Necessary and sufficient conditions of the existence of homoclinic trajectories and cascade of bifurcations in Lorenz-like systems: birth of strange attractor and 9 homoclinic bifurcations

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• 作者：G. A. Leonov
• 关键词：Homoclinic trajectories ; Strange attractor ; Blue sky catastrophe ; Cascade of bifurcations ; Fishing principle ; Lorenz ; like system ; Chen system ; Lu system ; Shimizu–Morioka system ; Saddle value ; Separatrix of saddle point
• 刊名：Nonlinear Dynamics
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：84
• 期：2
• 页码：1055-1062
• 全文大小：2,211 KB
• 参考文献：1.Homburg, A.J., Sandstede, B.: Homoclinic Bifurcations in Vector Fields. In: Handbook of Dynamical Systems, vol. 3, pp. 379–524. Elsevier, Amsterdam (2010)
  2.Belykh, V.N.: Bifurcation of separatrices of a saddle point of the Lorenz system. Differ. Equ. 20(10), 1184–1191 (1984)MathSciNet MATH
  3.Hastings, S.P., Troy, W.C.: A shooting approach to the Lorenz equations. Bull. Am. Math. Soc. 27, 298–303 (1992)MathSciNet CrossRef MATH
  4.Hastings, S.P., Troy, W.C.: A proof that the Lorenz equations have a homoclinic orbits. J. Differ. Equ. 113(1), 166–188 (1994)MathSciNet CrossRef MATH
  5.Hastings, S.P., Troy, W.C.: A shooting approach to chaos in the Lorenz equations. J. Differ. Equ. 127(1), 41–53 (1996)MathSciNet CrossRef MATH
  6.Leonov, G.A.: Estimation of loop-bifurcation parameters for a saddle—point separatrix of a Lorenz system. Differ. Equ. 24(6), 634–638 (1988). (Translated from Differential’nya Uravneniya)MATH
  7.Leonov, G.A.: On estimates of the bifurcation values of the parameters of a Lorenz system. Russ. Math. Surv. 43(3), 216–217 (1988)MathSciNet CrossRef MATH
  8.Leonov, G.A.: On homoclinic bifurcation in the Lorenz system. Vestnik St. Petersburg Univ. Math. 32(1), 13–15 (1999)MATH
  9.Chen, X.: Lorenz equations. Pt. I. Existence and nonexistence of homoclinic orbits. SIAM J. Math. Anal. 27(4), 1057–1069 (1996)MathSciNet CrossRef MATH
  10.Leonov, G.A.: The Tricomi problem on the existence of homoclinic orbits in dissipative systems. J. Appl. Math. Mech. 77(3), 29600304 (2013)MathSciNet CrossRef
  11.Leonov, G.A.: Strange Attractors and Classical Stability Theory. St. Petersburg University Press, St. Petersburg (2008)MATH
  12.Shimizu, T., Morioka, N.: On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model. Phys. Lett. A 76, 201–204 (1980)MathSciNet CrossRef
  13.Shilnikov, A.L.: On bifurcations of the Lorenz attractor in the Shimizu–Morioka model. Phys. D 62, 338–346 (1993)MathSciNet CrossRef
  14.Tigan, G.: On a method of finding homoclinic and heteroclinic orbits in multidimensional dynamical systems. Appl. Math. Inf. Sci. 4(3), 383–394 (2010)
  15.Tigan, G., Turaev, D.: Analytical search for homoclinic bifurcations in the Shimizu–Morioka model. Phys. D 240, 985–989 (2011)MathSciNet CrossRef MATH
  16.Leonov, G.A.: General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu–Morioka, Lu and Chen systems. Phys. Lett. A 376, 3045–3050 (2012)MathSciNet CrossRef MATH
  17.Leonov, G.A.: The Tricomi problem for the Shimizu–Morioka dynamical system. Dokl. Math. 86(3), 850–853 (2012)MathSciNet CrossRef MATH
  18.Leonov, G.A.: Formulas for the Lyapunov dimension of attractors of the generalized Lorenz systems. Dokl. Math. 87(3), 13–18 (2013)MathSciNet CrossRef
  19.Leonov, G.A.: Shilnikov chaos in Lorenz-like systems. Int. J. Bifurc. Chaos 23(3), art. num. 1350058 (2013)
  20.Lu, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, 1789–1812 (2002)MathSciNet CrossRef MATH
  21.Leonov, G.A., Kuznetsov, N.V.: On differences and similarities in the analysis of Lorenz Chen and Lu systems. Appl. Math. Comput. 256, 334–343 (2015)MathSciNet CrossRef
  22.Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9(7), 1465–1466 (1999)MathSciNet CrossRef MATH
  23.Leonov, G.A.: Attractors, limit cycles and homoclinic orbits of low-dimensional quadratic systems. Analytical methods. Can. Appl. Math. Q. 17(1), 121–159 (2009)MathSciNet MATH
  24.Leonov, G.A.: Criteria for the existence of homoclinic orbits of systems Lu and Chen. Dokl. Math. 87(2), 220–223 (2012)CrossRef MATH
  25.Leonov, G.A.: Rössler systems: estimates for the dimension of attractors and homoclinic orbits. Dokl. Math. 89(3), 369–371 (2014)MathSciNet CrossRef MATH
  26.Leonov, G.A.: Fishing principle for homoclinic and heteroclinic trajectories. Nonlinear Dyn. 78, 2751–2758 (2014)MathSciNet CrossRef MATH
  27.Leonov, G.A., Kuznetsov, N.V., Mokaev, T.N.: Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys. J. Spec. Top. 224, 1421–1458 (2015)CrossRef
  28.Holmes, P.J., Morsden, J.E.: A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam. Arch. Ration. Mech. Anal. 76(2), 135–166 (1981)MathSciNet CrossRef MATH
  29.Hu, W.P., Deng, Z.C., Wang, B., Ouyang, H.J.: Chaos in embedded single-walled carbon nanotube. Nonlinear Dyn. 72(1–2), 389–298 (2013)MathSciNet CrossRef
  30.Leonov, G.A., Burkin, I.M., Shepeljavyi, A.I.: Frequency Methods in Oscillation Theory. Kluwer Academic Publishers, Dordrecht (1996)CrossRef MATH
  31.Leonov, G.A.: Sets of transversal curves for two-dimensional systems of differential equations. Vestnik St. Petersburg Univ. Math. 39(4), 219–245 (2006)MathSciNet
  
• 作者单位：G. A. Leonov (1)
  
  1. Institute of Problems in Mechanical Engineering, RAS, Saint Petersburg State University, Saint Petersburg, Russia
  
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X

文摘

Necessary and sufficient conditions of the existence of homoclinic trajectories in Lorenz-like systems are obtained. The Shimizu–Morioka, Lu, and Chen systems are considered. For special Lorenz-like system with zero saddle value, cascade of bifurcations is obtained. Birth of strange attractor, blue sky catastrophe and 9 homoclinic bifurcations are discovered.