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Well-posedness results for fractional semi-linear wave equations
Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities
Department of Mathematics, University of Oviedo, Federico García Lorca 18, 33007 Oviedo, Spain |
A discussion on local bifurcations of codimension one and two is presented for generic unfoldings of Hopf-Bogdanov-Takens singularities of codimension three. Among all identified bifurcations, we focus on Hopf-Zero and Hopf-Hopf bifurcations, since, in certain cases, they can explain the emergence of chaotic dynamics. Moreover, numerical simulations are provided to illustrate that strange attractors appear at least when the second order normal form of the unfolding is considered.
References:
[1] |
A. Algaba, E. Freire and E. Gamero,
Hypernormal form for the Hopf-zero bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1988), 1857-1887.
doi: 10.1142/S0218127498001583. |
[2] |
A. Algaba, E. Freire, E. Gamero and A. J. Rodríguez-Luis,
On a codimension-three unfolding of the interaction of degenerate Hopf and pitchfork bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1333-1362.
doi: 10.1142/S0218127499000936. |
[3] |
A. Algaba, E. Freire, E. Gamero and A. J. Rodríguez-Luis,
A three-parameter study of a degenerate case of the Hopf-pitchfork bifurcation, Nonlinearity, 12 (1999), 1177-1206.
doi: 10.1088/0951-7715/12/4/324. |
[4] |
A. Algaba, E. Freire, E. Gamero and A. J. Rodríguez-Luis,
A tame degenerate Hopf-pitchfork bifurcation in a modified van der Pol-Duffing oscillator, Nonlinear Dynam., 22 (2000), 249-269.
doi: 10.1023/A:1008328027179. |
[5] |
A. Algaba, M. Merino, E. Freire, E. Gamero and A. J. Rodríguez-Luis,
On the Hopf-pitchfork bifurcation in the Chua's equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 291-305.
doi: 10.1142/S0218127400000190. |
[6] |
I. Baldomá, O. Castejón and T. M. Seara,
Exponentially small heteroclinic breakdown in the generic Hopf-Zero singularity, J. Dyn. Diff. Equat., 25 (2013), 335-392.
doi: 10.1007/s10884-013-9297-2. |
[7] |
I. Baldomá, O. Castejón and T. M. Seara,
Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (Ⅰ), J. Nonlinear Sci., 28 (2018), 1551-1627.
doi: 10.1007/s00332-018-9458-x. |
[8] |
I. Baldomá, O. Castejón and T. M. Seara,
Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (Ⅱ): The generic case, J. Nonlinear Sci., 28 (2018), 1489-1549.
doi: 10.1007/s00332-018-9459-9. |
[9] |
I. Baldomá, S. Ibáñez and T. M. Seara, Hopf-Zero singularities truly unfold chaos, preprint, arXiv: 1903.09023. Google Scholar |
[10] |
P. G. Barrientos, S. Ibáñez and J. A. Rodríguez,
Heteroclinic cycles arising in generic unfoldings of nilpotent singularities, J. Dyn. Diff. Equat., 23 (2011), 999-1028.
doi: 10.1007/s10884-011-9230-5. |
[11] |
P. G. Barrientos, S. Ibáñez and J. A. Rodríguez,
Robust cycles unfolding from conservative bifocal homoclinic orbits, Dyn. Syst., 31 (2016), 546-579.
doi: 10.1080/14689367.2016.1170763. |
[12] |
H. W. Broer and G. Vegter,
Subordinate Šil'nikov bifurcations near some singularities of vector fields having low codimension, Ergodic Theory Dynam. Systems, 4 (1984), 509-525.
doi: 10.1017/S0143385700002613. |
[13] |
A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer and B. Sautois,
New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), 147-175.
doi: 10.1080/13873950701742754. |
[14] |
F. Drubi, S. Ibáñez and J. Á. Rodríguez,
Coupling leads to chaos, J. Differential Equations, 239 (2007), 371-385.
doi: 10.1016/j.jde.2007.05.024. |
[15] |
F. Drubi, S. Ibáñez and J. Á. Rodríguez,
Hopf-pitchfork singularities in coupled systems, Phys. D, 240 (2011), 825-840.
doi: 10.1016/j.physd.2010.12.013. |
[16] |
F. Drubi, S. Ibáñez and D. Rivela, A formal classification of Hopf-Bogdanov-Takens singularities of codimension three, J. Math. Anal. Appl., 480 (2019), 123408.
doi: 10.1016/j.jmaa.2019.123408. |
[17] |
F. Dumortier, S. Ibáñez, H. Kokubu and C. Simó,
About the unfolding of a Hopf-zero singularity, Discrete Contin. Dyn. Syst., 33 (2013), 4435-4471.
doi: 10.3934/dcds.2013.33.4435. |
[18] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[19] |
A. J. Homburg,
Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbits to saddle-focus equilibria, Nonlinearity, 15 (2002), 1029-1050.
doi: 10.1088/0951-7715/15/4/304. |
[20] |
S. Ibáñez and J. A. Rodríguez,
Shilnikov bifurcations in generic $4$-unfoldings of a codimension-$4$ singularity, J. Differential Equations, 120 (1995), 411-428.
doi: 10.1006/jdeq.1995.1116. |
[21] |
S. Ibáñez and J. A. Rodríguez,
Shilnikov configurations in any generic unfolding of the nilpotent singularity of codimension three on ${\mathbb R}^3$, J. Differential Equations, 208 (2005), 147-175.
doi: 10.1016/j.jde.2003.08.006. |
[22] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998. |
[23] |
W. F. Langford and K. J. Zhan,
Interactions of Andronov-Hopf and Bogdanov-Takens bifurcations, The Arnoldfest, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 24 (1999), 365-383.
|
[24] |
L. Mora and M. Viana,
Abundance of strange attractors, Acta Math., 171 (1993), 1-71.
doi: 10.1007/BF02392766. |
[25] |
A. Pumariño and J. A. Rodríguez, Coexistence and Persistence of Strange Attractors, Lecture Notes in Mathematics, 1658. Springer-Verlag, Berlin, 1997.
doi: 10.1007/BFb0093337. |
[26] |
A. Pumariño and J. A. Rodríguez,
Coexistence and persistence of infinitely many strange attractors, Ergodic Theory Dynam. Systems, 21 (2001), 1511-1523.
doi: 10.1017/S0143385701001730. |
[27] |
L. P. Šil'nikov,
A case of the existence of a denumerable set of periodic motions, Dokl. Akad. Nauk SSSR, 160 (1965), 558-561.
|
[28] |
L. P. Šil'nikov, A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type, Math. USSR Sb., 10 (1970), 91-102. Google Scholar |
[29] |
A. Steindl,
Numerical investigation of the Hopf-Bogdanov-Takens mode interaction for a fluid-conveying tube, Procedia Engineering, 199 (2017), 857-862.
doi: 10.1016/j.proeng.2017.09.024. |
[30] |
F. Takens, Singularities of vector fields, Publ. Math. IHES, (1974), 47–100. |
[31] |
C. Tresser,
About some theorems by L. P. Šil'nikov, Ann. Inst. H. Poincaré Phys. Théor., 40 (1984), 441-461.
|
show all references
References:
[1] |
A. Algaba, E. Freire and E. Gamero,
Hypernormal form for the Hopf-zero bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1988), 1857-1887.
doi: 10.1142/S0218127498001583. |
[2] |
A. Algaba, E. Freire, E. Gamero and A. J. Rodríguez-Luis,
On a codimension-three unfolding of the interaction of degenerate Hopf and pitchfork bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1333-1362.
doi: 10.1142/S0218127499000936. |
[3] |
A. Algaba, E. Freire, E. Gamero and A. J. Rodríguez-Luis,
A three-parameter study of a degenerate case of the Hopf-pitchfork bifurcation, Nonlinearity, 12 (1999), 1177-1206.
doi: 10.1088/0951-7715/12/4/324. |
[4] |
A. Algaba, E. Freire, E. Gamero and A. J. Rodríguez-Luis,
A tame degenerate Hopf-pitchfork bifurcation in a modified van der Pol-Duffing oscillator, Nonlinear Dynam., 22 (2000), 249-269.
doi: 10.1023/A:1008328027179. |
[5] |
A. Algaba, M. Merino, E. Freire, E. Gamero and A. J. Rodríguez-Luis,
On the Hopf-pitchfork bifurcation in the Chua's equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 291-305.
doi: 10.1142/S0218127400000190. |
[6] |
I. Baldomá, O. Castejón and T. M. Seara,
Exponentially small heteroclinic breakdown in the generic Hopf-Zero singularity, J. Dyn. Diff. Equat., 25 (2013), 335-392.
doi: 10.1007/s10884-013-9297-2. |
[7] |
I. Baldomá, O. Castejón and T. M. Seara,
Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (Ⅰ), J. Nonlinear Sci., 28 (2018), 1551-1627.
doi: 10.1007/s00332-018-9458-x. |
[8] |
I. Baldomá, O. Castejón and T. M. Seara,
Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (Ⅱ): The generic case, J. Nonlinear Sci., 28 (2018), 1489-1549.
doi: 10.1007/s00332-018-9459-9. |
[9] |
I. Baldomá, S. Ibáñez and T. M. Seara, Hopf-Zero singularities truly unfold chaos, preprint, arXiv: 1903.09023. Google Scholar |
[10] |
P. G. Barrientos, S. Ibáñez and J. A. Rodríguez,
Heteroclinic cycles arising in generic unfoldings of nilpotent singularities, J. Dyn. Diff. Equat., 23 (2011), 999-1028.
doi: 10.1007/s10884-011-9230-5. |
[11] |
P. G. Barrientos, S. Ibáñez and J. A. Rodríguez,
Robust cycles unfolding from conservative bifocal homoclinic orbits, Dyn. Syst., 31 (2016), 546-579.
doi: 10.1080/14689367.2016.1170763. |
[12] |
H. W. Broer and G. Vegter,
Subordinate Šil'nikov bifurcations near some singularities of vector fields having low codimension, Ergodic Theory Dynam. Systems, 4 (1984), 509-525.
doi: 10.1017/S0143385700002613. |
[13] |
A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer and B. Sautois,
New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), 147-175.
doi: 10.1080/13873950701742754. |
[14] |
F. Drubi, S. Ibáñez and J. Á. Rodríguez,
Coupling leads to chaos, J. Differential Equations, 239 (2007), 371-385.
doi: 10.1016/j.jde.2007.05.024. |
[15] |
F. Drubi, S. Ibáñez and J. Á. Rodríguez,
Hopf-pitchfork singularities in coupled systems, Phys. D, 240 (2011), 825-840.
doi: 10.1016/j.physd.2010.12.013. |
[16] |
F. Drubi, S. Ibáñez and D. Rivela, A formal classification of Hopf-Bogdanov-Takens singularities of codimension three, J. Math. Anal. Appl., 480 (2019), 123408.
doi: 10.1016/j.jmaa.2019.123408. |
[17] |
F. Dumortier, S. Ibáñez, H. Kokubu and C. Simó,
About the unfolding of a Hopf-zero singularity, Discrete Contin. Dyn. Syst., 33 (2013), 4435-4471.
doi: 10.3934/dcds.2013.33.4435. |
[18] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[19] |
A. J. Homburg,
Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbits to saddle-focus equilibria, Nonlinearity, 15 (2002), 1029-1050.
doi: 10.1088/0951-7715/15/4/304. |
[20] |
S. Ibáñez and J. A. Rodríguez,
Shilnikov bifurcations in generic $4$-unfoldings of a codimension-$4$ singularity, J. Differential Equations, 120 (1995), 411-428.
doi: 10.1006/jdeq.1995.1116. |
[21] |
S. Ibáñez and J. A. Rodríguez,
Shilnikov configurations in any generic unfolding of the nilpotent singularity of codimension three on ${\mathbb R}^3$, J. Differential Equations, 208 (2005), 147-175.
doi: 10.1016/j.jde.2003.08.006. |
[22] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998. |
[23] |
W. F. Langford and K. J. Zhan,
Interactions of Andronov-Hopf and Bogdanov-Takens bifurcations, The Arnoldfest, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 24 (1999), 365-383.
|
[24] |
L. Mora and M. Viana,
Abundance of strange attractors, Acta Math., 171 (1993), 1-71.
doi: 10.1007/BF02392766. |
[25] |
A. Pumariño and J. A. Rodríguez, Coexistence and Persistence of Strange Attractors, Lecture Notes in Mathematics, 1658. Springer-Verlag, Berlin, 1997.
doi: 10.1007/BFb0093337. |
[26] |
A. Pumariño and J. A. Rodríguez,
Coexistence and persistence of infinitely many strange attractors, Ergodic Theory Dynam. Systems, 21 (2001), 1511-1523.
doi: 10.1017/S0143385701001730. |
[27] |
L. P. Šil'nikov,
A case of the existence of a denumerable set of periodic motions, Dokl. Akad. Nauk SSSR, 160 (1965), 558-561.
|
[28] |
L. P. Šil'nikov, A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type, Math. USSR Sb., 10 (1970), 91-102. Google Scholar |
[29] |
A. Steindl,
Numerical investigation of the Hopf-Bogdanov-Takens mode interaction for a fluid-conveying tube, Procedia Engineering, 199 (2017), 857-862.
doi: 10.1016/j.proeng.2017.09.024. |
[30] |
F. Takens, Singularities of vector fields, Publ. Math. IHES, (1974), 47–100. |
[31] |
C. Tresser,
About some theorems by L. P. Šil'nikov, Ann. Inst. H. Poincaré Phys. Théor., 40 (1984), 441-461.
|



Case II | Case I | |
Case IV | Case III |
Case II | Case I | |
Case IV | Case III |
Case IVb | Case VIIa | Case VIIb | Case V | |
Case VIa | Case Ib | Case Ia | Case III |
Case IVb | Case VIIa | Case VIIb | Case V | |
Case VIa | Case Ib | Case Ia | Case III |
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