Infinitely many multi-pulses near a bifocal cycle

Alexandre A. P. Rodrigues

doi:10.3934/proc.2015.0954

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

Infinitely many multi-pulses near a bifocal cycle

Alexandre A. P. Rodrigues ^1,

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

Abstract
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Under a Creative Commons license

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.

Keywords: Multi-pulses, Bifocal cycles, Blenders., Hyperchaos.

Mathematics Subject Classification: Primary: 37C29; Secondary: 34C28, 37C27, 37C2.

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), 75-95.
[2]	C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143(2) (1996), 357-396.
[3]	R. L. Devaney, Homoclinic orbits in Hamiltonian systems, J. Diff. Equations, 21 (1976), 431-438.
[4]	R. L. Devaney, Blue sky catastrophes in reversible and Hamiltonian systems, Indiana Univ. Math. J., 2 (1977), 247-263.
[5]	A.C. Fowler and C. T. Sparrow, Bifocal homoclinic orbits in four dimensions, Nonlinearity, 4 (1991), 1159-1182.
[6]	P. Gaspard, R. Kapral and G. Nicolis, Bifurcation phenomena near homoclinic systems: a two parameter analysis, J. Statist. Phys., 35, N. 5-6 (1984), 697-727.
[7]	J. Harterich, Cascades of reversible homoclinic orbits to a saddle-focus equilibrium, Phys. D, 112, N. 1-2 (1998), 187-200.
[8]	A. J. Homburg, Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbit to a saddle-focus equilibria, Nonlinearity, 15 (2002), 1029-1050.
[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), Vol. 25(11), to appear.
[10]	J. Knobloch and T. Wagenknecht, Homoclinic snaking near a heteroclinic cycle in reversible systems, SIAM J. Appl. Dyn. Syst., 7 (4), (2008), 13971420.
[11]	I. S. Labouriau and A. A. P. Rodrigues, Global generic dynamics close to symmetry, Journal of Differential Equations, 253 (2012), 2527-2557.
[12]	C. Laing and P. Glendinning, Bifocal homoclinic bifurcations, Physica D, 102 (1997), 1-14.
[13]	J. Lamb and J. Roberts, Time-reversal symmetry in dynamical systems: a survey, Physica D, 112 (1998), 1-39.
[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), 139-155.
[15]	A. A. P. Rodrigues, Persistent switching near a heteroclinic model for the geodynamo problem, Chaos, Solitons & Fractals, 47 (2013), 73-86.
[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), 54-57.
[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, 123, (1970), 92-103.
[18]	A. Vanderbauwhede and B . Fiedler, Homoclinic period blow-up in reversible and conservative systems, Z. Angew. Math. Phys., 43, (1992), 292-318.
[19]	C. Tresser, About some theorems by L. P. Shilnikov, Ann. Inst. H. Poincaré, 40 (1984), 441-461.

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), 75-95.
[2]	C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143(2) (1996), 357-396.
[3]	R. L. Devaney, Homoclinic orbits in Hamiltonian systems, J. Diff. Equations, 21 (1976), 431-438.
[4]	R. L. Devaney, Blue sky catastrophes in reversible and Hamiltonian systems, Indiana Univ. Math. J., 2 (1977), 247-263.
[5]	A.C. Fowler and C. T. Sparrow, Bifocal homoclinic orbits in four dimensions, Nonlinearity, 4 (1991), 1159-1182.
[6]	P. Gaspard, R. Kapral and G. Nicolis, Bifurcation phenomena near homoclinic systems: a two parameter analysis, J. Statist. Phys., 35, N. 5-6 (1984), 697-727.
[7]	J. Harterich, Cascades of reversible homoclinic orbits to a saddle-focus equilibrium, Phys. D, 112, N. 1-2 (1998), 187-200.
[8]	A. J. Homburg, Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbit to a saddle-focus equilibria, Nonlinearity, 15 (2002), 1029-1050.
[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), Vol. 25(11), to appear.
[10]	J. Knobloch and T. Wagenknecht, Homoclinic snaking near a heteroclinic cycle in reversible systems, SIAM J. Appl. Dyn. Syst., 7 (4), (2008), 13971420.
[11]	I. S. Labouriau and A. A. P. Rodrigues, Global generic dynamics close to symmetry, Journal of Differential Equations, 253 (2012), 2527-2557.
[12]	C. Laing and P. Glendinning, Bifocal homoclinic bifurcations, Physica D, 102 (1997), 1-14.
[13]	J. Lamb and J. Roberts, Time-reversal symmetry in dynamical systems: a survey, Physica D, 112 (1998), 1-39.
[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), 139-155.
[15]	A. A. P. Rodrigues, Persistent switching near a heteroclinic model for the geodynamo problem, Chaos, Solitons & Fractals, 47 (2013), 73-86.
[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), 54-57.
[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, 123, (1970), 92-103.
[18]	A. Vanderbauwhede and B . Fiedler, Homoclinic period blow-up in reversible and conservative systems, Z. Angew. Math. Phys., 43, (1992), 292-318.
[19]	C. Tresser, About some theorems by L. P. Shilnikov, Ann. Inst. H. Poincaré, 40 (1984), 441-461.

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