Abstract
In this chapter we state and prove the basic existence and uniqueness theorems (in Sections 3.2 and 3.3, respectively) for the initial value problem (IVP) for systems of nonlinear ODEs. For the moment we consider only autonomous systems, say
where \(\mathbf{F}: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}\); or more generally, we may assume that F is defined only on an open subset \(\mathcal{U}\subset \mathbb{R}^{d}\). In Section 3.4, we discuss extensions of the theory to nonautonomous systems.
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Notes
- 1.
This example is merely a simplification of Exercise 4(b) of Chapter 1 that makes the calculations all but trivial.
- 2.
This function is not a solution of (3.4) for t < 0. Geometrically, it follows from the fact that x′ ≥ 0 that x(t) ≤ 0 when t < 0. Analytically, the issue is that the radical means the positive square root, so \(\sqrt{t^{2 } /4} = \vert t\vert /2\neq (d/dt)(t^{2}/4)\). This example provides an unwelcome, but salutary, reminder that explicit solutions of ODEs need to be checked thoroughly.
- 3.
- 4.
Incidentally, if F is locally Lipschitz on \(\mathcal{U}\), then the restriction of F to a compact subset of \(\mathcal{U}\) is Lipschitz; see Proposition 3.3.2. The proof is trickier than you might expect.
- 5.
This might be an appropriate place to repeat our warning from Chapter 1: the symbols x, y, etc., may denote a point in \(\mathbb{R}^{d}\) (as in (3.6)) or a vector-valued function of time (as in (3.7)), depending on context. Whenever you see one of these letters, ask yourself which usage is intended.
- 6.
In the Pearls, Section 3.6.1, we invoke a little analysis to construct a more dramatic example of this point.
- 7.
Note that the functions in (3.11) are vector-valued. Each component of this equation is simply the standard one-variable fundamental theorem of calculus.
- 8.
For a scalar function it is obvious from the interpretation of the integral as an area that
$$\displaystyle{\left \vert \int _{0}^{1}f(s)\,ds\right \vert \leq \int _{ 0}^{1}\vert f(s)\vert \,ds.}$$To derive the analogous result for a vector- or matrix-valued function, apply the triangle inequality to approximating Riemann sums and then take limits.
- 9.
Incidentally, the contraction mapping theorem extends easily to the more general context of a complete metric space. This result has no connection to the fact that X is a linear space. The only change needed is to replace ∥ x −y ∥ by the distance function d(x, y). However, to minimize formalism, we don’t introduce this more general, but unnecessary, context.
- 10.
In words, B(b, δ) is the ball in the Euclidean space \(\mathbb{R}^{d}\) of radius δ around the vector b, while S is the (closed) ball in the infinite-dimensional space \(\mathcal{C}([-\eta,\eta ], \mathbb{R}^{d})\) of radius δ around the constant function b. In symbols,
$$\displaystyle{\mathsf{S} =\{ \mathbf{x} \in \mathcal{C}([-\eta,\eta ], \mathbb{R}^{d}):\| \mathbf{x} -\mathbf{b}\| \leq \delta \}.}$$ - 11.
The fixed point x is a continuous function on the closed interval [−η, η], which is more than (3.7) requires.
- 12.
With this notation we may write a single formula that is valid for both t > 0 and t < 0.
- 13.
Here is an example to show that the annoying hypothesis “ x j (β) belongs to the domain of F ” is really necessary. It’s a bit technical, so don’t pursue this unless you, like us, are amused by such things. On the domain \(\mathcal{U} =\{ (x,y): x^{2} > y^{3}\}\) consider the ODE (x′,y′) = (f(x,y), 0), where f(x,y) = (1∕3)(x 2 − y 3 ) −1 . Then (x 1 (t),y 1 (t)) = (t 1∕3 , 0) for t ≤ 0 and (x 2 (t),y 2 (t)) = (t 1∕3 , 0) for t ≥ 0 satisfy all the hypotheses of the theorem (with β = 0) except the containment hypothesis, but the function defined by (3.29) is not differentiable at t = 0.
- 14.
For the record: a function g is called piecewise continuous on an interval \(\mathrm{i}\) if there is a finite set of points {a k : k = 1, 2, …, p} in \(\mathrm{i}\) such that (i) g is continuous on \(\mathrm{i} \sim \cup _{k}\{a_{k}\}\) and (ii) at each point a k , the one-sided limits of g exist (and are finite).
- 15.
Note that the absolute-value function is not differentiable, so it is possible that g is not differentiable. Thus, the weak hypothesis in Gronwall’s lemma simplifies the proof of the theorem.
This is just one instance of how proofs in ODEs, which is an old subject, have been polished over the years. Indeed, beware of reading through this and other proofs too quickly and missing the cleverness. For example, in this proof, before applying Gronwall’s lemma, we prepare for it by (i) restricting t to a large closed subinterval of [0, β) to obtain compactness and (ii) invoking Proposition 3.3.2 to derive a Lipschitz constant that works on all of [0, T]. After these preparations have been made, the proof may be appropriately described as a straightforward application of Gronwall’s lemma.
- 16.
- 17.
This system has discontinuous coefficients, but it is a case in which the discontinuities are not a problem (cf. Section 3.6.1).
- 18.
The letter \(\mathcal{O}\) is mnemonic for order; thus \(\mathcal{O}(\varepsilon ^{3})\) is shorthand for omitted terms that are of order ɛ 3 or higher. A more serious usage of this notation is introduced in Section 4.6.4
- 19.
These points apply to the context of Theorem 3.2.1. In more general contexts, such as stochastic differential equations, Picard iteration may be the preferable method.
- 20.
A minor clarification: pumping a swing is an example of parametric resonance in a nonlinear problem: a pendulum.
References
G. Birkhoff and G.-C. Rota, Ordinary differential equations, 4th edition, Wiley, New York, 1989.
M. Braun, Differential equations and their applications: An introduction to applied mathematics, 4th edition, Springer, New York, 1993.
J. H. Hubbard and B. H. West, Differential equations: A dynamical systems approach, Springer-Verlag, New York, 1991.
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Schaeffer, D.G., Cain, J.W. (2016). Nonlinear Systems: Local Theory. 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_3
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