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## 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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar [2] P. J. Morrison, Magnetic field lines, Hamiltonian dynamics, and nontwist systems, Phys. Plasmas, 7 (2000), 2279-2289.  doi: 10.1063/1.874062.  Google Scholar [3] B. Shi, Magnetic Confinement Fusion: Principles and Practices, Beijing, Atomic Energy Press (in Chinese), 1999. Google Scholar [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.   Google Scholar [5] H. Natiq, S. Banerjee, A. 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.  Google Scholar [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.  Google Scholar [7] D. Constantinescu, O. Dumbrajs, V. Igochine, K. Lackner, H. 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.   Google Scholar [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.  Google Scholar [9] A. A. Elsadany, Am 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] M. R. A. Gouveia, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] J. Hale, Theory of Functional Differential Equations, Second edition, 1977. Applied Mathematical Sciences, Vol. 3.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Communications in Mathematical Physics, 196 (1998), 19-51.  doi: 10.1007/s002200050412.  Google Scholar [26] K. Huang, S. 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.  Google Scholar [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.  Google Scholar [28] K. Huang, S. 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.  Google Scholar [29] K. Yagasaki, Nonintegrability of the unfolding of the Fold-Hopf bifurcation, Nonlinearity, 31 (2018), 341-350.  doi: 10.1088/1361-6544/aa92e8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] Z. Wang, Z. Wei, X. 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.  Google Scholar [35] B. Balachandran, T. Kalmár-Nagy and D. E. Gilsinn, Delay Differential Equations. Recent Advances and New Directions, Springer, New York, 2009.  Google Scholar [36] M. Liao, C. 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.  Google Scholar [37] L. Li, C. 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.  Google Scholar [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.  Google Scholar [39] I. Richards and H. K Youn, The Theory of Distributions: A Nontechnical Introduction, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511623837.  Google Scholar [40] G. Hu, W. 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.  Google Scholar [41] B. D. Hassard, N. 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.   Google Scholar

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.  Google Scholar [2] P. J. Morrison, Magnetic field lines, Hamiltonian dynamics, and nontwist systems, Phys. Plasmas, 7 (2000), 2279-2289.  doi: 10.1063/1.874062.  Google Scholar [3] B. Shi, Magnetic Confinement Fusion: Principles and Practices, Beijing, Atomic Energy Press (in Chinese), 1999. Google Scholar [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.   Google Scholar [5] H. Natiq, S. Banerjee, A. 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.  Google Scholar [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.  Google Scholar [7] D. Constantinescu, O. Dumbrajs, V. Igochine, K. Lackner, H. 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.   Google Scholar [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.  Google Scholar [9] A. A. Elsadany, Am 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] M. R. A. Gouveia, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] J. Hale, Theory of Functional Differential Equations, Second edition, 1977. Applied Mathematical Sciences, Vol. 3.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Communications in Mathematical Physics, 196 (1998), 19-51.  doi: 10.1007/s002200050412.  Google Scholar [26] K. Huang, S. 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.  Google Scholar [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.  Google Scholar [28] K. Huang, S. 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.  Google Scholar [29] K. Yagasaki, Nonintegrability of the unfolding of the Fold-Hopf bifurcation, Nonlinearity, 31 (2018), 341-350.  doi: 10.1088/1361-6544/aa92e8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] Z. Wang, Z. Wei, X. 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.  Google Scholar [35] B. Balachandran, T. Kalmár-Nagy and D. E. Gilsinn, Delay Differential Equations. Recent Advances and New Directions, Springer, New York, 2009.  Google Scholar [36] M. Liao, C. 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.  Google Scholar [37] L. Li, C. 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.  Google Scholar [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.  Google Scholar [39] I. Richards and H. K Youn, The Theory of Distributions: A Nontechnical Introduction, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511623837.  Google Scholar [40] G. Hu, W. 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.  Google Scholar [41] B. D. Hassard, N. 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.   Google Scholar
The equilibrium $E_{+} = (0,1,1)$ is asymptotically for system (7) with $\tau = 0.9<\tau_{0}$
Bifurcating periodic solution for system (7) at $E_{+} = (0,1,1)$ with $\tau = 1.1>\tau_{0}$
Bifurcation diagram at $E_{+}$ in $(\tau,x)$, $(\tau,y)$ and $(\tau,z)$ space, respectively
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 $h = 3$
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 $0<\delta\leq50$
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
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 $t\rightarrow+\infty$
Dynamics of system (6) on the sphere at infinity
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