Abstract
Ambrosetti and Prodi (A primer of nonlinear analysis, Cambridge University Press, Cambridge, 1993) formulated an abstract version of the Hopf bifurcation theorem and tried to deduce the well-known classical result from it. In this paper, we examine the Hopf bifurcation phenomena in the framework of a Sobolev space (rather than C r), having recourse to the Carleson–Hunt theory. Some more careful reasonings to evaluate the magnitudes of the Fourier coefficients seem to be required in order to implement the Ambrosetti–Prodi approach in their classical setting. We incidentally try to fortify their way of proof from the standpoint of classical Fourier analysis.
Received: July 12, 2010
Revised: October 19, 2010
JEL classification: E32
Mathematics Subject Classification (2010): 34C23, 34C25, 42A20
I would like to express my sincere thanks to Professors S. Kusuoka, P.H. Rabinowitz and Y. Takahashi for their invaluable comments on the earlier draft.
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Notes
- 1.
Related topics are discussed in Ambrosetti–Prodi [1] pp. 17–21.
- 2.
Let \(\mathcal{V}\) and \(\mathcal{W}\) be a couple of Banach spaces. Assume that a function φ of an open subset U of \(\mathcal{V}\) into \(\mathcal{W}\) is Gâteaux-differentiable in a neighborhood V of x ∈ U. We denote by δφ(v) the Gâteaux-derivative of φ at v. If the function \(v\mapsto \delta \varphi (v)(V \rightarrow \mathcal{L}(\mathcal{V},\mathcal{W}))\) is continuous, then φ is Fréchet-differentiable.
- 3.
The continuity of the mapping (11) can be proved in the same manner as in the proof of Lemma 1. Assumption 1(i) is used again for the dominated convergence argument.
- 4.
This is a special case of the Rellich–Kondrachov compactness theorem. Evans [4] pp. 272–274.
- 5.
See, for instance, Zeidler [11] pp. 300–301.
- 6.
If a 2π-periodic function \(\varphi : \mathbb{R} \rightarrow \mathbb{R}\) is absolutely continuous and its derivative φ′ belongs to \({\mathcal{L}}^{2}([0, 2\pi ], \mathbb{R})\), then the Fourier series of φ uniformly converges to φ on \(\mathbb{R}\). cf. Katznelson [8] Theorem 6.2, pp. 33–34. The k-th Fourier coefficient of φ′ is given by \(ik\hat{\varphi }(k)\), where \(\hat{\varphi }(k)\) is the k-th Fourier coefficient of φ.
- 7.
span{ξ} denotes the subspace of \({\mathbb{C}}^{n}\) spanned by ξ.
- 8.
Put \(\mu p(t) + \nu q(t) = \mu (\gamma \cos t - \delta \sin t) + \nu (\gamma \sin t + \delta \cos t) = (\mu \gamma + \nu \delta )\cos t + (\nu \gamma - \mu \delta )\sin t = 0.\) Then we have
$$\left \{\begin{array}{@{}l@{\quad }l@{}} \mu \gamma + \nu \delta = 0,\quad \\ \nu \gamma - \mu \delta = 0.\quad \end{array} \right.$$It follows that
$$\left \{\begin{array}{@{}l@{\quad }l@{}} \mu \nu \gamma + {\nu }^{2}\delta = 0,\quad \\ \mu \nu \gamma - {\mu }^{2}\delta = 0.\quad \end{array} \right.$$Hence \(({\nu }^{2} + {\mu }^{2})\delta = 0.\) If μ≠0 or ν≠0, δ must be zero. And so ξ = γ, that is \(\xi =\bar{ \xi }\) (real vector). Thus we get a contradiction.
- 9.
Let \(\varphi : \mathbb{R} \rightarrow \mathbb{R}\) (we may replace \(\mathbb{R}\) by \({\mathbb{R}}^{n}\)) be a 2π-periodic function which is integrable on [ − π. π]. Furthermore we assume \(\hat{f}(0) = 0\) (\(\hat{f}(0)\) is the Fourier coefficient corresponding to k = 0). If we define
$$\Phi (t) ={ \int \nolimits \nolimits }_{0}^{t}\varphi (\tau )d\tau,$$Φ is a 2π-periodic continuous function and
$$\hat{\Phi }(k) = \frac{1} {ik}\hat{f}(k),\quad k\neq 0.$$See Katznelson [8] Theorem 1.6, p. 4 or Zygmund [12] Vol. 1, p. 42.
- 10.
For any \(({\alpha }_{0},{\beta }_{0}) \in {\mathbb{C}}^{n} \times \mathbb{C}\), there exist some \({\lambda }_{0} \in \mathbb{C}\) and \({\gamma }_{0} \in (i{\omega }^{{_\ast}}I - {A}_{{\mu }^{{_\ast}}})({\mathbb{C}}^{n})\) such that \({\alpha }_{0} = {\lambda }_{0}\xi + {\gamma }_{0}\). And such λ0 and γ0 are unique. Let (α0, β0) = (0, 0). Then we must have λ0 = 0 and γ0 = 0. The equation
$$\left (\begin{array}{*{10}c} (i{\omega }^{{_\ast}}I - {A}_{{\mu }^{{_\ast} }})\theta \\ \langle \eta,\theta \rangle \end{array} \right ) = \left (\begin{array}{*{10}c} 0\\ 0 \end{array} \right )$$has a unique solution \(\theta \,=\,0(\in {\mathbb{C}}^{n})\) because Ker\([i{\omega }^{{_\ast}}I - {A}_{{\mu }^{{_\ast}}}] \cap \mathrm{ Ker}\langle \eta,\cdot \rangle \,=\,\mathrm{Ker}[i{\omega }^{{_\ast}}\) \(I - {A}_{{\mu }^{{_\ast}}}] \cap [i{\omega }^{{_\ast}}I - {A}_{{\mu }^{{_\ast}}}]({\mathbb{C}}^{n})\,=\,\{0\}\). Thus we conclude that D (λ, θ) g(μ∗, iω∗, 0) is injective.
- 11.
iω∗ is a simple eigenvalue, again by Assumption 2.
- 12.
Look at the Fig. 2. For the sake of an intuitive exposition, the vectors η, ξ and κ are treated as real vectors. By \(\langle \eta,\xi \rangle =\parallel \eta \parallel \cdot \parallel \xi \parallel \cos \theta = 1\), it follows that \(\parallel \eta \parallel = 1/ \parallel \xi \parallel \cos \theta \). Hence
$$\Pi (\kappa ) = \xi \langle \eta,\kappa \rangle = (\parallel \kappa \parallel \cos \zeta / \parallel \xi \parallel \cos \theta ) \cdot \xi.$$Since ∥ κ ∥ cosζ = OA and ∥ ξ ∥ cosθ = OB,
$$\Pi (\kappa ) = \frac{OA} {OB} \cdot \xi.$$Here ζ is the angle between η and κ, and θ is the one between ξ and η. κ can be represented uniquely as \(\kappa = \alpha \eta + \beta z\) for some \(\alpha,\beta \in \mathbb{C}\) and \(z \in [i{\omega }^{{_\ast}}I - {A}_{{\mu }^{{_\ast}}}]({\mathbb{C}}^{n})\). On the other hand, η can be represented uniquely in the form \(\eta = a\xi + bz\prime\) for some \(a,b \in \mathbb{C}\) and \(z\prime \in [i{\omega }^{{_\ast}}I - {A}_{{\mu }^{{_\ast}}}]({\mathbb{C}}^{n})\). Since \(\parallel \eta {\parallel }^{2} =\langle a\xi + bz\prime,\eta \rangle = a\langle \xi,\eta \rangle + b\langle z\prime,\eta \rangle = a\), it follows that \(\eta =\parallel \eta {\parallel }^{2}\xi + bz\prime\). Hence we have
$$\kappa = \alpha \eta + \beta z = \alpha \parallel \eta {\parallel }^{2}\xi + (\alpha bz\prime + \beta z).$$Furthermore \(\Pi (\kappa )\,=\,\langle \eta,\xi \rangle \xi \,=\,\langle \eta,\alpha \,\parallel \,\eta \,{\parallel }^{2}\,\xi \,+\,(\alpha bz\prime P\,+\,\beta z)\rangle \xi \,=\,\alpha \,\parallel \,\eta \,{\parallel }^{2}\xi.(\langle \eta,\alpha bz\prime\,+\,\beta z\rangle = 0\) because \(\alpha bz\prime + \beta z \in [i{\omega }^{{_\ast}}I - {A}_{{\mu }^{{_\ast}}}]({\mathbb{C}}^{n}).\) ) Thus we obtain
$$\kappa = \Pi (\kappa ) + (\alpha bz\prime + \beta z).$$This is the direct sum of \({\mathbb{C}}^{n}\) corresponding to span{ξ} and \([i{\omega }^{{_\ast}}I - {A}_{{\mu }^{{_\ast}}}]({\mathbb{C}}^{n})\).
- 13.
Express each of the Fourier coefficients of y by the direct sum corresponding to span{ξ} and \([i{\omega }^{{_\ast}}I - {A}_{{\mu }^{{_\ast}}}]({\mathbb{C}}^{n})\). And delete all the terms which do not contribute to the former.
- 14.
See note 9 on page 54.
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Maruyama, T. (2011). On the Fourier analysis approach to the Hopf bifurcation theorem. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 15. Springer, Tokyo. https://doi.org/10.1007/978-4-431-53930-8_3
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