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February  2020, 25(2): 599-615. doi: 10.3934/dcdsb.2019256

Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities

Fátima Drubi , Santiago Ibáñez , and  David Rivela

Department of Mathematics, University of Oviedo, Federico García Lorca 18, 33007 Oviedo, Spain

* Corresponding author: mesa@uniovi.es

Received  February 2019 Revised  May 2019 Published  February 2020 Early access  November 2019

Fund Project: F. Drubi and S. Ibáñez are supported by the Spanish MICINN grant MTM2017-87697-P.

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.

Keywords: Singularities, unfoldings, Hopf-Bogdanov-Takens, chaotic behavior.
Mathematics Subject Classification: Primary: 58K45; Secondary: 37C10, 58K50.
Citation: Fátima Drubi, Santiago Ibáñez, David Rivela. Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 599-615. doi: 10.3934/dcdsb.2019256
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.

[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. 

[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.

[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. 

[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. 

Figure 1.  Primary bifurcations in the unfolding of a HBT singularity. A sphere $ \lambda_1^2+\lambda_2^2+\mu^2 = \delta $ is fixed and the front view corresponds to $ \lambda_1<0 $. Hence, points (respectively lines) correspond to bifurcation curves (respectively surfaces). We assume that $ \widehat C_{1, 1} <0 $
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Figure 3.  Examples of period-doubling and chaotic dynamics close to a Hopf-Zero bifurcation. In all cases, we have $ \lambda_2 = -0.1 $, $ \mu = -0.3 $ and $ \widehat C_{1, 1} = -1 $. Initial conditions are $ x = y = u = v = 0.04 $
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Figure 5.  Examples of period-doubling and chaotic dynamics close to a Hopf-Hopf bifurcation. In all cases, we have $ \lambda_2 = 0.21 $, $ \mu = -0.3 $ and $ \widehat C_{1, 1} = -1 $. Initial conditions are $ x = y = u = v = 0.04 $
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Figure 2.  Green line corresponds to the segment of parameters where simulations in Figure 3 have been done. Note that we start after the birth of an invariant torus when parameters cross the secondary Hopf bifurcation. This bifurcation curve has been obtained using Matcont [13]
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Figure 4.  Red line corresponds to the segment of parameters where simulations in Figure 5 have been done. Note that we start after the birth of an invariant torus when parameters cross the secondary Hopf bifurcation curve. This bifurcation curve has been obtained using Matcont [13]
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Table 1.  Cases of HZ bifurcation unfolded by a HBT singularity. With the appropriate choice for coefficients in the normal form of the HBT singularity, any case can be obtained. In Case Ⅲ, it can be expected the emergence of chaotic behavior when higher order terms are considered
$ \lambda_2\widehat C_{1, 1} <0 $ $ \lambda_2\widehat C_{1, 1}>0 $
$ \kappa=-1 $ Case II Case I
$ \kappa=1 $ Case IV Case III
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$ \lambda_2\widehat C_{1, 1} <0 $ $ \lambda_2\widehat C_{1, 1}>0 $
$ \kappa=-1 $ Case II Case I
$ \kappa=1 $ Case IV Case III
Table 2.  Cases of HH bifurcation unfolded by a HBT singularity. In Case Ⅵa, it can be expected the emergence of chaotic behavior when higher order terms are considered
$ \widehat C_{1, 1} <0 $ $ 0 <\widehat C_{1, 1} <\frac{1}{4} $ $ \frac{1}{4} <\widehat C_{1, 1} <\frac{1}{2} $ $ \frac{1}{2} <\widehat C_{1, 1} $
$ \kappa=-1 $ Case IVb Case VIIa Case VIIb Case V
$ \kappa=1 $ Case VIa Case Ib Case Ia Case III
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$ \widehat C_{1, 1} <0 $ $ 0 <\widehat C_{1, 1} <\frac{1}{4} $ $ \frac{1}{4} <\widehat C_{1, 1} <\frac{1}{2} $ $ \frac{1}{2} <\widehat C_{1, 1} $
$ \kappa=-1 $ Case IVb Case VIIa Case VIIb Case V
$ \kappa=1 $ Case VIa Case Ib Case Ia Case III
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