doi: 10.1007/s11071-015-2520-4.

[13]

Y. Liu, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the conjugate Lorenz-type system, Nonlinear Analysis Real World Applications, 13 (2012), 2466-2475.  doi: 10.1016/j.nonrwa.2012.02.011.

[14]

G. Meinsma, Elementary proof of the Routh-Hurwitz test, Systems & Control Letters, 25 (1995), 237-242.  doi: 10.1016/0167-6911(94)00089-E.

[15]

J. Hale, Theory of Functional Differential Equations, Second edition, 1977. Applied Mathematical Sciences, Vol. 3.

[16]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM Journal on Mathematical Analysis, 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086.

[17]

X. Sun, Y. Pei and B. Qin, Global existence and uniqueness of periodic waves in a population model with density-dependent migrations and Allee effect, International Journal of Bifurcation & Chaos, 27 (2017), 1750192, 10pp. doi: 10.1142/S0218127417501929.

[18]

X. Sun and Y. Pei, Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms, Discrete & Continuous Dynamical Systems-B, 24 (2019), 965-987.  doi: 10.3934/dcdsb.2018341.

[19]

X. Sun and Y. Pei, Exact bound on the number of zeros of Abelian integrals for two hyper-elliptic Hamiltonian systems of degree 4, Journal of Differential Equations, 267 (2019), 7369-7384.  doi: 10.1016/j.jde.2019.07.023.

[20]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[21]

J.-M. Ginoux, J. Llibre and K. Tchizawa, Canards existence in the hindmarsh-rose model, Mathematical Modelling Of Natural Phenomena, 14 (2019), Paper No. 409, 21 pp. doi: 10.1051/mmnp/2019012.

[22]

A. L. Shil'nikov, On bifurcations of the Lorenz attractor in the Shimizu-Morioka model, Physica D: Nonlinear Phenomena, 62 (1993), 338-346.  doi: 10.1016/0167-2789(93)90292-9.

[23]

J. J. Morales Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Mathematics, 179. Birkhäuser Verlag, Basel, 1999. doi: 10.1007/978-3-0348-8718-2.

[24]

M. Ayoul and N. T. Zung, Galoisian obstructions to non-Hamiltonian integrability, Comptes Rendus Mathematique, 348 (2010), 1323-1326.  doi: 10.1016/j.crma.2010.10.024.

[25]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Communications in Mathematical Physics, 196 (1998), 19-51.  doi: 10.1007/s002200050412.

[26]

K. HuangS. Shi and W. Li, Meromorphic and formal first integrals for the Lorenz system, Journal of Nonlinear Mathematical Physics, 25 (2018), 106-121.  doi: 10.1080/14029251.2018.1440745.

[27]

K. Huang, S. Shi and Z. Xu, Integrable deformations, bi-Hamiltonian structures and nonintegrability of a generalized Rikitake system, International Journal of Geometric Methods in Modern Physics, 16 (2019), 1950059, 17pp. doi: 10.1142/S0219887819500592.

[28]

K. HuangS. Shi and W. Li, Kovalevskaya exponents, weak painlevé property and integrability for quasi-homogeneous differential systems, Regular & Chaotic Dynamics, 25 (2020), 295-312.  doi: 10.1134/S1560354720030053.

[29]

K. Yagasaki, Nonintegrability of the unfolding of the Fold-Hopf bifurcation, Nonlinearity, 31 (2018), 341-350.  doi: 10.1088/1361-6544/aa92e8.

[30]

K. Huang, S. Shi and W. Li, Integrability analysis of the shimizu-morioka system, Communications in Nonlinear Science and Numerical Simulation, 84 (2020), 105101, 12pp. doi: 10.1016/j.cnsns.2019.105101.

[31]

J. J. Morales-Ruiz, J.-P. Ramis and C. Sim$\acute{o}$, Integrability of Hamiltonian systems and differential galois groups of higher variational equations, Annales Scientifiques de l'École Normale Supérieure, 406 (2006), 845-884. doi: 10.1016/j.ansens.2007.09.002.

[32]

J. Llibre and X. Zhang, Invariant algebraic surfaces of the Lorenz system, Journal of Mathematical Physics, 43 (2002), 1622-1645.  doi: 10.1063/1.1435078.

[33]

R. Oliveira and C. Valls, Global dynamical aspects of a generalized Chen-Wang differential system, Nonlinear Dynamics, 84 (2016), 1497-1516.  doi: 10.1007/s11071-015-2584-1.

[34]

Z. WangZ. WeiX. Xi and Y. Li, Dynamics of a 3D autonomous quadratic system with an invariant algebraic surface, Nonlinear Dynamics, 77 (2014), 1503-1518.  doi: 10.1007/s11071-014-1395-0.

[35]

B. Balachandran, T. Kalmár-Nagy and D. E. Gilsinn, Delay Differential Equations. Recent Advances and New Directions, Springer, New York, 2009.

[36]

M. LiaoC. Xu and X. Tang, Stability and Hopf bifurcation for a competition and cooperation model of two enterprises with delay, Communications in Nonlinear Science & Numerical Simulation, 19 (2014), 3845-3856.  doi: 10.1016/j.cnsns.2014.02.031.

[37]

L. LiC. Zhang and X. Yan, Stability and Hopf bifurcation analysis for a two-enterprise interaction model with delays, Communications in Nonlinear Science & Numerical Simulation, 30 (2016), 70-83.  doi: 10.1016/j.cnsns.2015.06.011.

[38]

R. Yang and J. Wei, Stability and bifurcation analysis of a diffusive prey-predator system in Holling type Ⅲ with a prey refuge, Nonlinear Dynamics, 79 (2015), 631-646.  doi: 10.1007/s11071-014-1691-8.

[39] I. Richards and H. K Youn, The Theory of Distributions: A Nontechnical Introduction, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511623837.
[40]

G. HuW. Li and X. Yan, Hopf bifurcations in a predator-prey system with multiple delays, Chaos, Solitons & Fractals, 42 (2009), 1273-1285.  doi: 10.1016/j.chaos.2009.03.075.

[41] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, volume 41 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge-New York, 1981. 

show all references

References:
[1]

R. Balescu, M. Vlad and F. Spineanu, Tokamap: A model of a partially stochastic toroidal magnetic field, In Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasmas (Carry-Le Rouet, 1997), volume 511 of Lecture Notes in Phys., pages 243-261. Springer, Berlin, 1998.

[2]

P. J. Morrison, Magnetic field lines, Hamiltonian dynamics, and nontwist systems, Phys. Plasmas, 7 (2000), 2279-2289.  doi: 10.1063/1.874062.

[3]

B. Shi, Magnetic Confinement Fusion: Principles and Practices, Beijing, Atomic Energy Press (in Chinese), 1999.

[4]

Zohm and Hartmut, The physics of edge localized modes (elms) and their role in power and particle exhaust, Plasma Physics & Controlled Fusion, 38 (1996), 1213-1223. 

[5]

H. NatiqS. BanerjeeA. P. Misra and M. R. M. Said, Degenerating the butterfly attractor in a plasma perturbation model using nonlinear controllers, Chaos Solitons Fractals, 122 (2019), 58-68.  doi: 10.1016/j.chaos.2019.03.009.

[6]

C. Kieu, Q. Wang and D. Yan, Dynamical transitions of the quasi-periodic plasma model, Nonlinear Dyn, 96 (2019), 323-338. doi: 10.1007/s11071-019-04792-2.

[7]

D. ConstantinescuO. DumbrajsV. IgochineK. LacknerH. Zohm and A. U. Team, Bifurcations and fast-slow dynamics in a low-dimensional model for quasi-periodic plasma perturbations, Romanian Reports in Physics, 67 (2015), 1049-1060. 

[8]

D. Constantinescu, O. Dumbrajs, V. Igochine, K. Lackner, R. Meyer-Spasche and H. Zohm, A low-dimensional model system for quasi-periodic plasma perturbations, Physics of Plasmas, 18 (2011), 062307. doi: 10.1063/1.3600209.

[9]

A. A. ElsadanyAm r Elsonbaty and H. N. Agiza, Qualitative dynamical analysis of chaotic plasma perturbations model, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 409-423.  doi: 10.1016/j.cnsns.2017.11.020.

[10]

E. A. González Velasco, Generic properties of polynomial vector fields at infinity, Transactions of the American Mathematical Society, 143 (1969), 201-222.  doi: 10.1090/S0002-9947-1969-0252788-8.

[11]

A. Cima and J. Llibre, Bounded polynomial vector fields, Transactions of the American Mathematical Society, 318 (1990), 557-579.  doi: 10.1090/S0002-9947-1990-0998352-5.

[12]

M. R. A. GouveiaM. Messias and C. Pessoa, Bifurcations at infinity, invariant algebraic surfaces, homoclinic and heteroclinic orbits and centers of a new Lorenz-like chaotic system, Nonlinear Dynamics, 84 (2016), 703-713.  doi: 10.1007/s11071-015-2520-4.

[13]

Y. Liu, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the conjugate Lorenz-type system, Nonlinear Analysis Real World Applications, 13 (2012), 2466-2475.  doi: 10.1016/j.nonrwa.2012.02.011.

[14]

G. Meinsma, Elementary proof of the Routh-Hurwitz test, Systems & Control Letters, 25 (1995), 237-242.  doi: 10.1016/0167-6911(94)00089-E.

[15]

J. Hale, Theory of Functional Differential Equations, Second edition, 1977. Applied Mathematical Sciences, Vol. 3.

[16]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM Journal on Mathematical Analysis, 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086.

[17]

X. Sun, Y. Pei and B. Qin, Global existence and uniqueness of periodic waves in a population model with density-dependent migrations and Allee effect, International Journal of Bifurcation & Chaos, 27 (2017), 1750192, 10pp. doi: 10.1142/S0218127417501929.

[18]

X. Sun and Y. Pei, Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms, Discrete & Continuous Dynamical Systems-B, 24 (2019), 965-987.  doi: 10.3934/dcdsb.2018341.

[19]

X. Sun and Y. Pei, Exact bound on the number of zeros of Abelian integrals for two hyper-elliptic Hamiltonian systems of degree 4, Journal of Differential Equations, 267 (2019), 7369-7384.  doi: 10.1016/j.jde.2019.07.023.

[20]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[21]

J.-M. Ginoux, J. Llibre and K. Tchizawa, Canards existence in the hindmarsh-rose model, Mathematical Modelling Of Natural Phenomena, 14 (2019), Paper No. 409, 21 pp. doi: 10.1051/mmnp/2019012.

[22]

A. L. Shil'nikov, On bifurcations of the Lorenz attractor in the Shimizu-Morioka model, Physica D: Nonlinear Phenomena, 62 (1993), 338-346.  doi: 10.1016/0167-2789(93)90292-9.

[23]

J. J. Morales Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Mathematics, 179. Birkhäuser Verlag, Basel, 1999. doi: 10.1007/978-3-0348-8718-2.

[24]

M. Ayoul and N. T. Zung, Galoisian obstructions to non-Hamiltonian integrability, Comptes Rendus Mathematique, 348 (2010), 1323-1326.  doi: 10.1016/j.crma.2010.10.024.

[25]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Communications in Mathematical Physics, 196 (1998), 19-51.  doi: 10.1007/s002200050412.

[26]

K. HuangS. Shi and W. Li, Meromorphic and formal first integrals for the Lorenz system, Journal of Nonlinear Mathematical Physics, 25 (2018), 106-121.  doi: 10.1080/14029251.2018.1440745.

[27]

K. Huang, S. Shi and Z. Xu, Integrable deformations, bi-Hamiltonian structures and nonintegrability of a generalized Rikitake system, International Journal of Geometric Methods in Modern Physics, 16 (2019), 1950059, 17pp. doi: 10.1142/S0219887819500592.

[28]

K. HuangS. Shi and W. Li, Kovalevskaya exponents, weak painlevé property and integrability for quasi-homogeneous differential systems, Regular & Chaotic Dynamics, 25 (2020), 295-312.  doi: 10.1134/S1560354720030053.

[29]

K. Yagasaki, Nonintegrability of the unfolding of the Fold-Hopf bifurcation, Nonlinearity, 31 (2018), 341-350.  doi: 10.1088/1361-6544/aa92e8.

[30]

K. Huang, S. Shi and W. Li, Integrability analysis of the shimizu-morioka system, Communications in Nonlinear Science and Numerical Simulation, 84 (2020), 105101, 12pp. doi: 10.1016/j.cnsns.2019.105101.

[31]

J. J. Morales-Ruiz, J.-P. Ramis and C. Sim$\acute{o}$, Integrability of Hamiltonian systems and differential galois groups of higher variational equations, Annales Scientifiques de l'École Normale Supérieure, 406 (2006), 845-884. doi: 10.1016/j.ansens.2007.09.002.

[32]

J. Llibre and X. Zhang, Invariant algebraic surfaces of the Lorenz system, Journal of Mathematical Physics, 43 (2002), 1622-1645.  doi: 10.1063/1.1435078.

[33]

R. Oliveira and C. Valls, Global dynamical aspects of a generalized Chen-Wang differential system, Nonlinear Dynamics, 84 (2016), 1497-1516.  doi: 10.1007/s11071-015-2584-1.

[34]

Z. WangZ. WeiX. Xi and Y. Li, Dynamics of a 3D autonomous quadratic system with an invariant algebraic surface, Nonlinear Dynamics, 77 (2014), 1503-1518.  doi: 10.1007/s11071-014-1395-0.

[35]

B. Balachandran, T. Kalmár-Nagy and D. E. Gilsinn, Delay Differential Equations. Recent Advances and New Directions, Springer, New York, 2009.

[36]

M. LiaoC. Xu and X. Tang, Stability and Hopf bifurcation for a competition and cooperation model of two enterprises with delay, Communications in Nonlinear Science & Numerical Simulation, 19 (2014), 3845-3856.  doi: 10.1016/j.cnsns.2014.02.031.

[37]

L. LiC. Zhang and X. Yan, Stability and Hopf bifurcation analysis for a two-enterprise interaction model with delays, Communications in Nonlinear Science & Numerical Simulation, 30 (2016), 70-83.  doi: 10.1016/j.cnsns.2015.06.011.

[38]

R. Yang and J. Wei, Stability and bifurcation analysis of a diffusive prey-predator system in Holling type Ⅲ with a prey refuge, Nonlinear Dynamics, 79 (2015), 631-646.  doi: 10.1007/s11071-014-1691-8.

[39] I. Richards and H. K Youn, The Theory of Distributions: A Nontechnical Introduction, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511623837.
[40]

G. HuW. Li and X. Yan, Hopf bifurcations in a predator-prey system with multiple delays, Chaos, Solitons & Fractals, 42 (2009), 1273-1285.  doi: 10.1016/j.chaos.2009.03.075.

[41] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, volume 41 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge-New York, 1981. 
Figure 1.  The equilibrium $ E_{+} = (0,1,1) $ is asymptotically for system (7) with $ \tau = 0.9<\tau_{0} $
Figure 2.  Bifurcating periodic solution for system (7) at $ E_{+} = (0,1,1) $ with $ \tau = 1.1>\tau_{0} $
Figure 3.  Bifurcation diagram at $ E_{+} $ in $ (\tau,x) $, $ (\tau,y) $ and $ (\tau,z) $ space, respectively
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 $
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 $
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
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 $
Figure 8.  Dynamics of system (6) on the sphere at infinity
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