December  2020, 7(2): 489-510. doi: 10.3934/jcd.2020020

Computing connecting orbits to infinity associated with a homoclinic flip bifurcation

Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand

* Corresponding author: b.krauskopf@auckland.ac.nz

Received  December 2019 Published  July 2020

Fund Project: AG is supported by the Dodd-Walls Centre for Photonic and Quantum Technologies; BK and HMO are supported by Royal Society of New Zealand Marsden Fund grant 16-UOA-286

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.

Citation: Andrus Giraldo, Bernd Krauskopf, Hinke M. Osinga. Computing connecting orbits to infinity associated with a homoclinic flip bifurcation. Journal of Computational Dynamics, 2020, 7 (2) : 489-510. doi: 10.3934/jcd.2020020
References:
[1]

P. AguirreB. 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.  Google Scholar

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L. A. Belyakov, Bifurcation set in a system with homoclinic saddle curve, Mat. Zametki, 28 (1980), 911-922.   Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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[11]

E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284.   Google Scholar

[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/. Google Scholar

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[16]

A. GiraldoB. 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.  Google Scholar

[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.  Google Scholar

[18]

A. J. HomburgH. 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.  Google Scholar

[19]

A. J. HomburgH. Kokubu and V. Naudot, Homoclinic-doubling cascades, Arch. Ration. Mech. Anal., 160 (2001), 195-243.  doi: 10.1007/s002050100159.  Google Scholar

[20]

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[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. Google Scholar

[22]

M. KisakaH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

Y. A. KuznetsovO. 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.  Google Scholar

[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.  Google Scholar

[27]

D. LinaroA. ChampneysM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

B. E. OldemanB. 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.  Google Scholar

[32]

B. Sandstede, Verzweigungstheorie Homokliner Verdopplungen, Ph.D thesis, University of Stuttgart in Stuttgart, Germany, 1993. Google Scholar

[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.  Google Scholar

show all references

References:
[1]

P. AguirreB. 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.  Google Scholar

[2]

P. AguirreB. 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.  Google Scholar

[3]

A. AlgabaM. C. Domínguez-MorenoM. 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.  Google Scholar

[4]

R. BarrioS. 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.  Google Scholar

[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.  Google Scholar

[6]

L. A. Belyakov, Bifurcation set in a system with homoclinic saddle curve, Mat. Zametki, 28 (1980), 911-922.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[9]

A. R. ChampneysY. 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.  Google Scholar

[10]

B. Deng, Homoclinic twisting bifurcations and cusp horseshoe maps, J. Dynam. Differential Equations, 5 (1993), 417-467.  doi: 10.1007/BF01053531.  Google Scholar

[11]

E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284.   Google Scholar

[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/. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. GiraldoB. 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.  Google Scholar

[16]

A. GiraldoB. 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.  Google Scholar

[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.  Google Scholar

[18]

A. J. HomburgH. 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.  Google Scholar

[19]

A. J. HomburgH. Kokubu and V. Naudot, Homoclinic-doubling cascades, Arch. Ration. Mech. Anal., 160 (2001), 195-243.  doi: 10.1007/s002050100159.  Google Scholar

[20]

A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations, J. Dynam. Differential Equations, 12 (2000), 807-850.  doi: 10.1023/A:1009046621861.  Google Scholar

[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. Google Scholar

[22]

M. KisakaH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

Y. A. KuznetsovO. 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.  Google Scholar

[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.  Google Scholar

[27]

D. LinaroA. ChampneysM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

B. E. OldemanB. 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.  Google Scholar

[32]

B. Sandstede, Verzweigungstheorie Homokliner Verdopplungen, Ph.D thesis, University of Stuttgart in Stuttgart, Germany, 1993. Google Scholar

[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.  Google Scholar

Figure 1.  Bifurcation diagram of system (2) showing: the curve of primary homoclinic bifurcation (brown), along which the homoclinic orbit changes at $ \mathbf{C}_{\rm in} $ from being orientable along $ \mathbf{H_o} $ to being non-orientable along $ \mathbf{H_t} $; curves $ \mathbf{SNP} $ and $ \mathbf{SNP^3} $ (green) of saddle-node bifurcation of periodic orbits; the first two curves $ \mathbf{PD} $ and $ \mathbf{PD^2} $ (red) of a cascade of period-doubling bifurcations; and the curves $ \mathbf{H^n} $ (increasingly darker shades of cyan) of $ n $-homoclinic bifurcations for $ n = 2, 3, 4, 5 $, and $ 6 $. On $ \mathbf{H^n} $ there are points $ \mathbf{C^n_{\rm O}} $ of orbit flip bifurcations (blue dots) and on $ \mathbf{H^2} $ there is a point $ \mathbf{C^2_{\rm I}} $ of inclination flip bifurcation (open dot). Panel (a) shows the $ (\alpha, \beta) $-plane, while panel (b) shows the $ (\alpha, \hat{\beta}) $-plane, where $ \hat{\beta} $ is the distance in the $ \beta $-coordinate from the curve $ \mathbf{H_{o/t}} $ of primary homoclinic bifurcation, which is now at $ \hat{\beta} = 0 $ (brown horizontal line). Panel (c) is an enlargement of the $ (\alpha, \hat{\beta}) $-plane near $ \mathbf{C}_{\rm in} $
Figure 2.  Phase portraits of system (2) along $ \mathbf{H_{t}} $, at $ \mathbf{C}_{\rm in} $ and along $ \mathbf{H_{o}} $ with enlargements near the saddle $ \mathbf{0} $ (top row). Shown are the saddle $ \mathbf{0} $, the homoclinic orbit $ \mathbf{\Gamma_{\rm HOM}} $ (brown curve) formed by one branch of $ W^s(\mathbf{0}) $, the other branch of $ W^s(\mathbf{0}) $ (cyan curve), a first part of $ W^{u}(\mathbf{0}) $ (red surface), and $ W^{uu}(\mathbf{0}) $ (magenta curve). Here $ (\alpha, \beta) = (5.8, 1.7010) $ in panel $ \mathbf{H_{o}} $, $ (\alpha, \beta) = (5.3573, 2.1917) $ in panel $ \mathbf{C}_{\rm in} $ and $ (\alpha, \beta) = (5.1, 2.717) $ in panel $ \mathbf{H_{t}} $
Figure 3.  The primary homoclinic orbit on $ \mathbf{H_t} $ and the $ n $-homoclinic orbits $ \mathbf{H^2} $ to $ \mathbf{H^6} $ of system (2) for $ \alpha = 5.3 $, shown in $ \mathbb{R}^3 $ in brown and increasingly darker shades of cyan to match the colors of the corresponding bifurcation curves in Fig. 1
Figure 4.  Dynamics at infinity for system (5), or system (4) with $ \bar{w} = 0 $, shown in the $ (\bar{x}, \bar{z}) $-plane in panel (a). Panel (b) shows the projection of panel (a) onto the corresponding Poincaré half-sphere with $ y_{\rm s} > 0 $ in the compactified $ (x_{\rm s}, y_{\rm s}, z_{\rm s}) $-cordinates
Figure 5.  Dynamics near the equilibrium $ (\bar{x}, \bar{z}, \bar{w}) = (0, 0, 0) $ of system (4). The behavior in the $ (x_{\rm B}, z_{\rm B}) $-plane, that is, the blow-up chart (6) with $ w_{\rm B} = 0 $, is shown in panel (a). It corresponds to the dynamics on a half-sphere around the origin in the $ (\bar{x}, \bar{z}, \bar{w}) $-space, as is illustrated in panel (b); compare also with Fig. 4(a)
Figure 6.  Numerical simulations suggest the existence of a cylinder-shaped separatrix $ S_{\rm c} $ of system (6) between trajectories that converge to the equilibrium $ (x_{\rm B}, z_{\rm B}, w_{\rm B}) = (0, -\alpha, 0) $, such as the orange trajectory, and those that do not, such as the blue trajectory. Panel (a) shows the $ (x_{\rm B}, z_{\rm B}, w_{\rm B}) $-space near $ (0, -\alpha, 0) $ and panel (b) the associated intersection sets with the plane defined by $ z_{\rm B} = -\alpha $
Figure 7.  The separatrix $ S_{\rm c} $ (purple surface) as represented locally by the cylinder $ C_{r^*} $, shown in the $ (\bar{x}, \bar{z}, \bar{w}) $-space of system (4). Panel (a) shows $ S_{\rm c} $ emerging from the blown-up half-sphere, while in panel (b), $ S_{\rm c} $ is a cone that emerges from the origin
Figure 8.  Set-up with Lin's method to compute a connecting orbit from $ \mathbf{q}_\infty $ to $ \mathbf{0} $ with two orbit segments that meet in the common Lin section $ \Sigma $ (green plane), illustrated in compactified Poincaré coordinates. Panel (a) shows the initially chosen orbit segments $ \mathbf{u} $ (cyan) to $ \mathbf{0} $ and $ \mathbf{u}_{\rm B} $ (magenta) from $ \mathbf{q}_\infty $ for $ \beta = 1.8 $ that define the Lin space $ Z $ (which appears curved in this representation); note that the Lin gap $ \eta $ is initially nonzero. Panel (b) shows the situation for $ \beta = 2.08874 $ where $ \eta = 0 $ and $ \mathbf{u} $ and $ \mathbf{u}_{\rm B} $ connect in $ \Sigma $ to form the heteroclinic connection; here, $ \alpha = 5.3 $
Figure 9.  Bifurcation diagram of system (2) with the additional curve $ \mathbf{Het^\infty} $ (magenta) of heteroclinic bifurcation involving the point $ \mathbf{q}_\infty $ at infinity. Panel (a) shows how $ W^s(\mathbf{0}) $ spirals towards infinity in the $ (x, y, z) $-space to form the heteroclinic connection on $ \mathbf{Het^\infty} $ for $ \alpha = 5.3 $ and $ \beta = 2.08874 $; see Fig. 3 for comparison. Panel (b) shows the overall bifurcation diagram in the $ (\alpha, \hat{\beta}) $-plane and panel (c) is an enlargement near the point $ \mathbf{C}_{\rm in} $; see Fig. 1 for details on the other bifurcation curves
Figure 10.  To the left of the curve $ \mathbf{Het^\infty} $ in the $ (\alpha, \hat{\beta}) $-plane, the stable manifold of $ W^s(\mathbf{0}) $ approaches, but does not connect to $ \mathbf{q}_\infty $, because it lies outside $ S_{\rm c} $ (a). To the right of $ \mathbf{Het^\infty} $, it lies inside $ S_{\rm c} $ and so connects to $ \mathbf{q}_\infty $. The illustration in compactified Poincaré coordinates is for $ \alpha = 5.3 $ with $ \beta = 2.9 $ in panel (a) and $ \beta = 2.8 $ in panel (b)
Figure 11.  Set-up with Lin's method to compute a connecting orbit from $ \mathbf{q}_\infty $ to a saddle periodic orbit $ \Gamma_o $ (green curve) with two orbit segments that meet in the common Lin section $ \Sigma $ (green plane), illustrated in compactified Poincaré coordinates for $ \alpha = 6.2 $ and $ \beta = 1.6 $. Panel (a) shows the initially chosen orbit segments $ \mathbf{u} $ (cyan) to $ \Gamma_o $ and $ \mathbf{u}_{\rm B} $ (magenta) from $ \mathbf{q}_\infty $ that define the Lin space $ Z $ (which appears curved in this representation); note that the Lin gap $ \eta $ is initially nonzero. Panel (b) shows the situation where $ \eta = 0 $ and $ \mathbf{u} $ and $ \mathbf{u}_{\rm B} $ connect in $ \Sigma $ to form the heteroclinic connection
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