2015, 2015(special): 954-964. doi: 10.3934/proc.2015.0954

Infinitely many multi-pulses near a bifocal cycle

1. 

Centro de Matemática da Universidade do Porto and Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto

Received  August 2014 Revised  April 2015 Published  November 2015

The entire dynamics in a neighbourhood of a reversible heteroclinic cycle involving a bifocus is far from being understood. In this article, using the well known theory of reversing symmetries, we prove that there are infinitely many pulses near a cycle involving two symmetric equilibria, a real saddle and a bifocus, giving rise to a complex network. We also conjecture that suspended blenders might appear in the neighbourhood of the network.
Citation: Alexandre A. P. Rodrigues. Infinitely many multi-pulses near a bifocal cycle. Conference Publications, 2015, 2015 (special) : 954-964. doi: 10.3934/proc.2015.0954
References:
[1]

M. A. D. Aguiar, I. S. Labouriau and A. A. P. Rodrigues, Switching near a heteroclinic network of rotating nodes,, Dyn. Syst., 25 (2010), 75.   Google Scholar

[2]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms,, Ann. of Math., 143 (1996), 357.   Google Scholar

[3]

R. L. Devaney, Homoclinic orbits in Hamiltonian systems,, J. Diff. Equations, 21 (1976), 431.   Google Scholar

[4]

R. L. Devaney, Blue sky catastrophes in reversible and Hamiltonian systems,, Indiana Univ. Math. J., 2 (1977), 247.   Google Scholar

[5]

A.C. Fowler and C. T. Sparrow, Bifocal homoclinic orbits in four dimensions,, Nonlinearity, 4 (1991), 1159.   Google Scholar

[6]

P. Gaspard, R. Kapral and G. Nicolis, Bifurcation phenomena near homoclinic systems: a two parameter analysis,, J. Statist. Phys., 35 (1984), 5.   Google Scholar

[7]

J. Harterich, Cascades of reversible homoclinic orbits to a saddle-focus equilibrium,, Phys. D, 112 (1998), 1.   Google Scholar

[8]

A. J. Homburg, Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbit to a saddle-focus equilibria,, Nonlinearity, 15 (2002), 1029.   Google Scholar

[9]

S. Ibáñez and A.A.P. Rodrigues, On the dynamics near a homoclinic network to a bifocus: switching and horseshoes,, Int. Journal Bif. Chaos, (2015).   Google Scholar

[10]

J. Knobloch and T. Wagenknecht, Homoclinic snaking near a heteroclinic cycle in reversible systems,, SIAM J. Appl. Dyn. Syst., 7 (2008).   Google Scholar

[11]

I. S. Labouriau and A. A. P. Rodrigues, Global generic dynamics close to symmetry,, Journal of Differential Equations, 253 (2012), 2527.   Google Scholar

[12]

C. Laing and P. Glendinning, Bifocal homoclinic bifurcations,, Physica D, 102 (1997), 1.   Google Scholar

[13]

J. Lamb and J. Roberts, Time-reversal symmetry in dynamical systems: a survey,, Physica D, 112 (1998), 1.   Google Scholar

[14]

L. M. Lerman, Homo and Heteroclinic orbits, hyperbolic subsets in a one-parameter unfolding of a Hamiltonian system with heteroclinic contour with two saddle-foci,, Regular and Chaotic Dynamical Systems, 2 (1997), 139.   Google Scholar

[15]

A. A. P. Rodrigues, Persistent switching near a heteroclinic model for the geodynamo problem,, Chaos, 47 (2013), 73.   Google Scholar

[16]

L. P. Shilnikov, The existence of a denumerable set of periodic motions in four dimensional space in an extended neighbourhood of a saddle-focus,, Sov. Math. Dokl., 172 (1967), 54.   Google Scholar

[17]

L. P. Shilnikov, On the question of the structure of an extended neighborhood of a structurally stable state of equilibrium of saddle-focus type,, Math. USSR Sb., 81 (1970), 92.   Google Scholar

[18]

A. Vanderbauwhede and B . Fiedler, Homoclinic period blow-up in reversible and conservative systems,, Z. Angew. Math. Phys., 43 (1992), 292.   Google Scholar

[19]

C. Tresser, About some theorems by L. P. Shilnikov,, Ann. Inst. H. Poincaré, 40 (1984), 441.   Google Scholar

show all references

References:
[1]

M. A. D. Aguiar, I. S. Labouriau and A. A. P. Rodrigues, Switching near a heteroclinic network of rotating nodes,, Dyn. Syst., 25 (2010), 75.   Google Scholar

[2]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms,, Ann. of Math., 143 (1996), 357.   Google Scholar

[3]

R. L. Devaney, Homoclinic orbits in Hamiltonian systems,, J. Diff. Equations, 21 (1976), 431.   Google Scholar

[4]

R. L. Devaney, Blue sky catastrophes in reversible and Hamiltonian systems,, Indiana Univ. Math. J., 2 (1977), 247.   Google Scholar

[5]

A.C. Fowler and C. T. Sparrow, Bifocal homoclinic orbits in four dimensions,, Nonlinearity, 4 (1991), 1159.   Google Scholar

[6]

P. Gaspard, R. Kapral and G. Nicolis, Bifurcation phenomena near homoclinic systems: a two parameter analysis,, J. Statist. Phys., 35 (1984), 5.   Google Scholar

[7]

J. Harterich, Cascades of reversible homoclinic orbits to a saddle-focus equilibrium,, Phys. D, 112 (1998), 1.   Google Scholar

[8]

A. J. Homburg, Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbit to a saddle-focus equilibria,, Nonlinearity, 15 (2002), 1029.   Google Scholar

[9]

S. Ibáñez and A.A.P. Rodrigues, On the dynamics near a homoclinic network to a bifocus: switching and horseshoes,, Int. Journal Bif. Chaos, (2015).   Google Scholar

[10]

J. Knobloch and T. Wagenknecht, Homoclinic snaking near a heteroclinic cycle in reversible systems,, SIAM J. Appl. Dyn. Syst., 7 (2008).   Google Scholar

[11]

I. S. Labouriau and A. A. P. Rodrigues, Global generic dynamics close to symmetry,, Journal of Differential Equations, 253 (2012), 2527.   Google Scholar

[12]

C. Laing and P. Glendinning, Bifocal homoclinic bifurcations,, Physica D, 102 (1997), 1.   Google Scholar

[13]

J. Lamb and J. Roberts, Time-reversal symmetry in dynamical systems: a survey,, Physica D, 112 (1998), 1.   Google Scholar

[14]

L. M. Lerman, Homo and Heteroclinic orbits, hyperbolic subsets in a one-parameter unfolding of a Hamiltonian system with heteroclinic contour with two saddle-foci,, Regular and Chaotic Dynamical Systems, 2 (1997), 139.   Google Scholar

[15]

A. A. P. Rodrigues, Persistent switching near a heteroclinic model for the geodynamo problem,, Chaos, 47 (2013), 73.   Google Scholar

[16]

L. P. Shilnikov, The existence of a denumerable set of periodic motions in four dimensional space in an extended neighbourhood of a saddle-focus,, Sov. Math. Dokl., 172 (1967), 54.   Google Scholar

[17]

L. P. Shilnikov, On the question of the structure of an extended neighborhood of a structurally stable state of equilibrium of saddle-focus type,, Math. USSR Sb., 81 (1970), 92.   Google Scholar

[18]

A. Vanderbauwhede and B . Fiedler, Homoclinic period blow-up in reversible and conservative systems,, Z. Angew. Math. Phys., 43 (1992), 292.   Google Scholar

[19]

C. Tresser, About some theorems by L. P. Shilnikov,, Ann. Inst. H. Poincaré, 40 (1984), 441.   Google Scholar

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