Abstract
In this chapter we relate the flow of an ODE \(\mathbf{x}^{{\prime}} = \mathbf{F}(\mathbf{x})\) near an equilibrium b ∗ to the flow of the linearization, by which we mean the equation w′ = A w, where A = DF(b ∗). There are two main theoretical results. (i) In Section 6.1, we assume \(\mathfrak{R}\lambda _{j}(A) <0\), which guarantees that all solutions of the linearization converge to the equilibrium; Theorem 6.1.1 shows that under this hypothesis, the full equation shares a version of this behavior, which is called asymptotic stability. (ii) In Section 6.6, the stable-manifold theorem (Theorem 6.6.1) characterizes the behavior of solutions when the Jacobian DF(b ∗) has eigenvalues with both positive and negative real parts.
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Notes
- 1.
- 2.
For example, the equilibrium (r, θ) = (1, 0) of (6.13) suffers from this degeneracy.
- 3.
- 4.
If you don’t like the geometric argument we use in deriving Points 3 and 4, you may just calculate detDF ∗ instead.
- 5.
Realistically, both chemicals should be allowed to diffuse. For the model problem (6.28), such a perturbation, if not too large, has little consequence. We ask you to verify this in Exercise 20. However, in a PDE formulation of the full problem, diffusion of the activator is an essential effect. In particular, it influences decisively the length scale of periodic structures that may arise from the Turing instability.
- 6.
A direct derivation of (6.35), (6.36) from Newton’s laws is tricky, but after the fact, let us interpret the equations in these terms. Equation (6.35) is the x-component of Newton’s second law for the motion of the center of mass of the total system. Equation (6.36) comes from computing moments around the pivot, but with one nonstandard ingredient: the term \(m\ell\cos \theta \,d^{2}\hat{x}/d\hat{t}^{2}\) is a fictitious torque reflecting the fact that moments are calculated about a point that may be accelerating. Note that the cart exerts an unknown force on the pendulum at the pivot. The above equations sidestep this issue; this force does not appear in (6.35), because an internal force does not contribute to the motion of the center of mass, and in (6.36) the moment of this force vanishes, because the length of the lever arm is zero.
- 7.
- 8.
Are you tempted to build a physical model and use it to test the predictions? Contemplating this, you will quickly see the wisdom in the quip due to V.I. Arnol’d, a distinguished Russian mathematician: “Mathematics is the part of physics where experiments are cheap.”
- 9.
- 10.
The general term “manifold” is defined in Section B.3.3, but in most examples the stable and unstable manifolds will be just curves in the plane.
- 11.
Recall that the orbit of a solution of an ODE means the curve traced out by the solution, considered merely as a subset of \(\mathbb{R}^{d}\), independent of any parametrization. By contrast, the term trajectory includes a specific parametrization that yields a solution of the ODE. Each orbit corresponds to infinitely many different trajectories, time translates of one another. Thus, it is more convenient to enumerate orbits than trajectories.
- 12.
Recall from the discussion of Jordan normal forms in Section C.1 that v is a generalized eigenvector of a matrix A with eigenvalue λ if (A −λ I)p v = 0 for some power p.
- 13.
- 14.
\(\mathcal{M}_{s}\) and \(\mathcal{M}_{u}\) always intersect at the equilibrium; we use “nontrivial” to refer to the existence of other intersection points. For (6.65), \(\mathcal{M}_{s}^{(glob)}\) and \(\mathcal{M}_{u}^{(glob)}\) coincide, but in higher-dimensional equations \(\mathcal{M}_{s}^{(glob)} \cap \mathcal{M}_{u}^{(glob)}\) may be a proper, but nontrivial, subset of both manifolds. Of course the intersection consists of one or more complete orbits.
- 15.
Strictly speaking, in this union we may take only those times t for which the initial value problem has a solution. This limitation may be expressed quite precisely in the flow notation of Section 4.5.2, but in our opinion such precision obscures more than it clarifies.
- 16.
In fact, using Theorem 6.1.1, you could now derive this behavior with less effort.
- 17.
“What else could it do?” we naively ask. Although phase portraits are often made using this kind of “logic,” be careful. Not seeing other alternatives might just reflect limitations of our imagination.
- 18.
Reflecting this behavior, in the case of a saddle point in the plane (d = 2, d s = 1), the stable and unstable manifolds (i.e., curves) were traditionally called separatrices.
- 19.
In case you were wondering: linearization is the issue that connects this problem to the rest of the present chapter: for most of the time while the solution is growing, θ is so close to vertical that solutions of the full equation and of the linearization could not be distinguished on a computer screen.
- 20.
Although we write \(\boldsymbol{\varphi }_{1}(\mathbb{R},\mathbf{b})\), only times such that \(\boldsymbol{\varphi }_{1}(t,\mathbf{b})\) is defined are to be substituted.
- 21.
- 22.
- 23.
See [12] for a careful treatment of the subject.
- 24.
Incidentally, there are more candidates for \(\mathcal{M}_{c}^{(l)}\) than suggested in the figure; see, for example, Figure 9.6(c).
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Schaeffer, D.G., Cain, J.W. (2016). Trajectories Near Equilibria. 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_6
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