# American Institute of Mathematical Sciences

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. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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. 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.  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. 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.  Google Scholar [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.  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. 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.  Google Scholar [19] A. J. Homburg, H. 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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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. 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.  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. 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.  Google Scholar [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.  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. 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.  Google Scholar [19] A. J. Homburg, H. 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. 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.  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. 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.  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. 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.  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. 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.  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
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 $\mathbf{C}_{\rm in}$
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 $\mathbf{H_{t}}$
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
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
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)
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 $z_{\rm B} = -\alpha$
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
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, $\alpha = 5.3$
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
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)
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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