Article Contents
Article Contents

Computing connecting orbits to infinity associated with a homoclinic flip bifurcation

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.

Mathematics Subject Classification: 37C29, 37G25, 37M21.

 Citation:

• 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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