Skip to main content

Advertisement

SpringerLink
Log in

The bounded sets, Hamilton energy, and competitive modes for the chaotic plasma system

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper estimates a new ultimate bound set (UBS) for the chaotic system caused by the interaction of the whistler and ion-acoustic waves with the plasma oscillation. The intrinsic Hamilton energy is estimated by using the Helmholtz theorem, and this kind of energy function is the most suitable Lyapunov function to discern its dynamic stability. It is found that the Hamilton energy is relative to the firing states of the dynamical system. In a stable state, the energy is also a constant, while a chaotic state is resulting from an oscillation in the energy. We use the Lagrange coefficient method to solve an optimization problem analytically so that we can find an accurate UBS for the plasma chaotic system. Further, we present the competitive modes (CMs) for the plasma system to investigate its different dynamical behaviors for different parameters. Simulation results for CMs confirm the theories presented about the Hamilton energy function and UBS.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Data availability

Not applicable.

References

  1. Pikovski, A.S., Rabinovich, M.I., Trakhtengerts, V.Y.: Onset of stochasticity in decay confinement of parametric instability. Sov. Phys. JETP. 47–4, 715–719 (1978)

    Google Scholar 

  2. Srivastava, M., Agrawal, S.K., Vishal, K., Das, S.: Chaos control of fractional order Rabinovich-Fabrikant system and synchronization between chaotic and chaos controlled fractional order Rabinovich-Fabrikant system. Appl. Math. Model. 38–13, 3361–3372 (2014)

    Article  MATH  Google Scholar 

  3. Yang, Y., Qi, G.: Comparing mechanical analysis with generalized-competitive-mode analysis for the plasma chaotic system. Phys. Lett. A. 383, 318–327 (2019)

    Article  MATH  Google Scholar 

  4. Wei, Z., Yu, P., Zhang, W., Yao, M.: Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system. Nonlinear Dyn. 82, 131–141 (2015)

    Article  MATH  Google Scholar 

  5. Xu, J., Xie, M.: Global solutions to nonisentropic hydrodynamic models for two-carrier plasmas. Nonlinear Anal. Real World Appl. 27, 107–123 (2016)

    Article  MATH  Google Scholar 

  6. Kuznetsov, N.V., Leonov, G.A., Vagaitsev, V.I.: Analytical-numerical method for attractor localization of generalized Chua’s system. IFAC Proc. 43, 29–33 (2010)

    Article  Google Scholar 

  7. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, I.: Localization of hidden Chua’s attractors. Phys. Lett. A. 375, 2230–2233 (2011)

    Article  MATH  Google Scholar 

  8. Singh, J.P., Roy, B.K., Kuznetsov, N.V.: Multistability and hidden attractors in the dynamics of permanent magnet synchronous motor. Int. J. Bifur. Chaos. 29, 1–17 (2019)

    Article  MATH  Google Scholar 

  9. Wei, Z., Zhang, W., Wang, Z., Yao, M.: Hidden attractors and dynamical behaviors in an extended Rikitake system. Int. J. Bifur. Chaos. 25, 1–11 (2015)

    Article  MATH  Google Scholar 

  10. Yang, Y., Qi, G., Hu, J., Faradja, P.: Finding method and analysis of hidden chaotic attractors for plasma chaotic system from physical and mechanistic perspectives. Int. J. Bifur. Chaos. 30–5, 2050072 (2020)

    Article  MATH  Google Scholar 

  11. Njitacke, Z.T., Awrejcewicz, J., Ramakrishnan, B., Rajagopal, K.: Jacques Kengne Hamiltonian energy computation and complex behavior of a small heterogeneous network of three neurons: circuit implementation. Nonlinear Dyn. 107, 2867–2886 (2022)

    Article  Google Scholar 

  12. Njitacke, Z.T., Takembo, C.N., Awrejcewicz, J., Fouda, H.P.E., Kengne, J.: Hamilton energy, complex dynamical analysis and information patterns of a new memristive FitzHugh-Nagumo neural network. Chaos Solit. Fractals. 160, 112211 (2022)

    Article  Google Scholar 

  13. Zhou, P., Hu, X.K., Zhu, Z.G., Ma, J.: What is the most suitable Lyapunov function? Chaos Solit. Fractals. 150, 111154 (2021)

    Article  MATH  Google Scholar 

  14. Sarasola, C., Torrealdea, F.J., D’Anjou, A., Moujahid, A., Grana, M.: Energy balance in feedback synchronization of chaotic systems. Phys. Rev. E. 69–1, 011606 (2004)

    Article  MATH  Google Scholar 

  15. Wang, M., Ma, S.: Hamilton energy control for the chaotic system with hidden attractors, Complexity. 5530557 (2021)

  16. Ma, J., Wu, F., Jin, W., Zhou, P., Hayat, T.: Calculation of Hamilton energy and control of dynamical systems with different types of attractors. Chaos. 27, 053108 (2017)

    Article  MATH  Google Scholar 

  17. Xie, Y., Zhou, P., Ma, J.: Energy balance and synchronization via inductive-coupling in functional neural circuits. Appl. Math. Model. 113, 175–187 (2023)

    Article  MATH  Google Scholar 

  18. Xie, Y., Yao, Z., Ma, J.: Phase synchronization and energy balance between neurons. Front Inform. Technol. Electron. Eng. 23, 1407–1420 (2022)

    Article  Google Scholar 

  19. El-Sayed, A.M.A., Nour, H.M., Elsaid, A., Matouk, A.E., Elsonbaty, A.: Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional order hyperchaotic system. Appl. Math. Model. 40(5–6), 3516–3534 (2016)

    Article  MATH  Google Scholar 

  20. Leonov, G., Bunin, A., Koksch, N.: Attractor localization of the Lorenz system. Z Angew Math. Mech. 67, 649–656 (1987)

    Article  MATH  Google Scholar 

  21. Leonov, G.: Lyapunov dimension formulas for Henon and Lorenz attractors. St. Petersburg Math. J. 13, 1–12 (2001)

    Google Scholar 

  22. Saberi Nik, H., Effati, S., Saberi-Nadjafi, J.: Ultimate bound sets of a hyperchaotic system and its application in chaos Synchronization. Complexity. 20–4, 30–44 (2015)

    MATH  Google Scholar 

  23. Saberi Nik, H., Effati, S., Saberi-Nadjafi, J.: New ultimate bound sets and exponential finite-time synchronization for the complex Lorenz system. J. Complex. 31–5, 715–730 (2015)

    Article  MATH  Google Scholar 

  24. Wang, P., Li, D., Wu, X., Lü, J., Yu, X.: Ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems. Int. J. Bifur. Chaos. 21–9, 1–9 (2011)

    MATH  Google Scholar 

  25. Wang, P., Zhang, Y., Tan, S., Wan, L.: Explicit ultimate bound sets of a new hyper-chaotic system and its application in estimating the Hausdorff dimension. Nonlinear Dyn. 74(1–2), 133–142 (2013)

    Article  MATH  Google Scholar 

  26. Vivek, D., Kanagarajan, K., Elsayed, E.M.: Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions. Med. J. M 15–15, 1–21 (2018)

    MATH  Google Scholar 

  27. Leng, X., Du, B., Gu, S., He, S.: Novel dynamical behaviors in fractional-order conservative hyperchaotic system and DSP implementation. Nonlinear Dyn. 109, 1167–1186 (2022)

    Article  Google Scholar 

  28. Yu, P., Yao, W., Chen, G.: Analysis on topological properties of the Lorenz and the Chen attractors using GCM. Int. J. Bifur. Chaos. 17, 2791–2796 (2007)

    Article  MATH  Google Scholar 

  29. Van Gorder, R.A.: Emergence of chaotic regimes in the generalized Lorenz canonical form: a competitive modes analysis. Nonlinear Dyn. 66(1–2), 153–160 (2011)

    Article  MATH  Google Scholar 

  30. Saberi Nik, H., Van Gorder, R.A.: Competitive modes for the Baier-Sahle hyperchaotic flow in arbitrary dimensions. Nonlinear Dyn. 74, 581–590 (2013)

    Article  MATH  Google Scholar 

  31. Choudhury, S.R., Van Gorder, R.A.: Competitive modes as reliable predictors of chaos versus hyperchaos and as geometric mappings accurately delimiting attractors. Nonlinear Dyn. 69, 2255–2267 (2012)

    Article  Google Scholar 

  32. Chien, F., Chowdhury, A.R., Saberi Nik, H.: Competitive modes and estimation of ultimate bound sets for a chaotic dynamical financial system. Nonlinear Dyn. 106, 3601–3614 (2021)

    Article  Google Scholar 

  33. Chien, F., Inc, M., Yosefzade, H.-R., Saberi Nik, H.: 0 Predicting the chaos and solution bounds in a complex dynamical system. Chaos Solit. Fractals. 153, 111474 (2021)

    Article  MATH  Google Scholar 

  34. Rasoolzadeh, A., Tavazoei, M.S.: Prediction of chaos in non-salient permanent-magnet synchronous machines. Phys. Lett. A. 377, 73–79 (2012)

    Article  Google Scholar 

  35. Pan, I., Das, S.: Evolving chaos: Identifying new attractors of the generalised Lorenz family. Appl. Math. Model. 57, 391–405 (2018)

  36. Kobe, D.H.: Helmholtz’s theorem revisited. Am J Phys. 54, 552–4 (1986)

    Article  Google Scholar 

Download references

Acknowledgements

The First author is supported by the National Natural Science Foundation of China (11601525). The second author extends her appreciation to the Al Mustaqbal University college for supporting this work for grant number (MUC-E\(_0\)122).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Fuli He

  2. Air conditioning and Refrigeration Techniques Engineering Department, Al-Mustaqbal University College, Babylon, 51001, Iraq

    Zahraa Kareem Abdullah

  3. Department of Mathematics and Statistics, University of Neyshabur, Neyshabur, Iran

    Hassan Saberi-Nik

  4. Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 90-537, Lodz, Poland

    Jan Awrejcewicz

Authors
  1. Fuli He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zahraa Kareem Abdullah
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hassan Saberi-Nik
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jan Awrejcewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

Corresponding authors

Correspondence to Fuli He or Hassan Saberi-Nik.

Ethics declarations

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, F., Abdullah, Z.K., Saberi-Nik, H. et al. The bounded sets, Hamilton energy, and competitive modes for the chaotic plasma system. Nonlinear Dyn 111, 4847–4862 (2023). https://doi.org/10.1007/s11071-022-08098-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-08098-8

Keywords

Search

Navigation