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Dynamical transitions of the quasi-periodic plasma model


This study examines the stability and transitions of the quasi-periodic plasma perturbation model from the perspective of the dynamical transition. By analyzing the principle of exchange of stability, it is shown that this model undergoes three different types of dynamical transitions as the model control parameter increases. For the first transition, the model exhibits a continuous transition type and subsequently bifurcates to two stable steady states. As this model parameter further increases, the second and third transition can be either continuous or catastrophic. For the continuous transition, the model bifurcates from the two steady states that are resulted from the first transition to a stable periodic solution. If the second transition is catastrophic, there exists a singular separation of a periodic solution, and a nontrivial attractor emerging suddenly for a subcritical value of the control parameters. Numerical experiments confirm both continuous and catastrophic types of transitions, which depend also on the other model parameters. The continuous transition region and catastrophic transition region are also numerically demonstrated. The method used in this paper can be generalized to study other ODE systems.

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  1. 1.

    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Chua, L.: Memristor: the missing circuit element. IEEE Trans. Circuit Theory 18(5), 507–519 (1971)

    Article  Google Scholar 

  3. 3.

    Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(2), 397–398 (1976)

    Article  MATH  Google Scholar 

  4. 4.

    Chen, G., Dong, X.: From Chaos to Order: Methodologies, Perspectives and Applications, Volume 24 of World Scientific Series on Nonlinear Science Series A. World Scientific, Singapore (1998)

    Book  Google Scholar 

  5. 5.

    Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifur. Chaos Appl. Sci. Eng. 12(3), 659–661 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Liao, X., Li, C., Zhou, S.: Hopf bifurcation and chaos in macroeconomic models with policy lag. Chaos Solitons Fractals 25(1), 91–108 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Algaba, A., Gamero, E., Rodrí guez Luis, A.J.: A bifurcation analysis of a simple electronic circuit. Commun. Nonlinear Sci. Numer. Simul. 10(2), 169–178 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Constantinescu, D., Dumbrajs, O., Igochine, V., Lackner, K., Meyer-Spasche, R., Zohm, H., Team, A.U.: A low-dimensional model system for quasi-periodic plasma. Phys. Plasmas 18(6), 062307 (2011)

    Article  Google Scholar 

  9. 9.

    Elsadany, A.A., Elsonbaty, A., Agiza, H.N.: Qualitative dynamical analysis of chaotic plasma perturbations model. Commun. Nonlinear Sci. Numer. Simul. 59, 409–423 (2018)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, Volume 112 of Applied Mathematical Sciences, 3rd edn. Springer, New York (2004)

    Book  Google Scholar 

  11. 11.

    Li, T.Y., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82(10), 985–992 (1975)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57(3, part 1), 617–656 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Smale, S.: The Mathematics of Time: Essays on Dynamical Systems, Economic Processes, and Related Topics. Springer, New York (1980)

    Book  MATH  Google Scholar 

  14. 14.

    Ma, T., Wang, S.: Bifurcation Theory and Applications. World Scientific Series on Nonlinear Science, vol. 53. World Scientific Publishing Co., Pte. Ltd., Hackensack (2005)

    Google Scholar 

  15. 15.

    Ma, T., Wang, S.: Phase Transition Dynamics. Springer, New York (2014)

    Book  MATH  Google Scholar 

  16. 16.

    Ma, T., Wang, S.: Stability and Bifurcation of Nonlinear Evolutions Equations. Science Press, Beijing (2007)

    Google Scholar 

  17. 17.

    Han, D.-Z., Hernandez, M., Wang, Q.: Dynamical transitions of a low-dimensional model for Rayleigh–Bénard convection under a vertical magnetic field. Chaos Solitons Fractals 114, 370–380 (2018)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Dijkstra, H., Sengul, T., Shen, J., Wang, S.: Dynamic transitions of quasi-geostrophic channel flow. SIAM J. Appl. Math. 75(5), 2361–2378 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Ma, T., Wang, A.: Rayleigh-Bénard convection: dynamics and structure in the physical space. Commun. Math. Sci. 5(3), 553–574 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Sengul, T., Shen, J., Wang, S.: Pattern formations of 2D Rayleigh–Bénard convection with no-slip boundary conditions for the velocity at the critical length scales. Math. Methods Appl. Sci. 38(17), 3792–3806 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Sengul, T., Wang, S.: Pattern formation in Rayleigh–Bénard convection. Commun. Math. Sci. 11(1), 315–343 (2013)

    MathSciNet  Article  MATH  Google Scholar 

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This research is supported in part by the National Science Foundation (NSF) Grant DMS-1515024, Indiana University Faculty Research fund, and the Office of Naval Research (ONR)’s Young Investigator Program Award and the National Science Foundation of China (NSFC) (No. 11771306). We thank three anonymous reviewers for their constructive comments and suggestions.

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Correspondence to Quan Wang.

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Kieu, C., Wang, Q. & Yan, D. Dynamical transitions of the quasi-periodic plasma model. Nonlinear Dyn 96, 323–338 (2019).

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  • Dynamic transition
  • Nonlinear dynamics
  • Continuous and catastrophic transition
  • Bifurcation
  • Plasma dynamics