ZERO-HOPF BIFURCATION IN NUCLEAR SPIN GENERATOR SYSTEM

Renxiang Shi; Jiang Yu

doi:10.11948/20180277

2021 Volume 11 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2021 > 11(1): 23-31

Next Article Previous Article

Renxiang Shi, Jiang Yu. ZERO-HOPF BIFURCATION IN NUCLEAR SPIN GENERATOR SYSTEM[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 23-31. doi: 10.11948/20180277

Citation:

Renxiang Shi, Jiang Yu. ZERO-HOPF BIFURCATION IN NUCLEAR SPIN GENERATOR SYSTEM[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 23-31. doi: 10.11948/20180277

ZERO-HOPF BIFURCATION IN NUCLEAR SPIN GENERATOR SYSTEM

Renxiang Shi^1, ,,
Jiang Yu²

1.
Department of Mathematics, Nanjing University of Chinese Medicine, 210023 Nanjing, China
2.
Department of Mathematics, Shanghai Jiaotong University, 200240 Shanghai, China

Corresponding author: Email address:srxahu@aliyun.com(R. Shi)
Fund Project: Research supported by the National Natural Science Foundations of China(Nos. 11701292; 11771282 and 11931016)

Abstract

Abstract

By computing we obtain that $ P_{1}(0, 0, 1) $ is a zero-Hopf equilibrium point of nuclear spin generator system. We prove that there exist two families of nuclear spin generator system which has the zero-Hopf equilibrium point $ P_{1}(0, 0, 1) $. Furthermore we prove that a limit cycle bifurcated from $ P_{1}(0, 0, 1) $ by averaging method of one order and second order respectively.
- Averaging method /
- nuclear spin generator /
- limit cycle /
- zero-Hopf bifurcation
MSC: 34C23, 34C29

FullText(HTML)

References

[1]	A. Buica and J. Llibre, Avariaging methods for finding period orbits via Brouwer degree, Bull. Sci. Math., 2004, 128, 7-22. doi: 10.1016/j.bulsci.2003.09.002 CrossRef Google Scholar
[2]	L. Cid-Montiel, J. Llibre and C. Stoica, Zero-hopf bifurcation in a hyperchaotic lorenz system, Nonliear Dyn., 2014, 75, 561-566. doi: 10.1007/s11071-013-1085-3 CrossRef Google Scholar
[3]	R. D. Euzebio and J. Llibre, Zero-hopf bifurcation in a Chua's system, Nonlinear Anal. Real Word Appl., 2017, 37, 31-40. doi: 10.1016/j.nonrwa.2017.02.002 CrossRef Google Scholar
[4]	R. D. Euzebio, J. Llibre and C. Vidal, Zero-hopf bifurcation in the Fitzhughnagumo system, Math. Method Appl. Sci., 2015, 38, 4289-4299. doi: 10.1002/mma.3365 CrossRef Google Scholar
[5]	J. Francoise and J. Llibre, Analytical study of a triple Hopf bifurcation in a tritrophic food chain model, Appl. Math. Comput., 2011, 217, 7146-7154. Google Scholar
[6]	M. Han, On the maximum number of periodic solutions of piecewise smooth periodic equations by average method, Journal of Applied Analysis and Computation, 2017, 7, 788-794. doi: 10.11948/2017049 CrossRef Google Scholar
[7]	M. A. Khan and S. Poria, Generalized synchronization of nuclear spin generator system and the application in secure communication, J. Dyn. Syst. Geom. Theor., 2012, 10, 53-59. Google Scholar
[8]	D. Li and K. Huang, Hopf-bifurcation in a three-dimensional system, Appl. Math. Mech., 1989, 10(11), 1011-1018. doi: 10.1007/BF02014548 CrossRef Google Scholar
[9]	J. Llibre and D. Xiao, Limit cycle bifurcation from a non-isolated zero-hopf equilbrium of three-dimensional differential systems, Pro. Amer. Math. Soc., 2014, 142, 2047-2062. doi: 10.1090/S0002-9939-2014-11923-X CrossRef Google Scholar
[10]	M. R. Molaei, Ö. Umut.Generalized synchronization of nuclear spin generator system, Chaos Solitons Fractals, 2008, 37, 227-232. doi: 10.1016/j.chaos.2006.08.035 CrossRef Google Scholar
[11]	J. Sanders, F. Verhulst and F.Murdock, Averaging method in Nonliear Dynamical system, 2nd edn., Springer, New York, 2007. Google Scholar
[12]	S. Sherman, A third-order nonlinear system arising from a nuclear spin generator, Contributions to Differential Equations, 1963, 2, 197-227. Google Scholar
[13]	Ö. Umut, Controlling chaos in nuclear spin generator system using backstepping design, Appl. Sci., 2009, 11, 151-160. Google Scholar
[14]	C. Valls, Intergrablity of a nulear spin genetor, Bull. Sci. Math., 2011, 135, 435-441. doi: 10.1016/j.bulsci.2011.04.004 CrossRef Google Scholar
[15]	Q. Yuan and X. Yang, Computer-assisted verification of chaos in the model of nuclear spin generator, Appl. Math. Comput., 2009, 213, 148-152. Google Scholar