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

Global invariant manifolds near a Shilnikov homoclinic bifurcation

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

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.
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.

[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.

[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.

[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.

[5]

M. R. Bassett and J. L. Hudson, Shil'nikov chaos during copper electrodissolution,, J. Phys. Chem., 92 (1988), 6963. doi: 10.1021/j100335a025.

[6]

L. A. Belyakov, A case of the generation of a periodic motion with homoclinic curves,, Mat. Zam., 15 (1974), 571.

[7]

L. A. Belyakov, Bifurcation of systems with homoclinic curve of a saddle-focus with saddle quantity zero,, Mat. Zam., 36 (1984), 681.

[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.

[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.

[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.

[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.

[12]

B. Deng and G. Hines, Food chain chaos due to Shilnikov's orbit,, Chaos, 12 (2002), 533. doi: 10.1063/1.1482255.

[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.

[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.

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E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems,, Congr. Numer., 30 (1981), 265.

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[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.

[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.

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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).

[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.

[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.

[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.

[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.

[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.

[25]

P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits,, J. Statist. Phys., 35 (1984), 645. doi: 10.1007/BF01010828.

[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.

[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.

[28]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,, $2^{nd}$ edition, (1986).

[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.

[30]

A. Gutek and J. van Mill, Continua that are locally a bundle of arcs,, Topology Proceedings, 7 (1982), 63.

[31]

A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations,, J. Dynam. Diff. Eqs., 12 (2000), 807. doi: 10.1023/A:1009046621861.

[32]

F. C. Hoppensteadt, An Introduction to the Mathematics of Neurons, Modeling in the Frequency Domain,, Cambridge University Press, (1997).

[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.

[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.

[35]

B. Krauskopf and H. M. Osinga, Two-dimensional global manifolds of vector fields,, Chaos, 9 (1999), 768. doi: 10.1063/1.166450.

[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.

[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.

[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.

[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.

[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.

[41]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory,, $3^{rd}$ edition, (2004).

[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.

[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.

[44]

J. R. Munkres, Topology,, $2^{nd}$ edition, (2000).

[45]

T. Noh, Shilnikov's chaos in the oxidation of formic acid with bismuth ion on Pt ring electrode,, Electrochimica Acta, 54 (2009), 3657.

[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.

[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.

[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.

[49]

I. M. Ovsyannikov and L. P. Shil'nikov, On systems with a saddle-focus homoclinic curve,, Math. USSR Sbornik, 58 (1987), 557.

[50]

T. Peacock and T. Mullin, Homoclinic bifurcations in a liquid crystal flow,, J. Fluid Mech., 432 (2001), 369.

[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.

[52]

M. A. F. Sanjuán, J. Kennedy, E. Ott and J. A. Yorke, Indecomposable continua and the characterization of strange sets in nonlinear dynamics,, Phys. Rev. Lett., 78 (1997), 1892.

[53]

L. P. Shilnikov, A case of the existence of a countable number of periodic orbits,, Sov. Math. Dokl., 6 (1965), 163.

[54]

L. P. Shilnikov, A contribution to the problem of the structure of an extended neighborhood of a rough state to a saddle-focus type,, Math. USSR-Sb, 10 (1970), 91.

[55]

L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part II,, World Scientific Series on Nonlinear Science, (2001). doi: 10.1142/9789812798558_0001.

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S. Wieczorek and B. Krauskopf, Bifurcations of $n-$homoclinic orbits in optically injected lasers,, Nonlinearity, 18 (2005), 1095. doi: 10.1088/0951-7715/18/3/010.

[60]

S. Wieczorek, B. Krauskopf and D. Lenstra, A unifying view of bifurcations in a semiconductor laser subject to optical injection,, Optics Communications, 172 (1999), 279. doi: 10.1016/S0030-4018(99)00603-3.

[61]

S. Wieczorek, B. Krauskopf and D. Lenstra, Multipulse excitability in a semiconductor laser with optical injection,, Physical Review Letters, 88 (2002), 1. doi: 10.1103/PhysRevLett.88.063901.

[62]

S. M. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, The dynamical complexity of optically injected semiconductor lasers,, Phys. Reports, 416 (2005), 1. doi: 10.1016/j.physrep.2005.06.003.

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show all references

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.

[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.

[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.

[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.

[5]

M. R. Bassett and J. L. Hudson, Shil'nikov chaos during copper electrodissolution,, J. Phys. Chem., 92 (1988), 6963. doi: 10.1021/j100335a025.

[6]

L. A. Belyakov, A case of the generation of a periodic motion with homoclinic curves,, Mat. Zam., 15 (1974), 571.

[7]

L. A. Belyakov, Bifurcation of systems with homoclinic curve of a saddle-focus with saddle quantity zero,, Mat. Zam., 36 (1984), 681.

[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.

[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.

[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.

[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.

[12]

B. Deng and G. Hines, Food chain chaos due to Shilnikov's orbit,, Chaos, 12 (2002), 533. doi: 10.1063/1.1482255.

[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.

[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.

[15]

E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems,, Congr. Numer., 30 (1981), 265.

[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.

[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.

[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.

[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).

[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.

[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.

[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.

[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.

[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.

[25]

P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits,, J. Statist. Phys., 35 (1984), 645. doi: 10.1007/BF01010828.

[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.

[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.

[28]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,, $2^{nd}$ edition, (1986).

[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.

[30]

A. Gutek and J. van Mill, Continua that are locally a bundle of arcs,, Topology Proceedings, 7 (1982), 63.

[31]

A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations,, J. Dynam. Diff. Eqs., 12 (2000), 807. doi: 10.1023/A:1009046621861.

[32]

F. C. Hoppensteadt, An Introduction to the Mathematics of Neurons, Modeling in the Frequency Domain,, Cambridge University Press, (1997).

[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.

[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.

[35]

B. Krauskopf and H. M. Osinga, Two-dimensional global manifolds of vector fields,, Chaos, 9 (1999), 768. doi: 10.1063/1.166450.

[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.

[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.

[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.

[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.

[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.

[41]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory,, $3^{rd}$ edition, (2004).

[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.

[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.

[44]

J. R. Munkres, Topology,, $2^{nd}$ edition, (2000).

[45]

T. Noh, Shilnikov's chaos in the oxidation of formic acid with bismuth ion on Pt ring electrode,, Electrochimica Acta, 54 (2009), 3657.

[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.

[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.

[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.

[49]

I. M. Ovsyannikov and L. P. Shil'nikov, On systems with a saddle-focus homoclinic curve,, Math. USSR Sbornik, 58 (1987), 557.

[50]

T. Peacock and T. Mullin, Homoclinic bifurcations in a liquid crystal flow,, J. Fluid Mech., 432 (2001), 369.

[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.

[52]

M. A. F. Sanjuán, J. Kennedy, E. Ott and J. A. Yorke, Indecomposable continua and the characterization of strange sets in nonlinear dynamics,, Phys. Rev. Lett., 78 (1997), 1892.

[53]

L. P. Shilnikov, A case of the existence of a countable number of periodic orbits,, Sov. Math. Dokl., 6 (1965), 163.

[54]

L. P. Shilnikov, A contribution to the problem of the structure of an extended neighborhood of a rough state to a saddle-focus type,, Math. USSR-Sb, 10 (1970), 91.

[55]

L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part II,, World Scientific Series on Nonlinear Science, (2001). doi: 10.1142/9789812798558_0001.

[56]

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