Abstract
In this epilogue, we offer brief overviews of several ODE topics not covered in the main body of the text.
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Notes
- 1.
- 2.
Using the terminology of Chapter 8, we may say that the BVP undergoes a saddle-node bifurcation as ℓ passes through ℓ 0.
- 3.
- 4.
Looking for points on the stable manifold of a saddle point of a nonlinear system of ODEs may be viewed as imposing a boundary condition at t = ∞, albeit not one that picks out a unique solution.
- 5.
This is the traditional application; to be more current, you might prefer to think of a long-distance power cable.
- 6.
For L to be linear, the boundary conditions must be homogeneous. The theory can be extended to inhomogeneous boundary conditions, but that is a digression we do not pursue.
- 7.
If you are familiar with probability theory, you may recognize that (10.18) could be formulated more simply in terms of conditional probabilities.
- 8.
The precise values of the o(h)-correction terms here and below are different. Part of the power of the order notation is to allow such sleight of hand in rigorous analysis.
- 9.
If you are trying to solve an IVP with (10.27), y 0 can be taken from the initial conditions, but y 1 must be determined by some other method. In principle, you could calculate y 1 with a higher-order one-step method, but there is no need: you do not sacrifice the higher-order accuracy of (10.27) if for one step only you compute y 1 with the less-accurate Euler approximation, y 1 = y 0 + hF(y 0).
- 10.
Can you possibly ignore a heavy-duty admonition like this one? We hope not!
- 11.
Huygens observed entrainment in 1666: over time, the pendulums of two clocks placed on the same board became synchronized, 180∘ out of phase. You can track down many other instances of it on the Internet.
- 12.
These properties and the properties quoted below are derived, for example, in Section 1.14 of [17].
- 13.
A technical issue: the existence of this conjugacy requires that Φ be at least \(\mathcal{C}^{2}\). Of course, this and more are satisfied here.
- 14.
Real graphs of this type are usually so flat that no structure can be seen. Besides being purely hypothetical, the vertical scale in the figure is enormously expanded.
- 15.
XPPAUT is one example of free software that can solve delay differential equations numerically.
- 16.
- 17.
- 18.
But see Exercise 7.
- 19.
In technical language, the set of exceptional points has Lebesgue measure zero. A brief introduction to measures from a dynamical-systems viewpoint is given in Section 11.4.1 of [67]. However, for this survey, the intuitive associations of this phrase suffice.
- 20.
The caveat “generic” is necessary, because, for example, if two initial conditions happen to lie on the same orbit, the two trajectories will be translates of each other and therefore not diverge.
- 21.
For the three-dimensional Lorenz system, there are actually three Lyapunov exponents; see Section 7.2 of [54] or Chapter 29 of [95]. Incidentally, the sum of these three exponents equals the trace of the matrix in (10.57), or − 13. 6666. Since this number is large and negative, we see from Proposition 7.9.1 that volumes contract rapidly as time evolves in the Lorenz equations.
- 22.
Some initial conditions, such as on the stable manifold of the origin, may produce different behavior. In computations, however, this is a nonissue: even if you were so profoundly unlucky as to choose “bad” initial conditions, because of round-off errors as the computation proceeds, exponential growth will leak into your results and then dominate the computation.
- 23.
Similar calculations were proposed in Exercise 9. 7.
- 24.
A minor technicality: depending on a, it might happen that f(λ) has zeros along this portion of Γ 3. If so, we may deform the middle portion of Γ 3 to avoid these zeros, and the variation is still \(\mathcal{O}(1)\).
- 25.
- 26.
The use of Fourier series in PDEs is covered, for example, in [80].
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Schaeffer, D.G., Cain, J.W. (2016). Epilogue. 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_10
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