American Institute of Mathematical Sciences

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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

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:
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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), 1803-1846. Google Scholar

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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), 1309-1344. 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), 1244-1262. 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), 1803-1846. 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), 1087-1100. doi: 10.1016/j.physd.2009.03.010.  Google Scholar

[5]

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[10]

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[13]

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[14]

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[15]

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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, F. Dercole, T. F. Fairgrieve, Y. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. H. Zhang, Department of Computer Science, Concordia University, Montreal, Canada, (2010). Available from http://cmvl.cs.concordia.ca/. 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), 1008-1041. 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), 805-822. doi: 10.1142/S0218127407017562.  Google Scholar

[22]

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[23]

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[24]

P. Gaspard, R. Kapral and G. Nicolis, Bifurcation phenomena near homoclinic systems: A two-parameter analysis, J. Statist. Phys., 35 (1984), 697-727. doi: 10.1007/BF01010829.  Google Scholar

[25]

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[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, Quebec City, Canada, (2001), AAS 01-31. Google Scholar

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[31]

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E. A. Jackson, The Lorenz system: II. The homoclinic convolution of the stable manifolds, Phys. Scr., 32 (1985), 469-475. 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), 180-201. 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), 768-774. 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), 546-569. doi: 10.1137/030600180.  Google Scholar

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