American Institute of Mathematical Sciences

Advanced
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

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

[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

[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.   Google Scholar

[53]

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

[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.   Google Scholar

[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.  Google Scholar

[56]

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering,, Adison-Wesley, (1994).  doi: 10.1063/1.4823332.  Google Scholar

[57]

G. A. K. van Voorn, B. W. Kooi and M. P. Boer, Ecological consequences of global bifurcations in some food chain models,, Math. Biosc., 226 (2010), 120.  doi: 10.1016/j.mbs.2010.04.005.  Google Scholar

[58]

K. Watada, T. Endo and H. Seishi, Shilnikov orbits in an autonomous third-order chaotic phase-locked loop,, IEEE Trans. on Circ. and Syst. I, 45 (1998), 979.  doi: 10.1109/81.721264.  Google Scholar

[59]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[63]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,, $2^{nd}$ edition, (2003).   Google Scholar

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.  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

[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.   Google Scholar

[53]

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

[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.   Google Scholar

[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.  Google Scholar

[56]

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering,, Adison-Wesley, (1994).  doi: 10.1063/1.4823332.  Google Scholar

[57]

G. A. K. van Voorn, B. W. Kooi and M. P. Boer, Ecological consequences of global bifurcations in some food chain models,, Math. Biosc., 226 (2010), 120.  doi: 10.1016/j.mbs.2010.04.005.  Google Scholar

[58]

K. Watada, T. Endo and H. Seishi, Shilnikov orbits in an autonomous third-order chaotic phase-locked loop,, IEEE Trans. on Circ. and Syst. I, 45 (1998), 979.  doi: 10.1109/81.721264.  Google Scholar

[59]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[63]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,, $2^{nd}$ edition, (2003).   Google Scholar

[1]

Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082
[2]

Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210
[3]

Rovella Alvaro, Vilamajó Francesc, Romero Neptalí. Invariant manifolds for delay endomorphisms. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35
[4]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[5]

Leonardo Mora. Homoclinic bifurcations, fat attractors and invariant curves. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1133-1148. doi: 10.3934/dcds.2003.9.1133
[6]

Martin Wechselberger, Warren Weckesser. Homoclinic clusters and chaos associated with a folded node in a stellate cell model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 829-850. doi: 10.3934/dcdss.2009.2.829
[7]

Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419
[8]

S. Secchi, C. A. Stuart. Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493
[9]

José F. Alves, Davide Azevedo. Statistical properties of diffeomorphisms with weak invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 1-41. doi: 10.3934/dcds.2016.36.1
[10]

Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133
[11]

George Osipenko. Indestructibility of invariant locally non-unique manifolds. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 203-219. doi: 10.3934/dcds.1996.2.203
[12]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967
[13]

Arturo Echeverría-Enríquez, Alberto Ibort, Miguel C. Muñoz-Lecanda, Narciso Román-Roy. Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (4) : 397-419. doi: 10.3934/jgm.2012.4.397
[14]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[15]

Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197
[16]

V. Afraimovich, T.R. Young. Multipliers of homoclinic orbits on surfaces and characteristics of associated invariant sets. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 691-704. doi: 10.3934/dcds.2000.6.691
[17]

Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203
[18]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
[19]

Flaviano Battelli, Claudio Lazzari. On the bifurcation from critical homoclinic orbits in n-dimensional maps. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 289-303. doi: 10.3934/dcds.1997.3.289
[20]

I. Baldomá, Àlex Haro. One dimensional invariant manifolds of Gevrey type in real-analytic maps. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 295-322. doi: 10.3934/dcdsb.2008.10.295

 Impact Factor: 

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences