Abstract
The main goal of this chapter is to give examples of five different types of global bifurcation. While the local bifurcations of the previous chapter were associated with stability changes in an equilibrium, the bifurcations in this chapter are associated with stability changes in a periodic solution. We make no pretense of completeness. It is not remotely possible to classify all possible global bifurcations. We introduce each type of bifurcation with an academic example that may be handled analytically, but for the more interesting examples that follow, we will rely heavily on computations (which we invite you to check) and on intuitive arguments. Even in cases for which proofs are available, we may omit them because we find them unrewarding. Consequently, this chapter is much less dense than the preceding one.
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Notes
- 1.
It may be instructive to compare Figures 9.3(c) and 9.4(a). In both figures, two different colored trajectories connect two equilibria. In Figure 9.3(c), both equilibria are saddle points; the trajectories are shown in purple because they are simultaneously (part of) the stable manifold of one equilibrium and the unstable manifold of the other. By contrast, in Figure 9.4(a), one of the equilibria is a sink; the trajectories are shown in red because they are the unstable manifold of the saddle point r = 1, θ = arccos(1 +μ), but they are in no way distinguished as regards the sink; they are only two of many orbits that converge to that equilibrium.
- 2.
The first two mechanisms are closely analogous to steady-state bifurcation and to Hopf bifurcation from equilibria, respectively. The third has no analogue in bifurcation from equilibria.
- 3.
As μ → −1, the inner orbit shrinks to the origin. On a superficial level this behavior resembles Hopf bifurcation, but this description is inappropriate, because (9.11) is singular at the origin.
- 4.
For modeling at a more fundamental level, we refer you to Keener and Sneyd [47], who give a beautiful introduction to electrophysiology from a mathematician’s perspective.
- 5.
For instance, a stimulus will fail to elicit an action potential if DI is too short. Such considerations are important for modeling, but for the purposes of this subsection they are a distraction that we sweep under the rug.
- 6.
Do you have enough artistic talent to make a perspective drawing that does justice to the three-dimensionality of this trajectory? If so, please send it to us, and we will put it on the web page, along with a grateful acknowledgment.
- 7.
- 8.
This bifurcation illustrates that, as mentioned above, mutual annihilation of limit cycles is not the only possibility when an eigenvalue of the Poincaré map crosses + 1. Cf. Exercise 3.
- 9.
We use this term informally without defining it precisely. Some features of chaotic behavior are explored in Section 10.6
- 10.
To be completely accurate, we should say that the projection of \(\mathcal{M}_{u}^{(+)}\) approaches the origin from the first quadrant. Here and below we gloss over this technical point to simplify the syntax.
- 11.
In other words, Γ − is the ω-limit of points on \(\mathcal{M}_{u}^{(+)}\). You might find it helpful to look at Exercise 5, which gives an analytically more tractable example of such behavior.
- 12.
If you are interested in neuroscience, you should definitely consult their book and/or [96]. For example, there are different types of bursting, and we model only one of them.
- 13.
Strictly speaking, we should say the purple (homoclinic) orbit and the equilibrium, but please cut us some slack.
- 14.
If it bothers you that the periodic forcing term J stim(t) is impulsive or that the right-hand side of the dh∕dt equation has jumps, feel free to employ smooth alternatives. If you do so, be sure that over each period, J stim(t) is nonzero only briefly, perhaps on the order of one to three milliseconds, but integrates to something on the order of say 0.5 (in order to provide a sufficient kick to v).
- 15.
How can we deduce this? In the first place, note that A ≈ 2. 4 in Figure 9.14(a), and at the bifurcation, it will be slightly larger, i.e., large enough to make 1 − A 2∕4 negative. Moreover, reflecting on the discussion following Exercise 7.16, we can see that for mechanism (i), the destabilizing eigenvalue of the Poincaré map would be + 1, while for mechanism (ii), it would be − 1; since the period is not doubled at the first bifurcation, we conclude that mechanism (i) is the driver. Period-doubling bifurcations of course do occur later, but they are less amenable to analysis; we do not attempt to locate them, even approximately.
- 16.
Note that this is an ODE with delay, a class of equations that is discussed in Section 10.5 Such equations exhibit some new phenomena; in particular, initial data must be specified along an entire time interval of length π∕ω. However, given such initial data, equations with delay can still be solved numerically using software such as XPPAUT.
- 17.
Although the description of fluid motion requires PDEs—the Navier–Stokes equation—in fact the analysis of bifurcation in many PDEs is closely analogous to the analysis of bifurcation in ODEs.
- 18.
Note the contrast: the stable and unstable manifolds of an equilibrium of an ODE cannot intersect transversely.
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Schaeffer, D.G., Cain, J.W. (2016). Examples of Global Bifurcation. In: Ordinary Differential Equations: Basics and Beyond. Texts in Applied Mathematics, vol 65. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6389-8_9
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