Abstract
As its title implies, this chapter is concerned with oscillatory solutions of ODEs. Solutions of the van der Pol system (1.36) plotted in Figure 1.7, are representative of the kind of behavior we focus on. Up to now, we have been forced to rely on the computer to study such phenomena. In this chapter, we introduce analytical techniques to predict and describe oscillatory behavior.
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Notes
- 1.
In Problem 3.10 we asked you to prove this, but we still give the easy proof here.
- 2.
Recall the distinction: orbit refers to the set \(\{\mathbf{x}(t): t \in \mathbb{R}\}\) considered as a subset of \(\mathbb{R}^{d}\) , independent of any parametrization; trajectory refers to the curve with the specific parametrization that satisfies the ODE x ′ = F ( x ).
- 3.
The simpler system r′ = 1 − r, θ′ = 1, has more or less the same behavior. However, unlike (7.5), if we attempt to write this equation in Cartesian coordinates, \(\mathbf{x}^{{\prime}} = \mathbf{F}(\mathbf{x})\), then F(x) is singular at the origin. It blows up like 1∕r there. In other words, the simpler ODE is an equation only on \(\mathbb{R}^{2} \sim \{\mathbf{0}\}\). Often we shall accept such a singularity in order to simplify an example.
- 4.
- 5.
Already in Section 4.3.3 we found it convenient to reduce a Euclidean coordinate modulo 2π and thereby interpret a planar ODE as one on the cylinder. In a similar spirit, the periodic solution of (7.4) in polar coordinates, r(t) ≡ 1, θ(t) = t, may be viewed as a closed curve on the cylinder (0, ∞) × S 1. Incidentally, as we show in Section 7.9.4, every ODE on \(\mathbb{T}^{2}\), in particular (7.6), can be embedded in an ODE on \(\mathbb{R}^{3}\).
- 6.
Ask yourself which method you find most appealing. Your preference provides guidance about possible career choices, or at least specializations within mathematics. If you like numerics best, consider scientific computation; if you like asymptotics best, consider traditional applied mathematics; if you like rigorous methods best, consider mathematical analysis.
- 7.
Asymptotics is only a secondary focus for this book. We refer you to [98] or [43] to go beyond our limited coverage, or better still, take a course. Asymptotics is a hard subject to learn without guidance from a pro.
If you wish to pursue one of the other two techniques, for numerical methods read Section 10.3 of this book and follow the references there, and for rigorous analysis see Section 6.7 of [54].
- 8.
Are we using Cartesian coordinates or polar coordinates to specify the equilibrium? It doesn’t matter, because of a convenient mathematical pun: (1, 0) specifies the same point in either coordinates.
- 9.
To make this more specific, the equilibrium (1, 0) of (7.18) is highly degenerate; indeed, all entries of the Jacobian DF vanish there.
- 10.
We write lowercase \(\boldsymbol{\gamma }(t)\) for the solution and uppercase Γ for the orbit.
- 11.
It is customary to say the Poincaré map, but in fact there are many. “The” is justified in the sense that all of these mappings may be transformed to one another (on appropriately small neighborhoods) by changes of coordinates.
- 12.
“Transverse” in this context reduces to the condition that \(\boldsymbol{\gamma }'(0)\) is not tangent to Σ.
Incidentally, it would not create any mathematical difficulties to assume that Σ was a curved surface, but writing precise equations in this case requires either the abstract language of differential geometry or some heavy-handed notation. Not liking either option, we have assumed that Σ is flat.
- 13.
In many examples it might seem as though we could drop the neighborhood \(\mathcal{N}\) and just make Σ small. This idea doesn’t work when \(\boldsymbol{\gamma }\) is unstable; see Exercise 21.
- 14.
Discrete-time dynamical system are important and interesting in their own right, independent of any connection to Poincaré maps. Chapter 10 of [81] has a brief, readable introduction to the behavior of one-dimensional maps, and [17] gives a more thorough, but still readable, treatment. (See also Sections 9.9.2 and 10.6(a) below.)
- 15.
Although we use Poincaré-map notation, we have not yet related this map to any periodic solution.
- 16.
In some cases this series may not converge—it could be only asymptotic (cf. Section 1.4 of [43] or Section 6.2 of [98]). Whether the series converges or not doesn’t really have much impact on the application of the method. In this example, one may deduce from the exact solution (7.44) that the series does actually converge in some neighborhood of zero.
- 17.
That is, the period of the forcing term equals the period of solutions of the homogeneous equation, x 1″ + x 1 = 0. The term resonance derives from Exercise 1.13.
- 18.
- 19.
This identity may also be interpreted as the Fourier series expansion of cos2 θsinθ.
- 20.
- 21.
Hint: It may be useful to reflect on the information from the next section that solutions slow down as the coefficient of the frictional term tends to infinity.
- 22.
We also refer to a fast–slow system such as (4.36) as a singular perturbation. In that case, the dimension of the system is reduced if ɛ = 0. The behavior of solutions of that problem was analyzed in Exercise 5.12.
- 23.
- 24.
The decay for van der Pol’s equation is much slower than for the linear equation. For these initial conditions, the effective coefficient of friction in van der Pol’s equation, β(x 2 − 1), is enhanced by a large factor, initially almost 25.
- 25.
For pedagogical reasons let’s explore the consequences of not scaling and ignoring the principle of dominant balance from Section 5.9.1; i.e., let us attempt to derive an approximate equation from (7.58) by dropping the first and third terms on the grounds that the middle term, which has the large coefficient, should dominate. This leads to a completely sterile equation, (x 2 − 1)x′ = 0, which has only trivial constant solutions.
Recall that the principle of dominant balance tells us to scale the equation so that two terms have comparable magnitude and dominate the third. The scaling τ = t∕β yields (7.62), which has this structure. The scaling τ = β t would balance two different terms; although our discussion doesn’t employ it, it is used in deriving the results quoted in Section 7.6.4.
- 26.
- 27.
- 28.
- 29.
In regular perturbation theory, such as in Section 7.5, higher-order calculations are routine, although often tedious. By contrast, the derivation of (7.69) is anything but routine. The fractional powers and logarithmic terms in (7.69), which are common in singular-perturbation expansions, warn of hidden complexities. The derivation of this result is based on a sophisticated application of matched asymptotic expansions. Exercise 5.12 gives a simple illustration of this technique; the general method is developed in [98] and in [43]; and the specific formula (7.69) is derived in [13].
Incidentally, some authors give a formula for T(β) with a different coefficient for β −1∕3; we assure you that equation (7.69) is correct.
- 30.
This τ has no connection to the scaled time in the Poincaré–Lindstedt method. We do not scale time in the present calculation.
- 31.
This map is an example of what is called an affine map: more precisely, it equals the sum of a linear map (i.e., b ↦ A b) and a constant term (i.e., b 0 − A b 0).
- 32.
The best reference for this theorem may be to look online. It is covered in many topology texts, but usually it’s near the end of the book where more of an investment is needed to extract the essential ideas.
- 33.
Such systems arise, among other contexts, in describing the linearization of the Poincaré map of a periodic solution of an ODE; this connection suggests the notation Π for the matrix in (7.97).
- 34.
Equation (7.109a) imposes a limit on the amplitude of vibration for stabilization, and because of our cavalier treatment of friction, it is unclear how this limit depends on β. This issue could be explored numerically, but of course if β = 0. 1, we know from Exercise 1.12 that α = 0. 1 is below this limit.
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Schaeffer, D.G., Cain, J.W. (2016). Oscillations in ODEs. 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_7
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