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2016 | OriginalPaper | Buchkapitel

8. Bifurcation from 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 Chapter 6 we studied the behavior of solutions of an ODE near a hyperbolic equilibrium point. In this chapter we turn to behavior near nonhyperbolic equilibria.

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Vorheriges Kapitel Oscillations in ODEs

Nächstes Kapitel Examples of Global Bifurcation

Up until now, a subscript star has been used to designate an isolated equilibrium of an ODE. In the present context, we have a curve of equilibria, and star designates a particular equilibrium of interest on that curve.

We ignore the equilibrium at x = π, which is unstable for all μ ≥ 0.

In the context of a parametrized family of ODEs like (8.1), the notation DF denotes the matrix of derivatives of F with respect to the state variables x ₁ ,…,x _d only. We write out derivatives with respect to parameters explicitly, such as ∂ F∕∂ μ.

The occurrence of additional solutions does not conflict with the result from Chapter 3 that under minimal hypotheses, the solution of the IVP is unique. The new solutions here are equilibrium solutions, possible ω-limits of trajectories.

The now unstable equilibrium at x = 0 for μ > 1 is not included in the counting; thus one does not speak of “trifurcation.” Somewhat inconsistently, the middle branch is not ignored in the term “pitchfork.”

The geometry in Figure 8.3(a) may seem a little artificial to you, but the behavior in this problem is actually representative of the collapse of a variety of structures. Such behavior is illustrated, for example, in Exercise 14. However, the more realistic geometry of the exercise is also more complicated to analyze. The artificially simple geometry in Figure 8.3(a) provides a gentler first encounter with subcritical bifurcation.

Incidentally, for the geometry in the figure, x is constrained to the interval (−π∕2, π∕2). Although we ignore this constraint in analyzing (8.8), one might question the physical significance of the equilibrium solution x = ±π outside this range.

Although x = ±π satisfies (8.8), we don’t show this solution in the figure.

If the nonzero eigenvalues of DF _∗ have negative real parts, then (8.11) implies that x _eq(μ) is stable for μ on one side of μ _∗ and unstable on the other. In other words, x _eq(μ) loses stability as μ crosses μ _∗. This is the most interesting case.

Besides (8.15), you should be aware of mischievous possibilities like x′ = φ(x) −μ x, where $\varphi (x) = e^{-1/x^{2} }\sin (1/x)$, defined for x = 0 by φ(0) = 0. The bifurcating equilibria wiggle back and forth infinitely many times above the point μ = 0.

For simplicity, we do not include the Allee effect. Locally near the bifurcation point, this effect would make little difference, as you can easily verify.

Despite this mismatch, we articulated Theorem 8.2.2 because we think it is informative for both pedagogical and historical reasons. The Lyapunov–Schmidt reduction of Section 8.5 applies to all types of steady-state bifurcation, including saddle-node bifurcations.

The term “saddle-node bifurcation” is natural for two-dimensional problems like (8.20). It is not natural, but is still used, for one-dimensional problems like x′ = −x ² +μ (cf. Table 8.1), as well as for problems with more than two variables.

More accurately, no steady-state bifurcation occurs; (8.23) is typically satisfied at a Hopf bifurcation point.

This equation repeats (8.10), but with a difference: here we make no assumption about equilibria of the ODE for μ ≠ μ _∗.

In general, a bifurcation problem may be reduced to a system with n variables, where n equals the dimension of kerDF _∗; see Chapter VII of [31].

Even the reduction of the Lorenz equation in Section 8.1.2, which we used to motivate the Lyapunov–Schmidt reduction, does not quite follow the standard procedure; cf. Exercise 5.

We’ll write x instead of x ₁ to cut down clutter in some upcoming equations.

In fact, the normal forms typically still apply even when higher-order terms are included; cf. Section II.9 of [31].

This convention is natural, because in applications it is more common for instability to appear as the parameters in the problem, which usually measure forcing, are increased.

Incidentally, our choice of − x ² +μ x for transcritical bifurcation is not important; any of the choices ± x ² ±μ x would work equally well. For that matter, ± (x ² −μ ²) are also possible, although we prefer a normal form in which x = 0 is a trivial solution branch.

The technical term is equivariant.

It is not hard to believe that the perturbation (8.55) splits the pitchfork into two components, but it is less clear whether for ɛ > 0 the negative μ-axis connects to the upper branch, as shown in Figure 8.8(b), or to the lower branch. The issue may be resolved with a Taylor-series expansion about the bifurcation point x _∗ = 0, μ _∗ = 1. Specifically, to lowest order, an equilibrium of (8.55) satisfies

$$\displaystyle{ x\,(\mu -1) + \varepsilon = 0, }$$

(8.56)

where we have neglected terms that are $\mathcal{O}(\vert x\vert ^{3})$ and $\varepsilon \mathcal{O}(\vert x\vert + \vert \mu - 1\vert )$. If ɛ > 0, the branch of the hyperbola (8.56) that is asymptotic to the negative μ-axis lies in the half-plane {x > 0}, and even with the higher-order terms, the solution of (8.55) will retain this behavior, as the figure indicates.

What might seem like the simplest unfolding, x′ = −x ³ +μ +α, doesn’t change anything; it just yields another hysteresis-point bifurcation at a slightly shifted location. Unfolding theory, the focus of Chapter III of [31], explains what perturbations in unfolding a singular bifurcation problem make an actual difference.

In Section 8.6.2 we discuss an application from chemical engineering with a hysteresis point in which such bistability can have disastrous consequences. Isola centers, which pose different risks, also appear in that application.

Since no symmetry is present in this problem, on grounds of genericity we expect the bifurcation to be transcritical. High-resolution computations confirm this expectation, but the actual bifurcation diagram is so nearly symmetric that even magnified 50 times, it is visually indistinguishable from a pitchfork. These circumstances give a warning about the limitations of arguments based on genericity.

It would be historically more accurate to refer to Andronov–Hopf bifurcation, but the shorter name has come into widespread use.

In connection with this behavior, we recommend that you revisit Exercise 2. 15.

In fact, (8.73) has greater generality than may be apparent; cf. Section 8.10.1.

The cell cycle is a prime example of such a biological clock. Although the cell cycle is vastly more complicated than the simple system (8.74), nevertheless, this model is considered to be a useful point of departure for studying the cell cycle (S. Haase, private communication). The model is mentioned in passing on pp. 564–565 of Winfree’s encyclopedic work [97].

Alternatively, they may be observed directly by solving the equations with “time running backward.” We invite you to do this computation.

It may be informative to compare (8.79) with the FitzHugh–Nagumo equation (Exercise 5. 11), which is a simplification of the Hodgkin–Huxley model. Both systems have the same x-nullclines, apart from scaling. While the y-nullclines of the FitzHugh–Nagumo equations are straight lines, the y-nullclines of (8.79) curve upward (see Figure 8.20), because a linear term is replaced by the exponential.

There’s a long story that we’re not telling you about how we chose these parameters. Let us remind you of the old adage, “A good teacher doesn’t tell you everything he/she knows.”

Equation ( 8.80c) has two roots $x_{{\ast}} = (1 \pm \sqrt{1 - 3\gamma })/3$. In ( 8.83a) we consider only the root with the plus sign, because the other case gives real eigenvalues for the Jacobian.

For other parameters, e.g., keeping α = 6 and γ = 0. 15 but reducing A to 0. 0001152, this determinant may vanish. In this case, we have what’s called a Takens–Bogdanov bifurcation (see Exercise 9).

Since $\overline{x} = x$, we may omit the bar over x.

It will appear below that there is no need for a term proportional to a ² in the expansion for x.

A time-honored example is Rayleigh–Bénard convection, the physical problem underlying the Lorenz equations in Section 8.1.2. Case Study 4 of [32] discusses information that can be derived using just the symmetries of the problem.

This geometry is more interesting than allowing diffusion between every pair of cells, because it approximates the geometry of the actual Turing instability. (Cf. [58], Section 2.2 of Volume II.)

Mode competition in a fluids problem, with more complicated phenomena than for plates, is analyzed in Case Study 6 of [32].

And subsequently to seven buckles, to eight buckles, and to complete collapse, but these jumps are beyond our focus.

In Stein’s experiment, compression was generated mechanically, but compression generated by thermal expansion, such as on a spacecraft during reentry, is also of interest.

Incidentally, Figure 8.24, a bifurcation diagram for equation (8.115) in the exercise, applies equally well to the buckling plate if the x- and y-modes in the figure are identified with the five-buckle and six-buckle modes of the plate, respectively.

This network is an example of what’s called a feed-forward loop (cf. Chapter 4 of [2]); i.e., the concentration of X is “fed forward” to influence a reaction rate of one of its products. Such feed-forward loops are embedded in many complex biological networks; the metabolic networks of [60] provide one example.

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Titel: Bifurcation from 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_8

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