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Study of the Hopf bifurcation in the Lorenz, Chen and Lü systems

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Abstract

In this paper, we perform a complete study of the Hopf bifurcations in the three-parameter Lorenz system, \(\dot{x} = \sigma (y-x),\,\dot{y} = \rho x - y - xz,\,\dot{z} = -bz + xy\), with \(\sigma , \rho , b \in \mathbb {R}\). On the one hand, we reobtain the results found in the literature for the Lorenz model when the three parameters are positive. On the other hand, we completely determine the loci of all the degeneracies exhibited by the Hopf bifurcation of the origin and of the nontrivial equilibria. In this last case, we demonstrate, among other things, that the first two Lyapunov coefficients simultaneously vanish in two codimension-three bifurcation points, giving rise in both cases to the coexistence of three periodic orbits involved in a cusp bifurcation. The analytical study that we carry out, where several complicated expressions have to be handled, successfully closes the problem of the Hopf bifurcations in the Lorenz system. Moreover, from our results, it is easy to obtain all the information on the Hopf bifurcations in the Chen and Lü systems, taking into account that they are, generically, particular cases of the Lorenz system, as can be seen by means of a linear scaling in time and state variables.

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References

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

    Article  Google Scholar 

  2. Sparrow, C.: The Lorenz Equations. Springer, New York (1982)

    MATH  Google Scholar 

  3. Kús, M.: Integrals of motion for the Lorenz system. J. Phys. A 16, L689–L691 (1983)

    Article  MATH  Google Scholar 

  4. Glendinning, P., Sparrow, C.: T-points: a codimension two heteroclinic bifurcation. J. Stat. Phys. 43, 479–488 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. Giacomini, H., Neukirch, S.: Integrals of motion and the shape of the attractor for the Lorenz model. Phys. Lett. A 227, 309–318 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, X.: Lorenz equations. Part I: Existence and nonexistence of homoclinic orbits. SIAM J. Math. Anal. 27, 1057–1069 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Tucker, W.: The Lorenz attractor exists. C. R. Acad. Sci. Paris, Sér. I 328, 1197–1202 (1999)

    Article  MATH  Google Scholar 

  8. Viana, M.: What’s new on Lorenz strange attractors? Math. Intell. 22, 6–19 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Leonov, G.A.: Bounds for attractors and the existence of homoclinic orbits in the Lorenz system. J. Appl. Math. Mech. 65, 19–32 (2001)

    Article  MathSciNet  Google Scholar 

  10. Doedel, E.J., Krauskopf, B., Osinga, H.M.: Global bifurcations of the Lorenz manifold. Nonlinearity 19, 2947–2972 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Doedel, E.J., Krauskopf, B., Osinga, H.M.: Global invariant manifolds in the transition to preturbulence in the Lorenz system. Indag. Math. 22, 222–240 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Barrio, R., Serrano, S.: A three-parametric study of the Lorenz model. Phys. D 229, 43–51 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Barrio, R., Serrano, S.: Bounds for the chaotic region in the Lorenz model. Phys. D 238, 1615–1624 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Barrio, R., Shilnikov, A., Shilnikov, L.: Kneadings, symbolic dynamics and painting Lorenz chaos. Int. J. Bifurc. Chaos 22, 1230016 (2012)

    Article  MathSciNet  Google Scholar 

  15. Algaba, A., Fernández-Sánchez, F., Merino, M., Rodríguez-Luis, A.J.: Analysis of the T-point-Hopf bifurcation in the Lorenz system. Submitted for publication (2013)

  16. Swinnerton-Dyer, P.: The invariant algebraic surfaces of the Lorenz system. Math. Proc. Camb. Philos. Soc. 132, 385–393 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Llibre, J., Zhang, X.: Invariant algebraic surfaces of the Lorenz system. J. Math. Phys. 43, 1622–1645 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kokubu, H., Roussarie, R.: Existence of a singularly degenerate heteroclinic cycle in the Lorenz system and its dynamical consequences: Part 1. J. Dyn. Differ. Equ. 16, 513–557 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cao, J., Zhang, X.: Dynamics of the Lorenz system having an invariant algebraic surface. J. Math. Phys. 48, 1–13 (2007)

    Article  MathSciNet  Google Scholar 

  20. Messias, M.: Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system. J. Phys. A 42, 115101 (2009)

    Article  MathSciNet  Google Scholar 

  21. Llibre, J., Messias, M., da Silva, P.R.: Global dynamics of the Lorenz system with invariant algebraic surfaces. Int. J. Bifurc. Chaos 20, 3137–3155 (2010)

    Article  MATH  Google Scholar 

  22. Roschin, M.: Dangerous stability boundaries in the Lorenz model. Prikl. Mat. Mekh. 42, 950–952 (1978)

    Google Scholar 

  23. Pade, J., Rauh, A., Tsarouhas, G.: Analytical investigation of the Hopf bifurcation in the Lorenz model. Phys. Lett. A 115, 93–96 (1986)

    Article  MathSciNet  Google Scholar 

  24. Tsarouhas, G., Pade, J.: The Hopf bifurcation in the Lorenz by the 2-timing method model. Physica A 138, 507–517 (1986)

  25. Algaba, A., Fernández-Sánchez, F., Merino, M., Rodríguez-Luis, A.J.: Centers on center manifolds in the Lorenz, Chen and Lü systems. Commun. Nonlinear Sci. Numer. Simul. 19, 772–775 (2014)

    Article  MathSciNet  Google Scholar 

  26. Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, 659–661 (2002)

    Article  MATH  Google Scholar 

  28. Algaba, A., Fernández-Sánchez, F., Merino, M., Rodríguez-Luis, A.J.: Chen’s attractor exists if Lorenz repulsor exists: The Chen system is a special case of the Lorenz system. Chaos 23, 033108 (2013)

  29. Algaba, A., Fernández-Sánchez, F., Merino, M., Rodríguez-Luis, A.J.: The Lü system is a particular case of the Lorenz system. Phys. Lett. A 377, 2771–2776 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    Book  MATH  Google Scholar 

  31. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (2004)

    Book  MATH  Google Scholar 

  32. Wiggins, S.: Introduction to Applied Dynamical Systems and Chaos. Springer, New York (2003)

    MATH  Google Scholar 

  33. Golubitsky, M., Langford, W.F.: Classification and unfoldings of degenerate Hopf bifurcations. J. Differ. Eq. 41, 375–415 (1981)

  34. Gamero, E., Freire, E., Ponce, E.: Normal forms for planar systems with nilpotent linear part. In: Seydel, R., Schneider, F. W., Küpper, T., Troger, H. (eds.) Bifurcation and Chaos: Analysis, Algorithms, Applications, International Series of Numerical Mathematics, vol. 97, pp. 123–127, Birkhäuser, Basel (1991)

  35. Kulpa, W.: The Poincaré–Miranda theorem. Am. Math. Mon. 104, 545–550 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  36. Doedel, E.J., Champneys, A.R., Dercole, F., Fairgrieve, T., Kuznetsov, Y., Oldeman, B.E., Paffenroth, R., Sandstede, B., Wang, X., Zhang, C.: AUTO-07P: continuation and bifurcation software for ordinary differential equations (with HomCont). Technical report, Concordia University, (2010)

  37. Chen, G.: The Chen system revisited. Dyn. Cont. Dis. Ser. B 20, 691–696 (2013)

    MATH  Google Scholar 

  38. Algaba, A., Fernández-Sánchez, F., Merino, M., Rodríguez-Luis, A.J.: Comments on “The Chen system revisited”. Dyn. Cont. Dis. Ser. B (2014, accepted)

  39. Lü, J.H., Zhou, T.S., Chen, G.R., Zhang, S.C.: Local bifurcation of the Chen system. Int. J. Bifurc. Chaos 12, 2257–2270 (2002)

    Article  MATH  Google Scholar 

  40. Li, C., Chen, G.: A note on Hopf bifurcation in Chen’s system. Int. J. Bifurc. Chaos 13, 1609–1615 (2003)

    Article  MATH  Google Scholar 

  41. Li, T.C., Chen, G.R., Tang, Y.: On stability and bifurcation of Chen’s system. Chaos Solitons Fract. 19, 1269–1282 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Chang, Y., Chen, G.: Complex dynamics in Chen’s system. Chaos Solitons Fract. 27, 75–86 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yu, Y., Zhang, S.: Hopf bifurcation in the Lü system. Chaos Solitons Fract. 17, 901–906 (2003)

    Article  MATH  Google Scholar 

  44. Yu, Y., Zhang, S.: Hopf bifurcation analysis of the Lü system. Chaos Solitons Fract. 21, 1215–1220 (2004)

    Article  MATH  Google Scholar 

  45. Lü, Z., Duan, L.: Codimension-2 Bautin bifurcation in the Lü system. Phys. Lett. A 366, 442–446 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Lü, Z., Duan, L.: Control of codimension-2 Bautin bifurcation in chaotic Lü system. Commun. Theor. Phys. 52, 631–636 (2009)

  47. Mello, L.F., Coelho, S.F.: Degenerate Hopf bifurcations in the Lü system. Phys. Lett. A 373, 1116–1120 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Mahdi, A., Pessoa, C., Shafer, D.S.: Centers on center manifolds in the Lü system. Phys. Lett. A 375, 3509–3511 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Departamento de Matemáticas, Centro de Investigación de Física Teórica y Matemática FIMAT, Universidad de Huelva, 21071, Huelva, Spain

    Antonio Algaba, María C. Domínguez-Moreno & Manuel Merino

  2. Departamento de Matemática Aplicada II, E.S. Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Sevilla, Spain

    Alejandro J. Rodríguez-Luis

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  1. Antonio Algaba
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  2. María C. Domínguez-Moreno
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  3. Manuel Merino
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  4. Alejandro J. Rodríguez-Luis
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Correspondence to Alejandro J. Rodríguez-Luis.

Additional information

This work has been partially supported by the Ministerio de Educación y Ciencia, Plan Nacional I+D+I co-financed with FEDER funds, in the frame of the project MTM2010-20907-C02 and by the Consejería de Economía, Innovación, Ciencia y Empleo de la Junta de Andalucía (FQM-276, TIC-0130 and P12-FQM-1658).

Appendix A: Expression of the coefficient \(a_2\)

Appendix A: Expression of the coefficient \(a_2\)

$$\begin{aligned} a_2 = \frac{\left( 1+\sigma \right) \left( \sigma +\rho \right) ^{2} \, N2(\sigma ,\rho )}{D2(\sigma ,\rho )} \, , \\ \end{aligned}$$

with

$$\begin{aligned}&N2(\sigma ,\rho ) = 243\,{\rho }^{20}+1458\,{\rho }^{19}+243\,{\rho }^{18}-2880\,{\sigma }^{13}\\&\quad -\,420336\,{\sigma }^{14}+14250222\,{\sigma }^{17}+8090019\,{\sigma }^{16}\\&\quad -\,39109068\,{\sigma }^{20}\\&\quad -\,12446514\,{\sigma }^{21}+1076328\,{\sigma }^{23} \\&\quad -\,18397233\,{\sigma }^{18}-153693\,{\sigma }^{22}\\&\quad +\,1944\,{\sigma }^{26}+40824\,{\sigma }^{25}\\&\quad +\,324999\,{\sigma }^{24}-206172\,{\sigma }^{15}-50590584\,{\sigma }^{19}\\&\quad -\,44346\,{\sigma }^{5}{\rho }^{10}-1691728\,{\sigma }^{9}{\rho }^{16}+ 41563054\,{\sigma }^{13}{\rho }^{2}\\ \end{aligned}$$
$$\begin{aligned}&\quad +\,17428272\,{\sigma }^{9}{\rho }^{15}-3494744\,{\sigma }^{10}{\rho }^{15}\\&\quad +\,2870663904\,{\sigma }^{8}{\rho }^{13}- 248592818\,{\sigma }^{10}{\rho }^{13}\\&\quad +\,5982446474\,{\sigma }^{9}{\rho }^{12}-361764876\,{\sigma }^{8}{\rho }^{14}\\&\quad -\,181778178\,{\sigma }^{10}{\rho }^{12} \\&\quad -\,106478878\,{\sigma }^{7}{\rho }^{15}-662223860\,{\sigma }^{9}{\rho }^{13}\\&\quad +\,2427162\,{\sigma }^{7}{\rho }^{17}\\&\quad +\,595646\,{\sigma }^{6}{\rho }^{18}\\&\quad -\,7710032\,{\sigma }^{8}{\rho }^{15}+1183696732\,{\sigma }^{14}{\rho }^{9}\\&\quad -\,355104690\,{\sigma }^{14}{\rho }^{10}-5922876\,{\sigma }^{11}{\rho }^{14}\\&\quad +\, 128482\,{\sigma }^{11}{\rho }^{15}-31500\,{\sigma }^{6}{\rho }^{19}\\&\quad +\, 101524524\,{\sigma }^{11}{\rho }^{13}-682800\,{\sigma }^{8}{\rho }^{17}\\&\quad -\,191732\,{\sigma }^{7}{\rho }^{18}\\&\quad -\,2177477478\,{\sigma }^{17}{\rho }^{2}\\&\quad +\,5239793644\,{\sigma }^{17}{\rho }^{3}+33615738\,{\sigma }^{14}{\rho }^{11}\\&\quad -\,398172288\,{\sigma }^{17}\rho +18586517951\,{\sigma }^{16}{\rho }^{4}\\&\quad +\,3157539096\,{\sigma }^{16}{\rho }^{5}\\&\quad -\,2547036178\,{\sigma }^{16}{\rho }^{2}-2212324718\,{\sigma }^{16}{\rho }^{3}\\&\quad -\,44692044\,{\sigma }^{17}{\rho }^{8}+881080\,{\sigma }^{17}{\rho }^{9}\\&\quad -\,736827206\,{\sigma }^{17}{\rho }^{6}\\&\quad +\,432672528\,{\sigma }^{17}{\rho }^{7}+8010399762\,{\sigma }^{17}{\rho }^{4}\\&\quad -\,4266703020\,{\sigma }^{17}{\rho }^{5}+62953068\,{\sigma }^{21}\rho \\&\quad +\,2501347\,{\sigma }^{16}{\rho }^{10}+96653011\,{\sigma }^{16}{\rho }^{8}\\&\quad -\,41853162\,{\sigma }^{16}{\rho }^{9}-8372433052\,{\sigma }^{16}{\rho }^{6}\\&\quad +\, 1606042608\,{\sigma }^{16}{\rho }^{7}+56354926590\,{\sigma }^{13}{\rho }^{6}\\&\quad -\,114808316\,{\sigma }^{13}{\rho }^{7}+12511468\,{\sigma }^{15}{\rho }^{10}\\&\quad -\,27105198966\,{\sigma }^{14}{\rho }^{7}+3518184654\,{\sigma }^{14}{\rho }^{8}\\&\quad +\,30959488890\,{\sigma }^{14}{\rho }^{5}\\&\quad +\,26156530216\,{\sigma }^{14}{\rho }^{6}+13575366\,{\sigma }^{13}{\rho }^{12}\\&\quad -\,13780750632\,{\sigma }^{14}{\rho }^{4}-850755420\,{\sigma }^{13}{\rho }^{10}\\&\quad -\,48595808\,{\sigma }^{13}{\rho }^{11}-24420245502\,{\sigma }^{13}{\rho }^{8}\\&\quad +\,8600609168\,{\sigma }^{13}{\rho }^{9}\\&\quad -\,14254787768\,{\sigma }^{13}{\rho }^{5}+46141328\,{\sigma }^{10}{\rho }^{14}\\&\quad +\,40662\,{\sigma }^{6}{\rho }^{9}-663488\,{\sigma }^{13}{\rho }^{13}+6881634\,{\sigma }^{8}{\rho }^{16}\\&\quad -\,3411972\,{\sigma }^{5}{\rho }^{11}-307050024\,{\sigma }^{10}{\rho }^{6}\\&\quad +\,84996\,{\sigma }^{5}{\rho }^{19}+417976\,{\sigma }^{15}{\rho }^{11} \end{aligned}$$
$$\begin{aligned}&\quad -\,1122254\,{\sigma }^{14}{\rho }^{12}+7511796716\,{\sigma }^{15}{\rho }^{4} \\&\quad +\,31496910086\,{\sigma }^{15}{\rho }^{5}-7864774566\,{\sigma }^{15}{\rho }^{3}\\&\quad +\,12552\,{\sigma }^{6}{\rho }^{8}-18741158\,{\sigma }^{6}{\rho }^{10}\\&\quad -\,155566028\,{\sigma }^{6}{\rho }^{11}+2729474184\,{\sigma }^{15}{\rho }^{8}\\&\quad -\,366540838\,{\sigma }^{15}{\rho }^{9}-13114981904\,{\sigma }^{15}{\rho }^{6}\\&\quad -\,5141838738\,{\sigma }^{15}{\rho }^{7}-23117\,{\sigma }^{12}{\rho }^{14}\\&\quad +\,119284241\,{\sigma }^{12}{\rho }^{12}-3673238\,{\sigma }^{12}{\rho }^{13}\\&\quad +\,10231704\,{\sigma }^{22}{\rho }^{2}+4360929650\,{\sigma }^{18}{\rho }^{3}\\&\quad +\,28440\,{\sigma }^{23}{\rho }^{2}-252144708\,{\sigma }^{19}\rho \\&\quad +\,4168296\,{\sigma }^{23}\rho + 23748978\,{\sigma }^{22}\rho \\&\quad +\,152101116\,{\sigma }^{18}{\rho }^{2}+27406500\,{\sigma }^{20}\rho \\&\quad +\,955021844\,{\sigma }^{19}{\rho }^{2}\\&\quad +\,305556083\,{\sigma }^{18}{\rho }^{4}-2291858\,{\sigma }^{18}{\rho }^{8}\\&\quad +\, 8546214\,{\sigma }^{20}{\rho }^{5}+35038972\,{\sigma }^{19}{\rho }^{6}\\&\quad +\,893097\,{\sigma }^{20}{\rho }^{6}+492046197\,{\sigma }^{20}{\rho }^{2}\\&\quad +\,1244164130\,{\sigma }^{19}{\rho }^{3}-1408826962\,{\sigma }^{18}{\rho }^{5}\\&\quad -\,2268\,{\sigma }^{5}{\rho }^{20}-218664\,{\sigma }^{23}{\rho }^{3}\\&\quad -\,4711986\,{\sigma }^{22}{\rho }^{3}\\&\quad +\,5265\,{\sigma }^{4}{\rho }^{20}+214183612\,{\sigma }^{18}{\rho }^{6}\\&\quad +\,85104450\,{\sigma }^{20}{\rho }^{3}-52203\,{\sigma }^{22}{\rho }^{4}\\&\quad -\,544345960\,{\sigma }^{19}{\rho }^{4}-127320270\,{\sigma }^{20}{\rho }^{4}\\&\quad -\,106811480\,{\sigma }^{19}{\rho }^{5} \\&\quad +\,11652172\,{\sigma }^{18}{\rho }^{7}-1569014\,{\sigma }^{19}{\rho }^{7}\\&\quad +\, 946536\,{\sigma }^{21}{\rho }^{5}\!+\!109939944\,{\sigma }^{21}{\rho }^{2}\\&\quad -\,8972310\,{\sigma }^{21}{\rho }^{4}-24544404\,{\sigma }^{21}{\rho }^{3}\\&\quad +\,406642780\,{\sigma }^{13}{\rho }^{3} +77686944\,{\sigma }^{15}\rho \\&\quad +\,260945730\,{\sigma }^{14}{\rho }^{2}-2102180664\,{\sigma }^{14}{\rho }^{3}\\&\quad -\,5506681304\,{\sigma }^{13}{\rho }^{4}+42315918\,{\sigma }^{16}\rho \\&\quad -\,320800116\,{\sigma }^{15}{\rho }^{2}-577843980\,{\sigma }^{18}\rho \\&\quad +\,105518 \,{\rho }^{16}{\sigma }^{10}-802459112\,{\sigma }^{11}{\rho }^{12}\\&\quad -\, 40231838\,{\sigma }^{9}{\rho }^{14}+6884382\,{\sigma }^{14}\rho \\&\quad -\,6136896\,{\sigma }^{7}{\rho }^{16}-5452\,{\sigma }^{4}{\rho }^{11}-372971\,{\sigma }^{4}{\rho }^{12}\\&\quad -\,3742006\,{\sigma }^{4}{\rho }^{13}-29385140\,{\sigma }^{5}{\rho }^{12}\\&\quad +\,405\,{\rho }^{20}{\sigma }^{6}+21914\,{\rho }^{18}{\sigma }^{8}\\&\quad +\,4500\,{\rho }^{19}{\sigma }^{7}+60056\,{\rho }^{17}{\sigma }^{9}\\&\quad +\,252\,{\sigma }^{2}{\rho }^{14} \end{aligned}$$
$$\begin{aligned}&\quad -\,1194\,{\sigma }^{2}{\rho }^{15}-17218\,{\sigma }^{3}{\rho }^{13}\\&\quad -\,259096\,{\sigma }^{3}{\rho }^{14}+792\,\sigma \,{\rho }^{16}\\&\quad +\,115857\,{\rho }^{16}{\sigma }^{2}+46260\,{\rho }^{18}\sigma \\&\quad +\,1411736\,{\rho }^{15}{\sigma }^{3}\\&\quad +\,235475038\,{\rho }^{15}{\sigma }^{6}-182007788\,{\rho }^{15}{\sigma }^{5}\\&\quad +\,6948\,{\rho }^{17}\sigma \\&\quad -\,855543400\,{\rho }^{14}{\sigma }^{6}+226762938\,{\rho }^{14}{\sigma }^{5}\\&\quad +\,903183998\,{\rho }^{13}{\sigma }^{6}+78206376\,{\rho }^{13}{\sigma }^{5}\\&\quad +\,357606828\,{\rho }^{12}{\sigma }^{6}+703818\,{\rho }^{17}{\sigma }^{2}\\&\quad -\,1836\,{\rho }^{19}\sigma +6720184\,{\rho }^{16}{\sigma }^{3}\\&\quad -\,16345833\,{\rho }^{16}{\sigma }^{6}\\&\quad +\,38632104\,{\rho }^{16}{\sigma }^{5}+5043721692\,{\sigma }^{9}{\rho }^{9}\\&\quad -\,1769371270\,{\sigma }^{8}{\rho }^{9}-18875150328\,{\sigma }^{9}{\rho }^{11}\\&\quad +\,8136348948\,{\sigma }^{8}{\rho }^{11}+18866302362\,{\sigma }^{9}{\rho }^{10}\\&\quad +\,2934248955\,{\sigma }^{8}{\rho }^{10}+343386\,{\sigma }^{24}\rho \\&\quad - \,36585\,{\sigma }^{24}{\rho }^{2}+9720\,{\sigma }^{25}\rho -3146904\,{\sigma }^{12}{\rho }^{2}\\&\quad -\,173856\,{\sigma }^{11}{\rho }^{2}-59232\,{\sigma }^{12}\rho \\&\quad +\,1920\,{\sigma }^{11}\rho +12412021\,{\sigma }^{4}{\rho }^{14}\\&\quad +\,117360888\,{\sigma }^{11}{\rho }^{4}+95721556\,{\sigma }^{12}{\rho }^{3}\\&\quad -\,1394200\,{\sigma }^{11}{\rho }^{3}\\&\quad -\,68899588\,{\sigma }^{7}{\rho }^{9}+42070780\,{\sigma }^{9}{\rho }^{6}\\&\quad -\,7550895841\,{\sigma }^{12}{\rho }^{6}-9057017632\,{\sigma }^{11}{\rho }^{6}\\&\quad +\,577176\,{\sigma }^{8}{\rho }^{6}+2062324\,{\sigma }^{7}{\rho }^{8}\\&\quad +\,1206232\,{\sigma }^{9}{\rho }^{5}-8576215106\,{\sigma }^{12}{\rho }^{5}\\&\quad -\,75177642\,{\sigma }^{11}{\rho }^{5} \end{aligned}$$
$$\begin{aligned}&\quad -\,6912\,{\sigma }^{8}{\rho }^{5}+133656\,{\sigma }^{7}{\rho }^{7}-53024\,{\sigma }^{9}{\rho }^{4}\\&\quad +\,268181575\,{\sigma }^{12}{\rho }^{4}+1199141210\,{\sigma }^{7}{\rho }^{11}\\&\quad -\,3999623050\,{\sigma }^{9}{\rho }^{8}-27781074212\,{\sigma }^{12}{\rho }^{8}\\&\quad +\,55804463784\,{\sigma }^{11}{\rho }^{8}-174989295\,{\sigma }^{8}{\rho }^{8}\\&\quad -\,602262212\,{\sigma }^{7}{\rho }^{10}-300136840\,{\sigma }^{9}{\rho }^{7}\\&\quad +\,66018168244\,{\sigma }^{12}{\rho }^{7}+892178244\,{\sigma }^{11}{\rho }^{7}\\&\quad +\,12664200\,{\sigma }^{8}{\rho }^{7}+974226168\,{\sigma }^{7}{\rho }^{14}\\&\quad -\,1371520092\,{\sigma }^{12}{\rho }^{11}+2415519882\,{\sigma }^{11}{\rho }^{11}\\&\quad -\,3071837374\,{\sigma }^{7}{\rho }^{13}+6762551850\,{\sigma }^{12}{\rho }^{10}\\&\quad +\,3977465124\,{\sigma }^{11}{\rho }^{10}+2956574860\,{\sigma }^{7}{\rho }^{12}\\&\quad -\,9202730924\,{\sigma }^{12}{\rho }^{9}-38346842490\,{\sigma }^{11}{\rho }^{9}\\&\quad +\,826600\,{\sigma }^{10}{\rho }^{4}-148000\,{\sigma }^{10}{\rho }^{3}\\&\quad +\,640\,{\sigma }^{10}{\rho }^{2}-8609818946\,{\sigma }^{8}{\rho }^{12}\\&\quad -\,3120090\,{\sigma }^{6}{\rho }^{17}-133308\,{\sigma }^{5}{\rho }^{17}\\&\quad -\,6936101986\,{\sigma }^{10}{\rho }^{7}+87787388\,{\sigma }^{10}{\rho }^{5}\\&\quad -\,113526\,{\sigma }^{4}{\rho }^{19}+76644\,{\sigma }^{3}{\rho }^{19}\\&\quad -\,31690082304\,{\sigma }^{10}{\rho }^{10}\\&\quad +\,36227210346\,{\sigma }^{10}{\rho }^{9}+5288748972\,{\sigma }^{10}{\rho }^{8}\\&\quad +\,8042440102\,{\sigma }^{10}{\rho }^{11}-825272\,{\sigma }^{5}{\rho }^{18}\\&\quad -\,170652\,{\rho }^{18}{\sigma }^{2}-28281632\,{\rho }^{16}{\sigma }^{4}\\&\quad +\,45027924\,{\rho }^{15}{\sigma }^{4}-6480\,{\sigma }^{3}{\rho }^{20}\\&\quad -\,2010072\,{\sigma }^{13}\rho \\&\quad +\,80032\,{\rho }^{18}{\sigma }^{3}+4455\,{\rho }^{20}{\sigma }^{2}\\&\quad +\,443561\,{\rho }^{18}{\sigma }^{4}-2951402\,{\rho }^{17}{\sigma }^{3}\\&\quad -\,20736\,{\rho }^{19}{\sigma }^{2}+3689616\,{\rho }^{17}{\sigma }^{4}\\&\quad -\,1620\,{\rho }^{20}\sigma , \end{aligned}$$

and

$$\begin{aligned} D2(\sigma ,\rho )&= 288\sigma \sqrt{-{\varDelta }} \left( \rho -1 \right) ^{3} \left( 4\sigma ^2(\rho -1)^2 \right. \\&\left. -\,9(\sigma +\rho )^2 {\varDelta } \right) \left( \sigma ^2(\rho -1)^2 \right. \\&\left. -\, (\sigma +\rho )^2 {\varDelta } \right) ^3\! \left( 4\sigma ^2(\rho \!-\!1)^2 - (\sigma \!+\!\rho )^2 {\varDelta } \right) ^3. \end{aligned}$$

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Algaba, A., Domínguez-Moreno, M.C., Merino, M. et al. Study of the Hopf bifurcation in the Lorenz, Chen and Lü systems. Nonlinear Dyn 79, 885–902 (2015). https://doi.org/10.1007/s11071-014-1709-2

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