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Time-resolved denoising using model order reduction, dynamic mode decomposition, and kalman filter and smoother
Computing connecting orbits to infinity associated with a homoclinic flip bifurcation
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand |
We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in $ \mathbb{R}^3 $ that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary $ n $-homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity.
We present an adaptation of the technique known as Lin's method that enables us to compute such connecting orbits to infinity. We first perform a weighted directional compactification of $ \mathbb{R}^3 $ with a subsequent blow-up of a non-hyperbolic saddle at infinity. We then set up boundary-value problems for two orbit segments from and to a common two-dimensional section: the first is to a finite saddle in the regular coordinates, and the second is from the vicinity of the saddle at infinity in the blown-up chart. The so-called Lin gap along a fixed one-dimensional direction in the section is then brought to zero by continuation. Once a connecting orbit has been found in this way, its locus can be traced out as a curve in a parameter plane.
References:
[1] |
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.
doi: 10.1137/130912542. |
[2] |
P. Aguirre, B. Krauskopf and H. M. Osinga,
Global invariant manifolds near a Shilnikov homoclinic bifurcation, J. Comput. Dyn., 1 (2014), 1-38.
doi: 10.3934/jcd.2014.1.1. |
[3] |
A. Algaba, M. C. Domínguez-Moreno, M. Merino and A. Rodríguez-Luis,
Study of a simple 3D quadratic system with homoclinic flip bifurcations of inward twist case $ \mathbf{C}_{\rm in} $, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 324-337.
doi: 10.1016/j.cnsns.2019.05.005. |
[4] |
R. Barrio, S. Ibáñez and L. Pérez,
Hindmarsh–Rose model: Close and far to the singular limit, Phys. Lett. A, 381 (2017), 597-603.
doi: 10.1016/j.physleta.2016.12.027. |
[5] |
R. Barrio, M. A. Martínez, S. Serrano and A. Shilnikov, Macro- and micro-chaotic structures in the Hindmarsh–Rose model of bursting neurons, Chaos, 24 (2014), 11pp.
doi: 10.1063/1.4882171. |
[6] |
L. A. Belyakov,
Bifurcation set in a system with homoclinic saddle curve, Mat. Zametki, 28 (1980), 911-922.
|
[7] |
L. A. Belyakov,
Bifurcations of systems with a homoclinic curve of the saddle-focus with a zero saddle value, Mat. Zametki, 36 (1984), 681-689.
|
[8] |
A. R. Champneys and Y. A. Kuznetsov,
Numerical detection and continuation of codimension-two homoclinic bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 785-822.
doi: 10.1142/S0218127494000587. |
[9] |
A. R. Champneys, Y. A. Kuznetsov and B. Sandstede,
A numerical toolbox for homoclinic bifurcation analysis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 867-887.
doi: 10.1142/S0218127496000485. |
[10] |
B. Deng,
Homoclinic twisting bifurcations and cusp horseshoe maps, J. Dynam. Differential Equations, 5 (1993), 417-467.
doi: 10.1007/BF01053531. |
[11] |
E. Doedel,
AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284.
|
[12] |
E. J. Doedel and B. E. Oldeman, AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations, Department of Computer Science, Concordia University, Montreal, Canada, 2010. Available from: http://www.cmvl.cs.concordia.ca/. |
[13] |
F. Dumortier, Local study of planar vector fields: Singularities and their unfoldings, in Structures in Dynamics: Finite Dimensional Deterministic Studies, Studies in Mathematical Physics, 2, Elsevier Science Publishers, Amsterdam, 1991,161–241.
doi: 10.1016/B978-0-444-89257-7.50011-5. |
[14] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006.
doi: 10.1007/978-3-540-32902-2. |
[15] |
A. Giraldo, B. Krauskopf and H. M. Osinga,
Saddle invariant objects and their global manifolds in a neighborhood of a homoclinic flip bifurcation of case B, SIAM J. Appl. Dyn. Syst., 16 (2017), 640-686.
doi: 10.1137/16M1097419. |
[16] |
A. Giraldo, B. Krauskopf and H. M. Osinga,
Cascades of global bifurcations and chaos near a homoclinic flip bifurcation: A case study, SIAM J. Appl. Dyn. Syst., 17 (2018), 2784-2829.
doi: 10.1137/17M1149675. |
[17] |
E. A. González Velasco,
Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc., 143 (1969), 201-222.
doi: 10.1090/S0002-9947-1969-0252788-8. |
[18] |
A. J. Homburg, H. Kokubu and M. Krupa,
The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit, Ergodic Theory Dynam. Systems, 14 (1994), 667-693.
doi: 10.1017/S0143385700008117. |
[19] |
A. J. Homburg, H. Kokubu and V. Naudot,
Homoclinic-doubling cascades, Arch. Ration. Mech. Anal., 160 (2001), 195-243.
doi: 10.1007/s002050100159. |
[20] |
A. J. Homburg and B. Krauskopf,
Resonant homoclinic flip bifurcations, J. Dynam. Differential Equations, 12 (2000), 807-850.
doi: 10.1023/A:1009046621861. |
[21] |
A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems, 3, Elsevier, New York, 2010,381–509. |
[22] |
M. Kisaka, H. Kokubu and H. Oka,
Bifurcations to $N$-homoclinic orbits and $N$-periodic orbits in vector fields, J. Dynam. Differential Equations, 5 (1993), 305-357.
doi: 10.1007/BF01053164. |
[23] |
B. Krauskopf and T. Rieß,
A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits, Nonlinearity, 21 (2008), 1655-1690.
doi: 10.1088/0951-7715/21/8/001. |
[24] |
B. Krauskopf, H. M. Osinga and J. Galán-Vioque, Numerical Continuation Methods for Dynamical Systems. Path Following and Boundary Value Problems, Understanding Complex Systems, Springer, Dordrecht, 2007.
doi: 10.1007/978-1-4020-6356-5. |
[25] |
Y. A. Kuznetsov, O. De Feo and S. Rinaldi,
Belyakov homoclinic bifurcations in a tritrophic food chain model, SIAM J. Appl. Math., 62 (2001), 462-487.
doi: 10.1137/S0036139900378542. |
[26] |
X.-B. Lin,
Using Mel'nikov's method to solve Šilnikov's problems, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 295-325.
doi: 10.1017/S0308210500031528. |
[27] |
D. Linaro, A. Champneys, M. Desroches and M. Storace,
Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh–Rose burster, SIAM J. Appl. Dyn. Syst., 11 (2012), 939-962.
doi: 10.1137/110848931. |
[28] |
K. Matsue,
On blow-up solutions of differential equations with Poincaré-type compactifications, SIAM J. Appl. Dyn. Syst., 17 (2018), 2249-2288.
doi: 10.1137/17M1124498. |
[29] |
M. Messias, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system, J. Phys. A, 42 (2009), 18pp.
doi: 10.1088/1751-8113/42/11/115101. |
[30] |
M. Messias,
Dynamics at infinity of a cubic Chua's system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 333-340.
doi: 10.1142/S0218127411028453. |
[31] |
B. E. Oldeman, B. Krauskopf and A. R. Champneys,
Numerical unfoldings of codimension-three resonant homoclinic flip bifurcations, Nonlinearity, 14 (2001), 597-621.
doi: 10.1088/0951-7715/14/3/309. |
[32] |
B. Sandstede, Verzweigungstheorie Homokliner Verdopplungen, Ph.D thesis, University of Stuttgart in Stuttgart, Germany, 1993. |
[33] |
B. Sandstede,
Constructing dynamical systems having homoclinic bifurcation points of codimension two, J. Dynam. Differential Equations, 9 (1997), 269-288.
doi: 10.1007/BF02219223. |
show all references
References:
[1] |
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.
doi: 10.1137/130912542. |
[2] |
P. Aguirre, B. Krauskopf and H. M. Osinga,
Global invariant manifolds near a Shilnikov homoclinic bifurcation, J. Comput. Dyn., 1 (2014), 1-38.
doi: 10.3934/jcd.2014.1.1. |
[3] |
A. Algaba, M. C. Domínguez-Moreno, M. Merino and A. Rodríguez-Luis,
Study of a simple 3D quadratic system with homoclinic flip bifurcations of inward twist case $ \mathbf{C}_{\rm in} $, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 324-337.
doi: 10.1016/j.cnsns.2019.05.005. |
[4] |
R. Barrio, S. Ibáñez and L. Pérez,
Hindmarsh–Rose model: Close and far to the singular limit, Phys. Lett. A, 381 (2017), 597-603.
doi: 10.1016/j.physleta.2016.12.027. |
[5] |
R. Barrio, M. A. Martínez, S. Serrano and A. Shilnikov, Macro- and micro-chaotic structures in the Hindmarsh–Rose model of bursting neurons, Chaos, 24 (2014), 11pp.
doi: 10.1063/1.4882171. |
[6] |
L. A. Belyakov,
Bifurcation set in a system with homoclinic saddle curve, Mat. Zametki, 28 (1980), 911-922.
|
[7] |
L. A. Belyakov,
Bifurcations of systems with a homoclinic curve of the saddle-focus with a zero saddle value, Mat. Zametki, 36 (1984), 681-689.
|
[8] |
A. R. Champneys and Y. A. Kuznetsov,
Numerical detection and continuation of codimension-two homoclinic bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 785-822.
doi: 10.1142/S0218127494000587. |
[9] |
A. R. Champneys, Y. A. Kuznetsov and B. Sandstede,
A numerical toolbox for homoclinic bifurcation analysis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 867-887.
doi: 10.1142/S0218127496000485. |
[10] |
B. Deng,
Homoclinic twisting bifurcations and cusp horseshoe maps, J. Dynam. Differential Equations, 5 (1993), 417-467.
doi: 10.1007/BF01053531. |
[11] |
E. Doedel,
AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284.
|
[12] |
E. J. Doedel and B. E. Oldeman, AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations, Department of Computer Science, Concordia University, Montreal, Canada, 2010. Available from: http://www.cmvl.cs.concordia.ca/. |
[13] |
F. Dumortier, Local study of planar vector fields: Singularities and their unfoldings, in Structures in Dynamics: Finite Dimensional Deterministic Studies, Studies in Mathematical Physics, 2, Elsevier Science Publishers, Amsterdam, 1991,161–241.
doi: 10.1016/B978-0-444-89257-7.50011-5. |
[14] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006.
doi: 10.1007/978-3-540-32902-2. |
[15] |
A. Giraldo, B. Krauskopf and H. M. Osinga,
Saddle invariant objects and their global manifolds in a neighborhood of a homoclinic flip bifurcation of case B, SIAM J. Appl. Dyn. Syst., 16 (2017), 640-686.
doi: 10.1137/16M1097419. |
[16] |
A. Giraldo, B. Krauskopf and H. M. Osinga,
Cascades of global bifurcations and chaos near a homoclinic flip bifurcation: A case study, SIAM J. Appl. Dyn. Syst., 17 (2018), 2784-2829.
doi: 10.1137/17M1149675. |
[17] |
E. A. González Velasco,
Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc., 143 (1969), 201-222.
doi: 10.1090/S0002-9947-1969-0252788-8. |
[18] |
A. J. Homburg, H. Kokubu and M. Krupa,
The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit, Ergodic Theory Dynam. Systems, 14 (1994), 667-693.
doi: 10.1017/S0143385700008117. |
[19] |
A. J. Homburg, H. Kokubu and V. Naudot,
Homoclinic-doubling cascades, Arch. Ration. Mech. Anal., 160 (2001), 195-243.
doi: 10.1007/s002050100159. |
[20] |
A. J. Homburg and B. Krauskopf,
Resonant homoclinic flip bifurcations, J. Dynam. Differential Equations, 12 (2000), 807-850.
doi: 10.1023/A:1009046621861. |
[21] |
A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems, 3, Elsevier, New York, 2010,381–509. |
[22] |
M. Kisaka, H. Kokubu and H. Oka,
Bifurcations to $N$-homoclinic orbits and $N$-periodic orbits in vector fields, J. Dynam. Differential Equations, 5 (1993), 305-357.
doi: 10.1007/BF01053164. |
[23] |
B. Krauskopf and T. Rieß,
A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits, Nonlinearity, 21 (2008), 1655-1690.
doi: 10.1088/0951-7715/21/8/001. |
[24] |
B. Krauskopf, H. M. Osinga and J. Galán-Vioque, Numerical Continuation Methods for Dynamical Systems. Path Following and Boundary Value Problems, Understanding Complex Systems, Springer, Dordrecht, 2007.
doi: 10.1007/978-1-4020-6356-5. |
[25] |
Y. A. Kuznetsov, O. De Feo and S. Rinaldi,
Belyakov homoclinic bifurcations in a tritrophic food chain model, SIAM J. Appl. Math., 62 (2001), 462-487.
doi: 10.1137/S0036139900378542. |
[26] |
X.-B. Lin,
Using Mel'nikov's method to solve Šilnikov's problems, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 295-325.
doi: 10.1017/S0308210500031528. |
[27] |
D. Linaro, A. Champneys, M. Desroches and M. Storace,
Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh–Rose burster, SIAM J. Appl. Dyn. Syst., 11 (2012), 939-962.
doi: 10.1137/110848931. |
[28] |
K. Matsue,
On blow-up solutions of differential equations with Poincaré-type compactifications, SIAM J. Appl. Dyn. Syst., 17 (2018), 2249-2288.
doi: 10.1137/17M1124498. |
[29] |
M. Messias, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system, J. Phys. A, 42 (2009), 18pp.
doi: 10.1088/1751-8113/42/11/115101. |
[30] |
M. Messias,
Dynamics at infinity of a cubic Chua's system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 333-340.
doi: 10.1142/S0218127411028453. |
[31] |
B. E. Oldeman, B. Krauskopf and A. R. Champneys,
Numerical unfoldings of codimension-three resonant homoclinic flip bifurcations, Nonlinearity, 14 (2001), 597-621.
doi: 10.1088/0951-7715/14/3/309. |
[32] |
B. Sandstede, Verzweigungstheorie Homokliner Verdopplungen, Ph.D thesis, University of Stuttgart in Stuttgart, Germany, 1993. |
[33] |
B. Sandstede,
Constructing dynamical systems having homoclinic bifurcation points of codimension two, J. Dynam. Differential Equations, 9 (1997), 269-288.
doi: 10.1007/BF02219223. |











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