Global invariant manifolds near a Shilnikov homoclinic bifurcation

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January 2014, 1(1): 1-38. doi: 10.3934/jcd.2014.1.1

Global invariant manifolds near a Shilnikov homoclinic bifurcation

Pablo Aguirre ^1, , Bernd Krauskopf ^2, and Hinke M. Osinga ^2,

1.	Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile
2.	Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142

Received November 2011 Revised July 2012 Published April 2014

Abstract
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We consider a three-dimensional vector field with a Shilnikov homoclinic orbit that converges to a saddle-focus equilibrium in both forward and backward time. The one-parameter unfolding of this global bifurcation depends on the sign of the saddle quantity. When it is negative, breaking the homoclinic orbit produces a single stable periodic orbit; this is known as the simple Shilnikov bifurcation. However, when the saddle quantity is positive, the mere existence of a Shilnikov homoclinic orbit induces complicated dynamics, and one speaks of the chaotic Shilnikov bifurcation; in particular, one finds suspended horseshoes and countably many periodic orbits of saddle type. These well-known and celebrated results on the Shilnikov homoclinic bifurcation have been obtained by the classical approach of reducing a Poincaré return map to a one-dimensional map.
In this paper, we study the implications of the transition through a Shilnikov bifurcation for the overall organization of the three-dimensional phase space of the vector field. To this end, we focus on the role of the two-dimensional global stable manifold of the equilibrium, as well as those of bifurcating saddle periodic orbits. We compute the respective two-dimensional global manifolds, and their intersection curves with a suitable sphere, as families of orbit segments with a two-point boundary-value-problem setup. This allows us to determine how the arrangement of global manifolds changes through the bifurcation and how this influences the topological organization of phase space. For the simple Shilnikov bifurcation, we show how the stable manifold of the saddle focus forms the basin boundary of the bifurcating stable periodic orbit. For the chaotic Shilnikov bifurcation, we find that the stable manifold of the equilibrium is an accessible set of the stable manifold of a chaotic saddle that contains countably many periodic orbits of saddle type. In intersection with a suitably chosen sphere we find that this stable manifold is an indecomposable continuum consisiting of infinitely many closed curves that are locally a Cantor bundle of arcs.

Keywords: invariant manifolds, Shilnikov chaos., Homoclinic bifurcation.

Mathematics Subject Classification: Primary: 34C37, 34C45, 37D45; Secondary: 34C23, 37M9.

Citation: Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1

References:

[1]	P. Aguirre, E. Doedel, B. Krauskopf and H. M. Osinga, Investigating the consequences of global bifurcations for two-dimensional manifolds of vector fields,, Discrete Contin. Dyn. Syst. A, 29 (2011), 1309. doi: 10.3934/dcds.2011.29.1309. Google Scholar
[2]	P. Aguirre, E. González-Olivares and E. Sáez, Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect,, SIAM J. Appl. Math., 69 (2009), 1244. doi: 10.1137/070705210. Google Scholar
[3]	P. Aguirre, B. Krauskopf and H. M. Osinga, Global invariant manifolds near homoclinic orbits to a real saddle: (non)orientability and flip bifurcation,, SIAM J. Appl. Dyn. Syst., 12 (2013), 1803. Google Scholar
[4]	R. Barrio, F. Blesa and S. Serrano, Qualitative analysis of the Rössler equations: Bifurcations of limit cycles and chaotic attractors,, Physica D, 238* (2009), 1087. doi: 10.1016/j.physd.2009.03.010. Google Scholar*
[5]	M. R. Bassett and J. L. Hudson, Shil'nikov chaos during copper electrodissolution,, J. Phys. Chem., 92 (1988), 6963. doi: 10.1021/j100335a025. Google Scholar
[6]	L. A. Belyakov, A case of the generation of a periodic motion with homoclinic curves,, Mat. Zam., 15 (1974), 571. Google Scholar
[7]	L. A. Belyakov, Bifurcation of systems with homoclinic curve of a saddle-focus with saddle quantity zero,, Mat. Zam., 36 (1984), 681. Google Scholar
[8]	W.-J. Beyn, On well-posed problems for connecting orbits in dynamical systems,, in Chaotic Numerics* (Geelong*, (1994), 131. doi: 10.1090/conm/172/01802. Google Scholar
[9]	A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd, When Shil'nikov meets Hopf in excitable systems,, SIAM J. Appl. Dyn. Syst., 6 (2007), 663. doi: 10.1137/070682654. Google Scholar
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[12]	B. Deng and G. Hines, Food chain chaos due to Shilnikov's orbit,, Chaos, 12 (2002), 533. doi: 10.1063/1.1482255. Google Scholar
[13]	M. Desroches, B. Krauskopf and H. M. Osinga, The geometry of slow manifolds near a folded node,, SIAM J. Appl. Dyn. Syst., 7 (2008), 1131. doi: 10.1137/070708810. Google Scholar
[14]	A. Dhooge, W. Govaerts and Yu. A. Kuznetsov, MATCONT: A Matlab package for numerical bifurcation analysis of ODEs,, ACM Trans. Math. Software, 29 (2003), 141. doi: 10.1145/779359.779362. Google Scholar
[15]	E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems,, Congr. Numer., 30 (1981), 265. Google Scholar
[16]	E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations,, in Numerical Continuation Methods for Dynamical Systems* (eds. B. Krauskopf*, (2007), 1. doi: 10.1007/978-1-4020-6356-5_1. Google Scholar
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[18]	E. J. Doedel, B. Krauskopf and H. M. Osinga, Global invariant manifolds in the transition to preturbulence in the Lorenz system,, Indagationes Mathematicae, 22 (2011), 222. doi: 10.1016/j.indag.2011.10.007. Google Scholar
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[20]	J. P. England, B. Krauskopf and H. M. Osinga, Computing one-dimensional global manifolds of Poincaré maps by continuation,, SIAM J. Appl. Dyn. Sys., 4 (2005), 1008. doi: 10.1137/05062408X. Google Scholar
[21]	J. P. England, B. Krauskopf and H. M. Osinga, Computing two-dimensional global invariant manifolds in slow-fast systems,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 805. doi: 10.1142/S0218127407017562. Google Scholar
[22]	J. A. Feroe, Homoclinic orbits in a parametrized saddle-focus system,, Physica D, 62 (1993), 254. doi: 10.1016/0167-2789(93)90285-9. Google Scholar
[23]	M. Friedman and E. J. Doedel, Numerical computation and continuation of invariant manifolds connecting fixed points,, SIAM J. Numer. Anal., 28 (1991), 789. doi: 10.1137/0728042. Google Scholar
[24]	P. Gaspard, R. Kapral and G. Nicolis, Bifurcation phenomena near homoclinic systems: A two-parameter analysis,, J. Statist. Phys., 35 (1984), 697. doi: 10.1007/BF01010829. Google Scholar
[25]	P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits,, J. Statist. Phys., 35 (1984), 645. doi: 10.1007/BF01010828. Google Scholar
[26]	G. Gómez, W. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, Invariant manifolds, the spatial three-body problem and space mission design,, Astrodynamics Specialist Meeting, (2001), 01. Google Scholar
[27]	W. Govaerts and Y. A. Kuznetsov, Interactive continuation tools,, in Numerical Continuation Methods for Dynamical Systems* (eds. B. Krauskopf*, (2007), 51. doi: 10.1007/978-1-4020-6356-5_2. Google Scholar
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[29]	J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system,, SIAM J. Appl. Dyn. Syst., 9 (2010), 138. doi: 10.1137/090758404. Google Scholar
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[31]	A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations,, J. Dynam. Diff. Eqs., 12 (2000), 807. doi: 10.1023/A:1009046621861. Google Scholar
[32]	F. C. Hoppensteadt, An Introduction to the Mathematics of Neurons, Modeling in the Frequency Domain,, Cambridge University Press, (1997). Google Scholar
[33]	E. A. Jackson, The Lorenz system: II. The homoclinic convolution of the stable manifolds,, Phys. Scr., 32 (1985), 469. doi: 10.1088/0031-8949/32/5/001. Google Scholar
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[36]	B. Krauskopf and H. M. Osinga, Computing geodesic level sets on global (un)stable manifolds of vector fields,, SIAM J. Appl. Dyn. Sys., 2 (2003), 546. doi: 10.1137/030600180. Google Scholar
[37]	B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments,, in Numerical Continuation Methods for Dynamical Systems* (eds. B. Krauskopf*, (2007), 117. doi: 10.1007/978-1-4020-6356-5_4. Google Scholar
[38]	B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763. doi: 10.1142/S0218127405012533. Google Scholar
[39]	B. Krauskopf and T. Riess, A Lin's method approach to finding and continuing heteroclinic orbits connections involving periodic orbits,, Nonlinearity, 21 (2008), 1655. doi: 10.1088/0951-7715/21/8/001. Google Scholar
[40]	B. Krauskopf, K. Schneider, J. Sieber, S. Wieczorek and M. Wolfrum, Excitability and self-pulsations near homoclinic bifurcations in semiconductor lasers,, Optics Communications, 215* (2003), 367. doi: 10.1016/S0030-4018(02)02239-3. Google Scholar*
[41]	Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory,, $3^{rd}$ edition, (2004). Google Scholar
[42]	C. M. Lee, P. J. Collins, B. Krauskopf and H. M. Osinga, Tangency bifurcations of global Poincaré maps,, SIAM J. Appl. Dyn. Syst., 7 (2008), 712. doi: 10.1137/07069972X. Google Scholar
[43]	E. N. Lorenz, Deterministic nonperiodic flows,, J. Atmosph. Sci., 20 (1963), 130. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. Google Scholar
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[45]	T. Noh, Shilnikov's chaos in the oxidation of formic acid with bismuth ion on Pt ring electrode,, Electrochimica Acta, 54 (2009), 3657. Google Scholar
[46]	B. E. Oldeman, A. R. Champneys and B. Krauskopf, Homoclinic branch switching: A numerical implementation of Lin's method,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2977. doi: 10.1142/S0218127403008326. Google Scholar
[47]	B. E. Oldeman, B. Krauskopf and A. R. Champneys, Numerical unfoldings of codimension-three resonant homoclinic flip bifurcations,, Nonlinearity, 14 (2001), 597. doi: 10.1088/0951-7715/14/3/309. Google Scholar
[48]	H. M. Osinga, Nonorientable manifolds in three-dimensional vector fields,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 553. doi: 10.1142/S0218127403006777. Google Scholar
[49]	I. M. Ovsyannikov and L. P. Shil'nikov, On systems with a saddle-focus homoclinic curve,, Math. USSR Sbornik, 58 (1987), 557. Google Scholar
[50]	T. Peacock and T. Mullin, Homoclinic bifurcations in a liquid crystal flow,, J. Fluid Mech., 432* (2001), 369. Google Scholar*
[51]	A. M. Rucklidge, Chaos in a low-order model of magnetoconvection,, Physica D, 62 (1993), 323. doi: 10.1016/0167-2789(93)90291-8. Google Scholar
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References:

[1]	P. Aguirre, E. Doedel, B. Krauskopf and H. M. Osinga, Investigating the consequences of global bifurcations for two-dimensional manifolds of vector fields,, Discrete Contin. Dyn. Syst. A, 29 (2011), 1309. doi: 10.3934/dcds.2011.29.1309. Google Scholar
[2]	P. Aguirre, E. González-Olivares and E. Sáez, Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect,, SIAM J. Appl. Math., 69 (2009), 1244. doi: 10.1137/070705210. Google Scholar
[3]	P. Aguirre, B. Krauskopf and H. M. Osinga, Global invariant manifolds near homoclinic orbits to a real saddle: (non)orientability and flip bifurcation,, SIAM J. Appl. Dyn. Syst., 12 (2013), 1803. Google Scholar
[4]	R. Barrio, F. Blesa and S. Serrano, Qualitative analysis of the Rössler equations: Bifurcations of limit cycles and chaotic attractors,, Physica D, 238* (2009), 1087. doi: 10.1016/j.physd.2009.03.010. Google Scholar*
[5]	M. R. Bassett and J. L. Hudson, Shil'nikov chaos during copper electrodissolution,, J. Phys. Chem., 92 (1988), 6963. doi: 10.1021/j100335a025. Google Scholar
[6]	L. A. Belyakov, A case of the generation of a periodic motion with homoclinic curves,, Mat. Zam., 15 (1974), 571. Google Scholar
[7]	L. A. Belyakov, Bifurcation of systems with homoclinic curve of a saddle-focus with saddle quantity zero,, Mat. Zam., 36 (1984), 681. Google Scholar
[8]	W.-J. Beyn, On well-posed problems for connecting orbits in dynamical systems,, in Chaotic Numerics* (Geelong*, (1994), 131. doi: 10.1090/conm/172/01802. Google Scholar
[9]	A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd, When Shil'nikov meets Hopf in excitable systems,, SIAM J. Appl. Dyn. Syst., 6 (2007), 663. doi: 10.1137/070682654. Google Scholar
[10]	A. R. Champneys, Y. Kuznetsov and B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis,, Int. J. Bifurc. Chaos, 6 (1996), 867. doi: 10.1142/S0218127496000485. Google Scholar
[11]	M. G. Clerc, P. C. Encina and E. Tirapegui, Shilnikov bifurcation: Stationary quasi-reversal bifurcation,, Int. J. Bifurc. Chaos, 18 (2008), 1905. doi: 10.1142/S0218127408021440. Google Scholar
[12]	B. Deng and G. Hines, Food chain chaos due to Shilnikov's orbit,, Chaos, 12 (2002), 533. doi: 10.1063/1.1482255. Google Scholar
[13]	M. Desroches, B. Krauskopf and H. M. Osinga, The geometry of slow manifolds near a folded node,, SIAM J. Appl. Dyn. Syst., 7 (2008), 1131. doi: 10.1137/070708810. Google Scholar
[14]	A. Dhooge, W. Govaerts and Yu. A. Kuznetsov, MATCONT: A Matlab package for numerical bifurcation analysis of ODEs,, ACM Trans. Math. Software, 29 (2003), 141. doi: 10.1145/779359.779362. Google Scholar
[15]	E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems,, Congr. Numer., 30 (1981), 265. Google Scholar
[16]	E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations,, in Numerical Continuation Methods for Dynamical Systems* (eds. B. Krauskopf*, (2007), 1. doi: 10.1007/978-1-4020-6356-5_1. Google Scholar
[17]	E. J. Doedel, B. Krauskopf and H. M. Osinga, Global bifurcations of the Lorenz manifold,, Nonlinearity, 19 (2006), 2947. doi: 10.1088/0951-7715/19/12/013. Google Scholar
[18]	E. J. Doedel, B. Krauskopf and H. M. Osinga, Global invariant manifolds in the transition to preturbulence in the Lorenz system,, Indagationes Mathematicae, 22 (2011), 222. doi: 10.1016/j.indag.2011.10.007. Google Scholar
[19]	E. J. Doedel and B. E. Oldeman, AUTO-07p Version 0.7: Continuation and bifurcation software for ordinary differential equations,, with major contributions from A. R. Champneys, (2010). Google Scholar
[20]	J. P. England, B. Krauskopf and H. M. Osinga, Computing one-dimensional global manifolds of Poincaré maps by continuation,, SIAM J. Appl. Dyn. Sys., 4 (2005), 1008. doi: 10.1137/05062408X. Google Scholar
[21]	J. P. England, B. Krauskopf and H. M. Osinga, Computing two-dimensional global invariant manifolds in slow-fast systems,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 805. doi: 10.1142/S0218127407017562. Google Scholar
[22]	J. A. Feroe, Homoclinic orbits in a parametrized saddle-focus system,, Physica D, 62 (1993), 254. doi: 10.1016/0167-2789(93)90285-9. Google Scholar
[23]	M. Friedman and E. J. Doedel, Numerical computation and continuation of invariant manifolds connecting fixed points,, SIAM J. Numer. Anal., 28 (1991), 789. doi: 10.1137/0728042. Google Scholar
[24]	P. Gaspard, R. Kapral and G. Nicolis, Bifurcation phenomena near homoclinic systems: A two-parameter analysis,, J. Statist. Phys., 35 (1984), 697. doi: 10.1007/BF01010829. Google Scholar
[25]	P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits,, J. Statist. Phys., 35 (1984), 645. doi: 10.1007/BF01010828. Google Scholar
[26]	G. Gómez, W. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, Invariant manifolds, the spatial three-body problem and space mission design,, Astrodynamics Specialist Meeting, (2001), 01. Google Scholar
[27]	W. Govaerts and Y. A. Kuznetsov, Interactive continuation tools,, in Numerical Continuation Methods for Dynamical Systems* (eds. B. Krauskopf*, (2007), 51. doi: 10.1007/978-1-4020-6356-5_2. Google Scholar
[28]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,, $2^{nd}$ edition, (1986). Google Scholar
[29]	J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system,, SIAM J. Appl. Dyn. Syst., 9 (2010), 138. doi: 10.1137/090758404. Google Scholar
[30]	A. Gutek and J. van Mill, Continua that are locally a bundle of arcs,, Topology Proceedings, 7 (1982), 63. Google Scholar
[31]	A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations,, J. Dynam. Diff. Eqs., 12 (2000), 807. doi: 10.1023/A:1009046621861. Google Scholar
[32]	F. C. Hoppensteadt, An Introduction to the Mathematics of Neurons, Modeling in the Frequency Domain,, Cambridge University Press, (1997). Google Scholar
[33]	E. A. Jackson, The Lorenz system: II. The homoclinic convolution of the stable manifolds,, Phys. Scr., 32 (1985), 469. doi: 10.1088/0031-8949/32/5/001. Google Scholar
[34]	J. Kennedy, How indecomposable continua arise in dynamical systems,, Annals of the New York Academy of Sciences, 704* (1993), 180. doi: 10.1111/j.1749-6632.1993.tb52522.x. Google Scholar*
[35]	B. Krauskopf and H. M. Osinga, Two-dimensional global manifolds of vector fields,, Chaos, 9 (1999), 768. doi: 10.1063/1.166450. Google Scholar
[36]	B. Krauskopf and H. M. Osinga, Computing geodesic level sets on global (un)stable manifolds of vector fields,, SIAM J. Appl. Dyn. Sys., 2 (2003), 546. doi: 10.1137/030600180. Google Scholar
[37]	B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments,, in Numerical Continuation Methods for Dynamical Systems* (eds. B. Krauskopf*, (2007), 117. doi: 10.1007/978-1-4020-6356-5_4. Google Scholar
[38]	B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763. doi: 10.1142/S0218127405012533. Google Scholar
[39]	B. Krauskopf and T. Riess, A Lin's method approach to finding and continuing heteroclinic orbits connections involving periodic orbits,, Nonlinearity, 21 (2008), 1655. doi: 10.1088/0951-7715/21/8/001. Google Scholar
[40]	B. Krauskopf, K. Schneider, J. Sieber, S. Wieczorek and M. Wolfrum, Excitability and self-pulsations near homoclinic bifurcations in semiconductor lasers,, Optics Communications, 215* (2003), 367. doi: 10.1016/S0030-4018(02)02239-3. Google Scholar*
[41]	Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory,, $3^{rd}$ edition, (2004). Google Scholar
[42]	C. M. Lee, P. J. Collins, B. Krauskopf and H. M. Osinga, Tangency bifurcations of global Poincaré maps,, SIAM J. Appl. Dyn. Syst., 7 (2008), 712. doi: 10.1137/07069972X. Google Scholar
[43]	E. N. Lorenz, Deterministic nonperiodic flows,, J. Atmosph. Sci., 20 (1963), 130. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. Google Scholar
[44]	J. R. Munkres, Topology,, $2^{nd}$ edition, (2000). Google Scholar
[45]	T. Noh, Shilnikov's chaos in the oxidation of formic acid with bismuth ion on Pt ring electrode,, Electrochimica Acta, 54 (2009), 3657. Google Scholar
[46]	B. E. Oldeman, A. R. Champneys and B. Krauskopf, Homoclinic branch switching: A numerical implementation of Lin's method,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2977. doi: 10.1142/S0218127403008326. Google Scholar
[47]	B. E. Oldeman, B. Krauskopf and A. R. Champneys, Numerical unfoldings of codimension-three resonant homoclinic flip bifurcations,, Nonlinearity, 14 (2001), 597. doi: 10.1088/0951-7715/14/3/309. Google Scholar
[48]	H. M. Osinga, Nonorientable manifolds in three-dimensional vector fields,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 553. doi: 10.1142/S0218127403006777. Google Scholar
[49]	I. M. Ovsyannikov and L. P. Shil'nikov, On systems with a saddle-focus homoclinic curve,, Math. USSR Sbornik, 58 (1987), 557. Google Scholar
[50]	T. Peacock and T. Mullin, Homoclinic bifurcations in a liquid crystal flow,, J. Fluid Mech., 432* (2001), 369. Google Scholar*
[51]	A. M. Rucklidge, Chaos in a low-order model of magnetoconvection,, Physica D, 62 (1993), 323. doi: 10.1016/0167-2789(93)90291-8. Google Scholar
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