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2016 | OriginalPaper | Buchkapitel

6. Trajectories Near Equilibria

verfasst von : David G. Schaeffer, John W. Cain

Erschienen in: Ordinary Differential Equations: Basics and Beyond

Verlag: Springer New York

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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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Vorheriges Kapitel Nondimensionalization and Scaling

Nächstes Kapitel Oscillations in ODEs

A similar integral equation, (3.16), arose in proving the existence theorem. The advantage of (6.6) over (3.16) is twofold: (i) the decay in (6.4) assists the convergence of the integral for large t, and (ii) in the integrand, r(x) is small when x is close to zero.

For example, the equilibrium (r, θ) = (1, 0) of (6.13) suffers from this degeneracy.

In Section 4.4.2 we proved global existence for the activator–inhibitor equations assuming κ < ∞, and global existence for (6.22) was posed as Exercise 4.3(c).

If you don’t like the geometric argument we use in deriving Points 3 and 4, you may just calculate detDF _∗ instead.

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.

This section is a standalone, moderately difficult application of Theorem 6.1.1. If you feel you need to press ahead, the section may be skipped or read later without loss of continuity.

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.

We find this choice surprising, because it seems to make the system more unstable. Indeed, if C is given by (6.41) and A = B = 0, then two of the roots of (6.40) have positive real parts.

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.”

In Exercise 13(c) we relate this Lyapunov function to behavior of the augmented Lotka–Volterra equation that we discussed in Section 1.6.

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.

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.

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.

Equation (6.61) is an example of what’s called a Hamiltonian system, a class of ODEs that is defined in Exercise 10. From now on, we will use the letter H for the energy of Hamiltonian systems.

\(\mathcal{M}_{s}\) 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.

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.

In fact, using Theorem 6.1.1, you could now derive this behavior with less effort.

“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.

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.

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.

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.

If you continually refer back to example (6.59) throughout the technical results that follow, the main payoff arrives once you reach equation (6.107). Namely, you’ll find that ψ(c) = −c ²∕3 (scalar-valued for this toy example), in agreement with formula (6.60) for the stable manifold.

If you object to using such heavy theory to prove such a basic result, we remind you that Exercise 18 outlines a direct proof of Proposition 6.2.1.

See [12] for a careful treatment of the subject.

Incidentally, there are more candidates for \(\mathcal{M}_{c}^{(l)}\) than suggested in the figure; see, for example, Figure 9.6(c).

[12]

J. Carr, Applications of centre manifold theory, Springer-Verlag, New York, 1982.MATH

[33]

J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag, New York, 1983.CrossRefMATH

[36]

P. Hartman, On local homeomorphisms of Euclidean spaces, Boletín de la Sociedad Matemática Mexicana 5 (1960), 220–241.MathSciNetMATH

[37]

_________ , Ordinary differential equations, John Wiley and Sons, New York, 1964.MATH

[54]

J. D. Meiss, Differential dynamical systems, SIAM, Philadelphia, 2007.CrossRefMATH

[57]

J. D. Murray, Mathematical biology. I: An introduction, Springer, New York, 2002.

[58]

_________ , Mathematical biology. II: Spatial models and biomedical applications, Springer, New York, 2003.

[63]

L. Perko, Differential equations and dynamical systems, 3rd edition, Springer-Verlag, New York, 2001.CrossRefMATH

[88]

S. T. Thornton and J. B. Marion, Classical dynamics of particles and systems, Thomson Brooks/Cole, 2004.

[95]

_________ , Introduction to applied nonlinear dynamical systems and chaos, 2nd edition, Springer, New York, 2003.MATH

Titel: Trajectories Near Equilibria
verfasst von: David G. Schaeffer
John W. Cain
Verlag: Springer New York
Buch: Ordinary Differential Equations: Basics and Beyond
Print ISBN: 978-1-4939-6387-4

Electronic ISBN: 978-1-4939-6389-8

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-1-4939-6389-8_6

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