A note on the Hopf stability formula under nonresonance conditions

Wolf–Jürgen Beyn^{∗}

email: beyn@mathematik.uni-bielefeld.de

Thorsten Göke^{∗}

email: goeke@mathematik.uni-bielefeld.de

Department of Mathematics, University of Bielefeld Postfach 100131, D–33501 Bielefeld, Germany

Abstract

The Hopf stability formula determines the super- or subcriticality of periodic branches at a Hopf bifurcation point as well as the stability of the emanating periodic orbits. The standard approach to this formula uses the center manifold reduction and hence requires that there are no further eigenvalues on the imaginary axis except for the complex conjugate pair. In this paper we give a complete derivation of this formula under the weaker nonresonance condition which only excludes integer multiples of the purely imaginary pair. We show that the shooting type approach in Amann (1990) [1] can be corrected in such a way that a full algorithmic treatment of the stability formula becomes possible.

Keywords:

Hopf bifurcation, stability formula, nonresonance condition, numerical algorithm

AMS subject classification: Primary 58F14. Secondary 34C23.

∗supported by DFG Schwerpunktprogramm “Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme”.

1 Introduction

In this paper we consider a parametrized autonomous system

ẋ = f(λ, x)  (1.1)

under the classical assumptions of the Hopf bifurcation theorem

A1 f ∈ C^{k}(Λ × X, ℝ^{n}) with k ≥ 2 is defined on open subsets X ⊂ ℝ^{n} and Λ ⊂ ℝ.

A2 There exists x_{0} ∈ X with f(·, x_{0}) = 0.

A3 There exist λ_{0} ∈ Λ and T_{0} ∈ ℝ_{+} \\{0\} such that iω_{0} := i\frac{2π}{T_{0}} is a simple eigenvalue of B := D_{2}f(λ_{0}, x_{0}) and iω_{0}ℤ ∩ σ(B) = {±iω_{0}}.

A4 β := (\frac{d}{dλ}Reμ)(λ_{0}) ≠ 0 holds for the smooth continuation μ(λ) ∈ σ(D_{2}f(λ, x_{0})) of the eigenvalue μ(λ_{0}) = iω_{0} of B, which is defined on a neighborhood V ⊂ Λ of λ_{0}.

It is well known that under these conditions a C^{k−1}-branch (γ(s), λ(s)) of T (s)-periodic orbits bifurcates at s = 0 from (x_{0}, λ_{0}, T_{0}) and that this branch is unique in a certain restricted sense ([1], [2]). Moreover, the behavior of the nontrivial Floquet multiplier on the branch is determined by the signs of λ̇(s) and β from A4 ([1] Theorem 27.11, [2]).

In order to really determine super- or subcriticality (i.e. λ̇(s) > 0 or λ̇(s) < 0) one needs to calculate λ̈(0) in terms of the derivatives of the function f.

This is usually achieved by center manifold reduction ([5], [6], [7]). The normalized coefficient in front of the cubic term of the two-dimensional normal form is then called the first Liapunov coefficient (cf. [6] Chapter 3). However, this reduction requires that there are no other eigenvalues except ±iω_{0} on the imaginary axis, which is stronger than the nonresonance condition in A3.

In contrast to this [1] uses a normalized shooting formulation to compute λ̈(0) under the nonresonance condition. However, we found that the final formulae in [1] (27.14, 27.15) still involve integrals and are not correct in dimension ≥ 3.

Therefore we give in this paper a derivation of a complete algorithm for λ̈(0) which works under the nonresonance condition. It is not surprising that the final formula agrees with the formula obtained via the center manifold reduction [5], [6].

In section 2 we extract the necessary preparations for the Hopf bifurcation theorem from [1] and we state the main result. The calculations leading to the algorithm are described in section 3. Finally, in section 4 we apply the algorithm to an example with two pairs of purely imaginary eigenvalues.

2 The Main Result

For later reference we quote here the Hopf bifurcation theorem from Amann (1990) [1]. Theorem 1 : Under the assumptions A1 to A4 there exist some ϵ > 0 and a function (x(·), T (·), λ(·)) ∈ C^{k−1}((−ϵ, ϵ), X × ℝ_{+} \ {0} × Λ) with the following properties: H1 (x(0), T (0), λ(0)) = (x_{0}, T_{0}, λ_{0}). H2 For every s ∈ ( 0, ϵ) the solution y(·, s) of ẏ = f(λ(s), y), y(0) = x(s) defines a T (s)-

periodic orbit γ(s) := {y(t, s) | t ∈ ℝ}. H3 γ(s) ≠ γ(t) for all 0 < s < t < ϵ. H4 In some neighborhood of (x_{0}, T_{0}, λ_{0}) every periodic orbit belongs to the family

{γ(s) | 0 < s < ϵ}. In the following we describe the algorithm for computing λ̈(0) for k ≥ 3 (in A1). Due to A3, we can choose eigenvectors e_{1} + ie_{2} of B = D_{2}f(λ_{0}, x_{0}) and e′_{1} + ie′_{2} of B^{T} corresponding to the eigenvalue iω_{0} and satisfying

L1 ⟨e′_{1}, e_{1}⟩ = −⟨e′_{2}, e_{2}⟩ = 1 and ⟨e′_{1}, e_{2}⟩ = ⟨e′_{2}, e_{1}⟩ = 0, L2 N ⊕ R = ℝ^{n}, N = Span_{IR}{e_{1}, e_{2}}, R = Span_{IR}{e′_{1}, e′_{2}}^{⊥}.

By using these vectors we let v^{1} := ⟨e′_{1}, v⟩ and v^{2} := −⟨e′_{2}, v⟩ be the coordinates of any vector v ∈ ℝ^{n} with respect to e_{1}, e_{2}. Consider

\begin{aligned}φ : & ℝ^{2} × R × (Λ − λ_{0}) & → & ℝ^{n} \\ & (x, y, z, λ) & ↦ & f(λ_{0} + λ, xe_{1} + ye_{2} + z + x_{0}) \end{aligned}

and for i, j, l ∈ {1, 2} define

\begin{aligned}φ_{i} & := & D_{i}φ(0) & = & D_{2}f(λ_{0}, x_{0})[e_{i}] ∈ ℝ^{n}, \\ φ_{ij} & := & D_{i}D_{j}φ(0) & = & D_{2}^{2}f(λ_{0}, x_{0})[e_{i}, e_{j}] ∈ ℝ^{n}, \\ φ_{i3} & := & D_{i}D_{3}φ(0) & = & D_{2}^{2}f(λ_{0}, x_{0})[e_{i}, id_{R}] ∈ L(R, ℝ^{n}), \\ φ_{i4} & := & D_{i}D_{4}φ(0) & = & D_{1}D_{2}f(λ_{0}, x_{0})[e_{i}] ∈ ℝ^{n}, \\ φ_{ijl} & := & D_{i}D_{j}D_{l}φ(0) & = & D_{2}^{3}f(λ_{0}, x_{0})[e_{i}, e_{j}, e_{l}] ∈ ℝ^{n}. \end{aligned}

Then the main result is:

Theorem 2 : Under the assumptions of Theorem 1 we have

H5 β := (\frac{d}{dλ}Reμ)(λ_{0}) = \frac{1}{2}(φ_{14}^{1} + φ_{24}^{2})

and the following linear system has a unique solution ζ_{11}, ζ_{12}, ζ_{22} ∈ ℝ^{n}:

( \begin{matrix}B & 0 & 2ω_{0}I \\ 0 & B & −2ω_{0}I \\ −ω_{0}I & ω_{0}I & B \end{matrix} ) ( \begin{matrix}ζ_{11} \\ ζ_{22} \\ ζ_{12} \end{matrix} ) = − ( \begin{matrix}φ_{11} − φ_{11}^{1}e_{1} − φ_{11}^{2}e_{2} \\ φ_{22} − φ_{22}^{1}e_{1} − φ_{22}^{2}e_{2} \\ φ_{12} − φ_{12}^{1}e_{1} − φ_{12}^{2}e_{2} \end{matrix} ) .  (2.1)

Let A1 hold with k ≥ 3 and define the coefficient ψ by

ψ := \frac{1}{16}((φ_{111}^{1} + φ_{122}^{1} + φ_{112}^{2} + φ_{222}^{2})

+(φ_{13}^{1}(3ζ_{11} + ζ_{22}) + 2φ_{23}^{1}ζ_{12} + 2φ_{13}^{2}ζ_{12} + φ_{23}^{2}(ζ_{11} + 3ζ_{22}))

+\frac{1}{ω_{0}} ( − φ_{11}^{1}φ_{12}^{1} + φ_{11}^{1}φ_{11}^{2} − φ_{12}^{1}φ_{22}^{1} − φ_{22}^{1}φ_{22}^{2} + φ_{11}^{2}φ_{12}^{2} + φ_{12}^{2}φ_{22}^{2})).

Then the following properties are satisfied for the periodic branch from Theorem 1:

H6 λ̈(0) = −\frac{2ψ}{β}, i.e. the bifurcation is supercritical for \frac{ψ}{β} < 0 and subcritical for \frac{ψ}{β} > 0.

H7 If ψ < 0 and Re(σ(D_{2}f(λ_{0}, x_{0})) \ {±iω_{0}}) < 0 hold, then the periodic orbits are asymptotically stable and they are unstable if ψ > 0.

Remarks: The value of ψ depends on the eigenvectors chosen, but its sign is independent of these choices.

The result can be used to approximate initial values of periodic orbits near the Hopf bifurcation by (λ_{0} − \frac{ψ}{β} s^{2}, x_{0} + se_{1} + ζ_{11}s^{2}) = (λ(s), x(s)) + O(|s|^{3}).

In Hassard, Kazarinov and Wan (1981) [5] the center manifold is used to construct a function b^{p}(ϵ) = ∑_{i=1}^{[k/2]} b_{2i}^{p}ϵ^{2i} +o(ϵ^{k}) such that the first nonvanishing b_{2i}^{p} determines the stability of the emanating orbits. By equation (2.9a) in [5] we get b_{2}^{p} = −βλ̈(0) = 2ψ. Therefore we have the same stability condition as in [5].

In Kuznetsov (1995) [6] the system is normalized by dividing by ω_{0} and it can be shown, that the formula for ψ divided by ω_{0} is the same as formula (5.59) in [6].

The nonresonance conditionA3 is used to prove the Hopf bifurcation theorem. For the determination of ψ we only use that 0 and ±2iω_{0} are not in the spectrum of B.

Theorem 2 is a corrected version of [1] 27.14, 27.15. It will be proved in section 3 and the major deviations from [1] will be indicated in a final remark.

The present paper is a condensed version of parts of Göke (1995) [4] where also the complete details are given.

The proof of theorem 1 is given in [1] and we repeat some ingredients of the proof, which are needed for the calculation of β and ψ in theorem 2.

First transform (λ_{0}, x_{0}) into the origin and normalize the period by setting

f̃(σ, x) := \frac{τ + T_{0}}{2π} f(λ + λ_{0}, x + x_{0}) with σ := (λ, τ).  (2.2)

x + x_{0} is an initial value of a (τ + T_{0})-periodic solution of ẏ = f(λ + λ_{0}, y), if and only if x satisfies

g(σ, x) := x − u(2π, x, σ) = 0,  (2.3)

where u(·, x, σ) is the solution of ẏ = f̃(σ, y), y(0) = x.

The derivative of g at (0, 0) is given by

D_{2}g(0, 0) = id − D_{2}u(2π, 0, 0) = id − e^{2πL}  (2.4)

with L := D_{2}f̃(0, 0) = \frac{T_{0}}{2π} D_{2}f(λ_{0}, x_{0}). This follows by integrating the variational equation

D_{2}u̇(t, x, σ) = D_{2}f̃(σ, u(t, x, σ))D_{2}u(t, x, σ) and D_{2}u(0, x, σ) = id  (2.5)

at (σ, x) = (0, 0).

The following relations are easy to obtain from L1, L2 and the nonresonance condition A3 with the help of the spectral mapping theorem:

L3 Le_{1} = −e_{2}, Le_{2} = e_{1}, L^{T} e′_{1} = −e′_{2} and L^{T} e′_{2} = e′_{1}.

L4 N = Span_{IR}{e_{1}, e_{2}} = ker(D_{2}g(0, 0)).

L5 R = Span_{IR}{e′_{1}, e′_{2}}^{⊥} = im(D_{2}g(0, 0)).

L6 N and R are invariant under L, e^{tL} and D_{2}g(0, 0).

L7 The restriction D_{2}g(0, 0)|_{R} ∈ L(R) is an isomorphism.

L8 Qv := v^{1}e_{1} + v^{2}e_{2} = ⟨e′_{1}, v⟩e_{1} − ⟨e′_{2}, v⟩e_{2} and P := I − Q are projectors on N, R.

A Liapunov–Schmidt reduction shows, by using implicit function theorem, that there exists a function η ∈ C^{k}(Σ × U, V), with suitable neighborhoods Σ ⊂ Λ × ℝ, U ⊂ N and V ⊂ R of 0, which satisfies

P g(σ, u + η(σ, u)) = 0 ∀ (σ, u) ∈ Σ × U.  (2.6)

If we define h ∈ C^{k}(Σ × U, N) by h(σ, u) := Qg(σ, u + η(σ, u)), we obtain

g(σ, u + η(σ, u)) = 0 ⟺ h(σ, u) = 0.  (2.7)

Because of g(·, 0) = 0 we get

η(·, 0) = 0 and D_{1}η(0, 0) = 0.  (2.8)

By the equivalence (2.7) we have reduced (2.3) to solving h(σ, u) = 0 for (σ, u). This is done in [1] in Chapter 26. In Theorem 26.25 it is shown, that

β = (\frac{d}{dλ}Reμ) (λ_{0}) = \frac{1}{2π} ∫_{0}^{2π}⟨e′_{1}, e^{−τL}D_{1}D_{2}f(λ_{0}, x_{0})[e^{τL}e_{1}]⟩ dτ  (2.9)

holds and hence A4 implies that this integral does not vanish.

This condition is used in [1] Theorem 26.19 to prove that there exist a δ > 0 and a continuous function

σ̂ : K_{N}(δ) → Λ × ℝ_{+} , K_{N}(δ) := {x ∈ N | ∥x∥ < δ}

such that h(σ̂(α), α) = 0 ∀ α ∈ K_{N}(δ). The function σ̂ is C^{k−1} in K_{N}(δ) \ {0} and so is the line function σ(s) = σ̂(se_{1}), s ∈ (−δ, δ).

The solutions of (2.3) are then given by the trivial branch Σ × {0} and by the manifold

M = {(σ̂(α), α + η(σ̂(α), α)) | α ∈ K_{N}(δ)}.

Since the range of η is in R we obtain

P x = η(σ, Qx) ∀ (σ, x) ∈ M.  (2.10)

With σ(s) := σ̂(se_{1}), s ∈ (−ϵ, ϵ) for sufficiently small ϵ < δ we define the functions which then satisfy H1–H4 by (λ(s), T (s)) := (λ_{0}, T_{0}) + σ(s) and x(s) := x_{0} + se_{1} + η(σ(s), se_{1}).

ϵ can be chosen so small that the periodic orbits are in M, i.e.

(σ(s), u(t, se_{1} + η(σ(s), se_{1}), σ(s))) ⊂ M ∀ t ∈ ℝ, s ∈ (−ϵ, ϵ).  (2.11)

Furthermore one can show σ̇(0) = 0 and hence λ̇(0) = 0.

3 The Stability Coefficient

In this section we calculate λ̈(0) which has the opposite sign of ψ/β and which determines the sign of λ̇(s) = sλ̈(0) + o(s) if λ̈(0) ≠ 0 and if s is sufficiently small. From the proof of [1] Lemma 27.13 follows λ̈(0) = \frac{γ}{3T_{0}β} with

γ := ⟨e′_{1}, D_{2}^{3}g(0, 0)[e_{1}]^{3}⟩ + 3⟨e′_{1}, D_{2}^{2}g(0, 0)[ζ_{11}, e_{1}]⟩,  (3.1)
ζ_{11} := −(D_{2}g(0, 0)|_{R})^{−1}P D_{2}^{2}g(0, 0)[e_{1}]^{2}.  (3.2)

The main work is to determine β (i.e. H5) and to prove ψ = −\frac{ω_{0}}{12π} γ. H6 and H7 then follow from Theorem 27.11 and Lemma 27.13 [1].

For abbreviation we often use s := sin(t) and c := cos(t) and the formulae

\begin{aligned}∫_{0}^{2π} cos^{2}(t) dt = ∫_{0}^{2π} sin^{2}(t) dt & = & π, \\ ∫_{0}^{2π} c^{4}dt = ∫_{0}^{2π} s^{4}dt = 3 ∫_{0}^{2π} c^{2}s^{2}dt & = & \frac{3}{4}π, \\ ∫_{0}^{2π}c^{6}dt = ∫_{0}^{2π}s^{6}dt = 5 ∫_{0}^{2π}c^{4}s^{2}dt = 5 ∫_{0}^{2π}c^{2}s^{4}dt & = & \frac{5}{8}π, \end{aligned}  (3.3)

as well as the fact that the integrals of odd functions over the period interval vanish.

First we determine β = (\frac{d}{dλ}Reμ) (λ_{0}) by using (2.9), (3.3), ⟨e′_{1}, e^{−tL}v⟩ = v^{1}c − v^{2}s and D_{1}D_{2}f(λ_{0}, x_{0})[e^{tL}e_{1}] = cφ_{14} − sφ_{24}.

β = \frac{1}{2π} ∫_{0}^{2π}⟨e′_{1}, e^{−τL}D_{1}D_{2}f(λ_{0}, x_{0})[e^{τL}e_{1}]⟩ dτ
= \frac{1}{2π} ∫_{0}^{2π}(c^{2}φ_{14}^{1} − scφ_{24}^{1}) − (scφ_{14}^{2} − s^{2}φ_{24}^{2}) dτ
= \frac{1}{2π} π(φ_{14}^{1} + φ_{24}^{2}) = \frac{1}{2}(φ_{14}^{1} + φ_{24}^{2}).

We evaluate D_{2}^{2}g(0, 0) = −D_{2}^{2}u(2π, 0, 0), D_{2}^{3}g(0, 0) = −D_{2}^{3}u(2π, 0, 0) in order to calculate γ. Therefore we differentiate (2.5) with respect to x and get

\begin{aligned}D_{2}^{2}u̇(t, x, σ)[z_{1}, z_{2}] & = & D_{2}f̃(σ, u(t, x, σ))D_{2}^{2}u(t, x, σ)[z_{1}, z_{2}], \\ & & +D_{2}^{2}f̃(σ, u(t, x, σ))[D_{2}u(t, x, σ)z_{1}, D_{2}u(t, x, σ)z_{2}] \\ D_{2}^{2}u(0, x, σ)[z_{1}, z_{2}] & = & 0. \end{aligned}  (3.4)

At (σ, x) = (0, 0), by the variation of constants formula, this system has the solution

D_{2}^{2}u(t, 0, 0)[z_{1}, z_{2}] = ∫_{0}^{t} e^{(t−τ)L}D_{2}^{2}f̃(0, 0)[e^{τL}z_{1}, e^{τL}z_{2}]dτ.  (3.5)

Similarly by differentiating (3.4) with respect to x at 0 we get

D_{2}^{3}u(t, 0, 0)[z]^{3} = ∫_{0}^{t} e^{(t−τ)L}D_{2}^{3}f̃(0, 0)[e^{τL}z]^{3}dτ
+3 ∫_{0}^{t} e^{(t−τ)L}D_{2}^{2}f̃(0, 0)[D_{2}^{2}u(τ, 0, 0)[z]^{2}, e^{τL}z]dτ

as the solution of ẏ = Ly +D_{2}^{3}f̃(0, 0)[e^{tL}z]^{3} +3D_{2}^{2}f̃(0, 0)[D_{2}^{2}u(t, 0, 0)[z]^{2}, e^{tL}z], y(0) = 0. This implies

D_{2}^{3}g(0, 0)[e_{1}]^{3} = − ∫_{0}^{2π} e^{(2π−τ)L}D_{2}^{3}f̃(0, 0)[e^{τL}e_{1}]^{3}dτ  (3.6 a)
−3 ∫_{0}^{2π} e^{(2π−τ)L}D_{2}^{2}f̃(0, 0)[D_{2}^{2}u(τ, 0, 0)[e_{1}]^{2}, e^{τL}e_{1}]dτ. (3.6 b)

Next we prove ζ_{11} = D_{2}^{2}η (0, 0)[e_{1}]^{2}. Differentiate P g(σ, u + η(σ, u)) = 0 (cf. (2.6)) with respect to u and get

P D_{2}g(σ, u + η(σ, u))[id_{N} + D_{2}η(σ, u)] = 0.  (3.7)

Evaluate this at (σ, u) = (0, 0) and use L4, L5, η (0, 0) = 0, im(D_{2}η (0, 0)) ⊂ R to get

(D_{2}g (0, 0)|_{R})D_{2}η (0, 0) û = 0 ∀ û ∈ N.

D_{2}g (0, 0)|_{R} is an isomorphism due to L7 and therefore

D_{2} η(0, 0) = 0.  (3.8)

Differentiating (3.7) once more with respect to u at (0, 0) and using (2.8), (3.8) we get

P D_{2}^{2}g (0, 0)|_{N×N} + P D_{2}g (0, 0)D_{2}^{2}η (0, 0) = 0

and, because D_{2}g (0, 0)|_{R} is an isomorphism,

D_{2}^{2}η (0, 0)[e_{1}]^{2} = −(D_{2}g (0, 0)|_{R})^{−1}P D_{2}^{2}g (0, 0)[e_{1}]^{2} = ζ_{11}.

Moreover we define ζ_{12} := D_{2}^{2}η (0, 0)[e_{1}, e_{2}], ζ_{22} := D_{2}^{2}η (0, 0)[e_{2}]^{2} and we will show that (2.1) holds with these settings.

In order to use z_{1} = ζ_{11} in (3.5) we need to know e^{tL}ζ_{11}, which is difficult to calculate. For that reason we prove

\begin{aligned}e^{tL}ζ_{11} & = & D_{2}u(t, 0, 0)D_{2}^{2}η (0, 0)[e_{1}]^{2} \\ & = & D_{2}^{2}η (0, 0)[D_{2}u(t, 0, 0)[e_{1}]]^{2} − P D_{2}^{2}u(t, 0, 0)[e_{1}]^{2}. \end{aligned}  (3.9)

We can use (2.10), (2.11) to get

P u(t, se_{1} + η (σ(s), se_{1}), σ(s)) = η (σ(s), Qu(t, se_{1} + η (σ(s), se_{1}), σ(s)))

for all t ∈ ℝ and all s ∈ (−ϵ, ϵ) and differentiate with respect to s

P D_{2}u(t, se_{1} + η (σ(s), se_{1}), σ(s))[e_{1} + D_{1}η (σ(s), se_{1})σ̇(s)
+D_{2}η (σ(s), se_{1})e_{1}] + P D_{3}u(t, se_{1} + η (σ(s), se_{1}), σ(s))σ̇(s)
= D_{1}η(σ(s), Qu(t, se_{1} + η (σ(s), se_{1}), σ(s)))σ̇(s)
+D_{2}η(σ(s), Qu(t, se_{1} + η (σ(s), se_{1}), σ(s)))
[QD_{2}u(t, se_{1} + η (σ(s), se_{1}), σ(s))[e_{1} + D_{1}η (σ(s), se_{1})σ̇(s)
+D_{2}η (σ(s), se_{1})e_{1}] + QD_{3}u(t, se_{1} + η (σ(s), se_{1}), σ(s))σ̇(s)].

If we differentiate twice at s = 0 and use D_{1}η (0, 0) = 0, D_{2}η (0, 0) = 0, D_{3}u(t, 0, 0) = 0 (compare A2) and σ̇(0) = 0 we get the equation

P D_{2}^{2}u(t, 0, 0)[e_{1}]^{2} + P D_{2}u(t, 0, 0)D_{2}^{2}η(0, 0)[e_{1}]^{2} = D_{2}^{2}η(0, 0)[QD_{2}u(t, 0, 0)[e_{1}]]^{2}.

In this formula N and R are invariant under D_{2}u(t, 0, 0) = e^{tL}, D_{2}^{2}η (0, 0)[e_{1}]^{2} ∈ R and e_{1} ∈ N. Hence we can ignore the projectors P, Q in front of D_{2}u(t, 0, 0) and end up with (3.9).

Combining (3.5) and (3.9) yields

\begin{aligned}D_{2}^{2}g(0, 0)[ζ_{11}, e_{1}] & = & −D_{2}^{2}u(2π, 0, 0)[D_{2}^{2}η (0, 0)[e_{1}]^{2}, e_{1}] \\ & = & − ∫_{0}^{2π} e^{(2π−τ)L}D_{2}^{2}f̃(0, 0)[e^{τL}D_{2}^{2}η(0, 0)[e_{1}]^{2}, e^{τL}e_{1}]dτ \\ & = & − ∫_{0}^{2π} e^{(2π−τ)L}D_{2}^{2}f̃(0, 0)[D_{2}^{2}η(0, 0)[e^{τL}e_{1}]^{2}, e^{τL}e_{1}]dτ \\ & & + ∫_{0}^{2π} e^{(2π−τ)L}D_{2}^{2}f̃(0, 0)[P D_{2}^{2}u(τ, 0, 0)[e_{1}]^{2}, e^{τL}e_{1}]dτ \end{aligned}  {(3.10 a) \atop (3.10 b)}

With Q = id − P, (3.1), (3.6) and (3.10) we can express γ by three integrals.

γ = ⟨e′_{1}, D_{2}^{3}g(0, 0)[e_{1}]^{3} + 3D_{2}^{2}g(0, 0)[ζ_{11}, e_{1}]⟩
= − ∫_{0}^{2π}⟨e′_{1}, e^{(2π−τ)L}D_{2}^{3}f̃(0, 0)[e^{τL}e_{1}]^{3}⟩dτ  (3.11 a)
−3 ∫_{0}^{2π}⟨e′_{1}, e^{(2π−τ)L}D_{2}^{2}f̃(0, 0)[QD_{2}^{2}u(τ, 0, 0)[e_{1}]^{2}, e^{τL}e_{1}]⟩dτ  (3.11 b)
−3 ∫_{0}^{2π}⟨e′_{1}, e^{(2π−τ)L}D_{2}^{2}f̃(0, 0)[D_{2}^{2}η(0, 0)[e^{τL}e_{1}]^{2}, e^{τL}e_{1}]⟩dτ.  (3.11 c)

Now we determine the three integrals (3.11 a)–(3.11 c). The integral (3.11 a) is calculated as in [1] (p. 434). With e^{tL}e_{1} = ce_{1} − se_{2} we find

D_{2}^{3}f̃(0, 0)[e^{τL}e_{1}]^{3} = \frac{1}{ω_{0}} D_{2}^{3}f(λ_{0}, x_{0}) (c^{3}[e_{1}]^{3} − 3c^{2}s[e_{1}^{2}, e_{2}] + 3cs^{2}[e_{1}, e_{2}^{2}] − s^{3}[e_{2}]^{3})
= \frac{1}{ω_{0}} (c^{3}φ_{111} − 3c^{2}sφ_{112} + 3cs^{2}φ_{122} − s^{3}φ_{222}).

and by using ⟨e′_{1}, e^{(2π−t)L|N}v⟩ = (v^{1}c − v^{2}s) for v ∈ ℝ^{n} we get

⟨e′_{1}, e^{(2π−τ)L}D_{2}^{3}f̃(0, 0)[e^{τL}e_{1}]^{3}⟩ = \frac{1}{ω_{0}} (c^{4}φ_{111}^{1} − 3c^{3}sφ_{112}^{1} + 3c^{2}s^{2}φ_{122}^{1} − cs^{3}φ_{222}^{1}

−c^{3}sφ_{111}^{2} + 3c^{2}s^{2}φ_{112}^{2} − 3cs^{3}φ_{122}^{2} + s^{4}φ_{222}^{2}).

Integrating from 0 to 2π and using (3.3) implies for the integral (3.11 a)

∫_{0}^{2π}⟨e′_{1}, e^{(2π−τ)L}D_{2}^{3}f̃(0, 0)[e^{τL}e_{1}]^{3}⟩dτ = \frac{3π}{4ω_{0}} (φ_{111}^{1} + φ_{122}^{1} + φ_{112}^{2} + φ_{222}^{2}).  (3.12)

To calculate the integral (3.11 b) we must evaluate (cf. (3.5))

QD_{2}^{2}u(t, 0, 0)[e_{1}]^{2} = ∫_{0}^{t} e^{(t−τ)L}QD_{2}^{2}f̃(0, 0)[e^{τL}e_{1}]^{2}dτ.  (3.13)

From e^{−τL|_{N}} v = (v^{1}c − v^{2}s)e_{1} + (v^{1}s + v^{2}c)e_{2} for v ∈ N we get

∫_{0}^{t} e^{−τL}QD_{2}^{2}f̃(0, 0)[e^{τL}e_{1}]^{2}dτ

= \frac{1}{ω_{0}} ∫_{0}^{t} e^{−τL}QD_{2}^{2}f(λ_{0}, x_{0}) (c^{2}[e_{1}]^{2} − 2cs[e_{1}, e_{2}] + s^{2}[e_{2}]^{2})dτ
= \frac{1}{ω_{0}} ∫_{0}^{t} e^{−τL}Q(c^{2}φ_{11} − 2csφ_{12} + s^{2}φ_{22})dτ
= \frac{1}{ω_{0}} ∫_{0}^{t} c^{3}φ_{11}^{1} − c^{2}s(2φ_{12}^{1} + φ_{11}^{2}) + cs^{2}(φ_{22}^{1} + 2φ_{12}^{2}) − s^{3}φ_{22}^{2} dτ · e_{1} (3.14 a)
+ \frac{1}{ω_{0}} ∫_{0}^{t} c^{3}φ_{11}^{2} + c^{2}s(φ_{11}^{1} − 2φ_{12}^{2}) + cs^{2}(φ_{22}^{2} − 2φ_{12}^{1}) + s^{3}φ_{22}^{1} dτ · e_{2}. (3.14 b)

Integration from 0 to t and using

∫_{0}^{t} cos^{2}(τ) sin(τ)dτ = −\frac{1}{3}(cos^{3}(t) − 1),
∫_{0}^{t} cos^{3}(τ)dτ = \frac{2}{3} sin^{3}(t) + cos^{2}(t) sin(t),
∫_{0}^{t} cos(τ) sin^{2}(τ)dτ = \frac{1}{3} sin^{3}(t),
∫_{0}^{t} sin^{3}(τ)dτ = −\frac{2}{3}(cos^{3}(t) − 1) − cos(t) sin^{2}(t)

gives for the coordinate of e_{1} (3.14 a)

\frac{1}{3ω_{0}} ((c^{3} − 1)(2φ_{12}^{1} + φ_{11}^{2} + 2φ_{22}^{2}) + 3c^{2}sφ_{11}^{1} + 3cs^{2}φ_{22}^{2} + s^{3}(2φ_{11}^{1} + φ_{22}^{1} + 2φ_{12}^{2}))

and for the coordinate of e_{2} (3.14 b)

\frac{1}{3ω_{0}} ((c^{3} − 1)(2φ_{12}^{2} − φ_{11}^{1} − 2φ_{22}^{1}) + 3c^{2}sφ_{11}^{2} − 3cs^{2}φ_{22}^{1} + s^{3}(2φ_{11}^{2} + φ_{22}^{2} − 2φ_{12}^{1})).

By multiplying e^{tL} and using e^{tL}v = (v^{1}c+v^{2}s)e_{1}+(−v^{1}s+v^{2}c)e_{2} for v ∈ N and c^{3}s+cs^{3} = cs we get for (3.13) QD_{2}^{2}u(t, 0, 0)[e_{1}]^{2} = a(t)e_{1} + b(t)e_{2} where

a(t) := \frac{1}{3ω_{0}} ((c^{4} − c)(2φ_{12}^{1} + φ_{11}^{2} + 2φ_{22}^{2}) + 2cs(φ_{11}^{1} + φ_{12}^{2} − φ_{22}^{1})
+3c^{2}s^{2}(φ_{22}^{2} + φ_{11}^{2}) + s^{4}(2φ_{11}^{2} + φ_{22}^{2} − 2φ_{12}^{1}) + s(φ_{11}^{1} − 2φ_{12}^{2} + 2φ_{22}^{1}))

and

b(t) := \frac{1}{3ω_{0}} ((c^{4} − c)(2φ_{12}^{2} − φ_{11}^{1} − 2φ_{22}^{1}) + 2cs(φ_{11}^{2} − φ_{12}^{1} − φ_{22}^{2})
−3c^{2}s^{2}(φ_{22}^{1} + φ_{11}^{1}) − s^{4}(2φ_{11}^{1} + φ_{22}^{1} + 2φ_{12}^{2}) + s(φ_{11}^{2} + 2φ_{12}^{1} + 2φ_{22}^{2})).

Now we compute with [QD_{2}^{2}(u(t, 0, 0)[e_{1}]^{2}, e^{tL}e_{1}] = a(t)c[e_{1}]^{2}+(b(t)c−a(t)s)[e_{1}, e_{2}]−b(t)s[e_{2}]^{2} and ⟨e′_{1}, e^{(2π−t)L}v⟩ = v^{1}c − v^{2}s the integrand in (3.11 b).

⟨e′_{1}, e^{(2π−t)L}D_{2}^{2}f̃(0, 0)[QD_{2}^{2}u(t, 0, 0)[e_{1}]^{2}, e^{tL}e_{1}]⟩
= ⟨e′_{1}, e^{(2π−t)L} \frac{1}{ω_{0}} D_{2}^{2}f(λ_{0}, x_{0})(a(t)c[e_{1}]^{2} + (b(t)c − a(t)s)[e_{1}, e_{2}] − b(t)s[e_{2}]^{2})⟩
= ⟨e′_{1}, e^{(2π−t)L} \frac{1}{ω_{0}} (a(t)cφ_{11} + (b(t)c − a(t)s)φ_{12} − b(t)sφ_{22})⟩
= \frac{1}{ω_{0}} (a(t)(c^{2}φ_{11}^{1} − cs(φ_{12}^{1}+φ_{11}^{2}) + s^{2}φ_{12}^{2}) + b(t)(c^{2}φ_{12}^{1} − cs(φ_{22}^{1}+φ_{12}^{2}) + s^{2}φ_{22}^{2})).

Integration from 0 to 2π and using (3.3) results in

3ω_{0}^{2} ∫_{0}^{2π} ⟨e′_{1}, e^{(2π−τ)L}D_{2}^{2}f̃(0, 0)[QD_{2}^{2}u(τ, 0, 0)[e_{1}]^{2}, e^{τL}e_{1}]⟩ dτ
= \frac{5}{8}π (φ_{11}^{1}φ_{12}^{1} + φ_{11}^{1}φ_{11}^{2} + 2φ_{11}^{1}φ_{22}^{2} + 2φ_{12}^{1}φ_{12}^{2} − 2φ_{12}^{1}φ_{22}^{1})
+\frac{1}{8}π (2φ_{12}^{2}φ_{12}^{1} + φ_{12}^{2}φ_{11}^{2} + 4φ_{12}^{2}φ_{22}^{2} + 2φ_{11}^{1}φ_{22}^{2}
+3φ_{11}^{1}φ_{11}^{2} − 2φ_{22}^{2}φ_{22}^{1} − 3φ_{12}^{1}φ_{22}^{1} − 3φ_{12}^{1}φ_{11}^{1})
+\frac{1}{8}π (3φ_{12}^{2}φ_{22}^{2} + 3φ_{12}^{2}φ_{11}^{2} + 2φ_{11}^{1}φ_{11}^{2} − 2φ_{11}^{1}φ_{22}^{2} − 4φ_{11}^{1}φ_{12}^{1}
−3φ_{22}^{2}φ_{22}^{1} − φ_{12}^{1}φ_{22}^{1} − 2φ_{12}^{1}φ_{12}^{2})
+\frac{5}{8}π (2φ_{12}^{2}φ_{11}^{2} − φ_{12}^{2}φ_{22}^{2} − 2φ_{12}^{2}φ_{12}^{1} − 2φ_{22}^{2}φ_{11}^{1} − φ_{22}^{2}φ_{22}^{1})
+\frac{1}{4}π (−2φ_{11}^{1}φ_{12}^{1} + 4φ_{12}^{1}φ_{22}^{1} − 2φ_{11}^{1}φ_{11}^{2} − 4φ_{11}^{2}φ_{12}^{2} + 2φ_{12}^{2}φ_{22}^{2} + 2φ_{22}^{1}φ_{22}^{2})

and thus for the integral (3.11 b)

∫_{0}^{2π} ⟨e′_{1}, e^{2π−τL}D_{2}^{2}f̃(0, 0)[QD_{2}^{2}u(τ, 0, 0)[e_{1}]^{2}, e^{τL}e_{1}]⟩ dτ

= \frac{π}{4ω_{0}^{2}} (−φ_{11}^{1}φ_{12}^{1} + φ_{11}^{1}φ_{11}^{2} − φ_{12}^{1}φ_{22}^{1} − φ_{22}^{1}φ_{22}^{2} + φ_{11}^{2}φ_{12}^{2} + φ_{12}^{2}φ_{22}^{2}).  (3.15)

Finally, for the integral (3.11 c), we may assume for a moment that we know the ζ_{ij} and determine them later. The definitions of ζ_{ij} and e^{τL}e_{1} = ce_{1} − se_{2} implies

ζ(τ) := D_{2}^{2}η(0, 0)[e^{τL}e_{1}]^{2} = c^{2}ζ_{11} − 2csζ_{12} + s^{2}ζ_{22}  (3.16)

and thus

D_{2}^{2}f̃(0, 0)[D_{2}^{2}η(0, 0)[e^{τL}e_{1}]^{2}, e^{τL}e_{1}] = \frac{1}{ω_{0}} D_{2}^{2}f(λ_{0}, x_{0})[ζ(τ), ce_{1} − se_{2}]
= \frac{1}{ω_{0}} (cφ_{13}ζ(τ) − sφ_{23}ζ(τ)).

Since ⟨e′_{1}, e^{(2π−τ)L}v⟩ = v^{1}c − v^{2}s, (3.16) and (3.3) we obtain

∫_{0}^{2π} ⟨e′_{1}, e^{(2π−τ)L}D_{2}^{2}f̃(0, 0)[D_{2}^{2}η(0, 0)[e^{τL}e_{1}]^{2}, e^{τL}e_{1}]⟩ dτ

= \frac{1}{ω_{0}} ∫_{0}^{2π}c^{2}φ_{13}^{1}ζ(τ) − sc(φ_{23}^{1} + φ_{13}^{2})ζ(τ) + s^{2}φ_{23}^{2}ζ(τ) dτ
= \frac{1}{ω_{0}} ∫_{0}^{2π}c^{4}φ_{13}^{1}ζ_{11} + s^{2}c^{2}(φ_{13}^{1}ζ_{22} + 2(φ_{23}^{1} + φ_{13}^{2})ζ_{12} + φ_{23}^{2}ζ_{11}) + s^{4}φ_{23}^{2}ζ_{22} dτ
= \frac{π}{4ω_{0}} (φ_{13}^{1}(3ζ_{11} + ζ_{22}) + 2φ_{23}^{1}ζ_{12} + 2φ_{13}^{2}ζ_{12} + φ_{23}^{2}(ζ_{11} + 3ζ_{22})).  (3.17)

Now we can combine the integrals (3.12), (3.15), (3.17)

γ = − \frac{3π}{4ω_{0}} ((φ_{111}^{1} + φ_{122}^{1} + φ_{112}^{2} + φ_{222}^{2})

+(φ_{13}^{1}(3ζ_{11} + ζ_{22}) + 2φ_{23}^{1}ζ_{12} + 2φ_{13}^{2}ζ_{12} + φ_{23}^{2}(ζ_{11} + 3ζ_{22}))

+\frac{1}{ω_{0}} ( − φ_{11}^{1}φ_{12}^{1} + φ_{11}^{1}φ_{11}^{2} − φ_{12}^{1}φ_{22}^{1} − φ_{22}^{1}φ_{22}^{2} + φ_{11}^{2}φ_{12}^{2} + φ_{12}^{2}φ_{22}^{2}))

and get ψ = −\frac{ω_{0}}{12π} γ. The proof is now finished if we have verified that ζ_{11}, ζ_{12}, ζ_{22} solve the linear system (2.1). In order to see this we start with differentiating (3.9) with respect to t and use (2.5) and (3.4) to get

LD_{2}u(t, 0, 0)D_{2}^{2}η(0, 0)[e_{1}]^{2} = 2D_{2}^{2}η(0, 0)[LD_{2}u(t, 0, 0)[e_{1}], D_{2}u(t, 0, 0)[e_{1}]]
−P LD_{2}^{2}u(t, 0, 0)[e_{1}]^{2} − P D_{2}^{2}f̃(0, 0)[D_{2}u(t, 0, 0)[e_{1}]]^{2}.

N and R are invariant under L, hence P L = LP, and with (3.9) we get

LD_{2}^{2}η(0, 0)[e^{tL}e_{1}]^{2} = −P D_{2}^{2}f̃(0, 0)[e^{tL}e_{1}]^{2} + 2D_{2}^{2}η(0, 0)[Le^{tL}e_{1}, e^{tL}e_{1}].

Inserting the equation (3.16), [e^{tL}e_{1}]^{2} = c^{2}[e_{1}]^{2} − 2sc[e_{1}, e_{2}] + s^{2}[e_{2}]^{2} with the definition of φ_{ij}^{3} and [Le^{tL}e_{1}, e^{tL}e_{1}] = −sc[e_{1}]^{2} − (c^{2} − s^{2})[e_{1}, e_{2}] + sc[e_{2}]^{2} with the definition of ζ_{ij} leads to

c^{2}Lζ_{11} − 2scLζ_{12} + s^{2}Lζ_{22}
= −2scζ_{11} − 2(c^{2} − s^{2})ζ_{12} + 2scζ_{22} − \frac{1}{ω_{0}} c^{2}φ_{11}^{3} + 2\frac{1}{ω_{0}} scφ_{12}^{3} − \frac{1}{ω_{0}} s^{2}φ_{22}^{3}.

We evaluate the equation at times t = 0, t = \frac{1}{2}π and t = \frac{1}{4}π to get

Lζ_{11} + 2ζ_{12} = − \frac{1}{ω_{0}} φ_{11}^{3},  (3.18)
Lζ_{22} − 2ζ_{12} = − \frac{1}{ω_{0}} φ_{22}^{3},  (3.19)
L(ζ_{11} − 2ζ_{12} + ζ_{22}) + 2(ζ_{11} − ζ_{22}) = \frac{1}{ω_{0}} (−φ_{11}^{3} + 2φ_{12}^{3} − φ_{22}^{3}).  (3.20)

Combining these equations in the form \frac{1}{2}((3.18) + (3.19) − (3.20)) yields

Lζ_{12} + (ζ_{22} − ζ_{11}) = − \frac{1}{ω_{0}} φ_{12}^{3}.  (3.21)

With (3.18), (3.19), (3.21), I := id and B = ω_{0}L we get the linear system (2.1), i.e.

( \begin{matrix}B & 0 & 2ω_{0}I \\ 0 & B & −2ω_{0}I \\ −ω_{0}I & ω_{0}I & B \end{matrix} ) ( \begin{matrix}ζ_{11} \\ ζ_{22} \\ ζ_{12} \end{matrix} ) = ( \begin{matrix}−φ_{11}^{3} \\ −φ_{22}^{3} \\ −φ_{12}^{3} \end{matrix} ) .

Since 0 is no eigenvalue of B, this is equivalent to

( \begin{matrix}B & 0 & 2ω_{0}I \\ 0 & B & −2ω_{0}I \\ −ω_{0}B & ω_{0}B & B^{2} \end{matrix} ) ( \begin{matrix}ζ_{11} \\ ζ_{22} \\ ζ_{12} \end{matrix} ) = ( \begin{matrix}−φ_{11}^{3} \\ −φ_{22}^{3} \\ −Bφ_{12}^{3} \end{matrix} )

and we can factorize the matrix in

( \begin{matrix}I & 0 & 0 \\ 0 & I & 0 \\ −ω_{0}I & ω_{0}I & I \end{matrix} ) ( \begin{matrix}I & 0 & 0 \\ 0 & I & 0 \\ 0 & 0 & B^{2} + 4ω_{0}^{2}I \end{matrix} ) ( \begin{matrix}I & 0 & 2ω_{0}I \\ 0 & I & −2ω_{0}I \\ 0 & 0 & I \end{matrix} ) ( \begin{matrix}B & 0 & 0 \\ 0 & B & 0 \\ 0 & 0 & I \end{matrix} ) . (3.22)

B^{2} + 4ω_{0}^{2}I is nonsingular, because ±2iω_{0} is not in the spectrum of B. Hence we can solve the linear system (2.1). □

Remarks:

R1 We describe in detail the deviations from Amann (1990) [1].

The assumption (80) φ_{11}^{3} = 0 in 27.14 [1] does not imply P D_{2}^{2}g(0, 0)[e_{1}]^{2} = 0 (cf. §26 (60)) and hence ζ_{11} = 0 cannot be concluded. If φ_{11}^{3} = 0 is replaced by the stronger assumption

P D_{2}^{2}f̃(0, 0)|_{N×N} = 0 (⇔ φ_{ij}^{3} = 0, i, j ∈ {1, 2})  (3.23)

then the proof of 27.14 remains valid. Condition (3.23) is needed to justify the computation of D_{2}^{2}f̃(0, 0)[e^{tL}e_{1}]^{2} in [1] on page 434. If assumption (3.23) is dropped the calculations on the pages 434–436 can still be used to compute QD_{2}^{2}f̃(0, 0)[e^{tL}e_{1}]^{2} and (3.11 b). In our approach the remainder term of (3.6 b) has been shifted into a different integral (3.10 b) and can be eliminated. Since assumption (3.23) is used to calculate ⟨e′_{1}, D_{2}^{3}g(0, 0)[e_{1}]^{3}⟩ it is not sufficient to add 3⟨e′_{1}, D_{2}^{2}g(0, 0)[ζ_{11}, e_{1}]⟩ in Remark 27.15 [1].

R2 As the factorization (3.22) shows the linear system (2.1) can be solved in a convenient way by solving two linear systems with B and one with B^{2} + 4ω_{0}^{2}I.

4 Example

In this section we discuss an example with two purely imaginary eigenvalues. The terms of second and third order are constructed in such a way that all terms of the formula for ψ are used for both eigenvalues.

The example is four dimensional with z = (u, v, w, x)^{T} ∈ ℝ^{4} and has a 2–dimensional parameter space λ = (ν, μ) ∈ ℝ^{2}. The ordinary differential equation is

ż = A(λ)z + b(z) =: f(λ, z)  (4.1)

with

A(λ) = ( \begin{matrix}ν − μ & 1 & 0 & 0 \\ −1 & ν − μ & 0 & 0 \\ 0 & 0 & ν + μ & π \\ 0 & 0 & −π & ν + μ \end{matrix} ) , b(z) = ( \begin{matrix}u^{2} + uv − u^{3} + w^{2} \\ uv − v^{2} + uw \\ w^{2} − wx + u^{2} + wv \\ −wx − x^{2} + x^{3} \end{matrix} ) .

The 2-dimensional parameter space is used because a Hopf–Hopf bifurcation is a codimension 2 phenomenon. We notice that Theorem 2 is still applicable in this situation, but for a complete description of the dynamics one would need the Poincaré normal form ([6] §8.6).

For the Hopf bifurcation theorem we set μ = μ_{0} constant and use Λ̄ = ℝ, X̄ = ℝ^{4} and f̄ = f(·, μ_{0}, ·). A1 and A2 hold with z_{0} = 0 and the eigenvalues are (ν − μ_{0}) ± i and (ν + μ_{0}) ± iπ. Therefore we have purely imaginary eigenvalues ±i at ν = μ_{0} and ±iπ at ν = −μ_{0} and the derivatives of the real parts of the eigenvalues with respect to ν are always β = 1. For that reason A3 and A4 hold and we have Hopf bifurcations at ν = μ_{0} and ν = −μ_{0}. In the special case of μ_{0} = 0 this is a Hopf–Hopf bifurcation for which A3 still holds.

Now we know, that there exist periodic orbits with periods 2 and 2π in a neighborhood of (λ, x) = (0, 0) and we calculate ψ for both eigenvalues at μ_{0} = 0.

First we calculate ψ for the eigenvalues ±i. In this case we can use e_{1} := (1, 0, 0, 0)^{T} and e_{2} := (0, 1, 0, 0)^{T} to define the right eigenvector e_{1} + ie_{2} and the left eigenvector e_{1}^{T} − ie_{2}^{T} for the eigenvalue i of A(0). L1 holds with these vectors and we calculate

1. φ_{111} = (−6, 0, 0, 0)^{T}, φ_{122} = 0 φ_{112} = 0, φ_{222} = 0 and

φ_{111}^{1} + φ_{122}^{1} + φ_{112}^{2} + φ_{222}^{2} = −6.

2. φ_{11} = (2, 0, 2, 0)^{T}, φ_{12} = (1, 1, 0, 0)^{T}, φ_{22} = (0, −2, 0, 0)^{T} and

−φ_{11}^{1}φ_{12}^{1} + φ_{11}^{1}φ_{11}^{2} − φ_{12}^{1}φ_{22}^{1} − φ_{22}^{1}φ_{22}^{2} + φ_{11}^{2}φ_{12}^{2} + φ_{12}^{2}φ_{22}^{2} = −4.

3. φ_{13}(0, 0, 1, 0)^{T} = (0, 1, 0, 0)^{T}, φ_{23}(0, 0, 1, 0)^{T} = (0, 0, 1, 0)^{T}, φ_{i3}(0, 0, 0, 1)^{T} = 0, and ζ_{11} = \frac{−4+2π^{2}}{π(4−π^{2})}(0, 0, 0, 1)^{T}, ζ_{22} = \frac{−4}{π(4−π^{2})}(0, 0, 0, 1)^{T}, ζ_{12} = \frac{−2}{4−π^{2}} (0, 0, 1, 0)^{T} is the solution of the linear system (2.1). So we get

φ_{13}^{1}(3ζ_{11} + ζ_{22}) + 2φ_{23}^{1}ζ_{12} + 2φ_{13}^{2}ζ_{12} + φ_{23}^{2}(ζ_{11} + 3ζ_{22}) = \frac{4}{π^{2} − 4}.

Alltogether we have

ψ := \frac{1}{16}(−6 + \frac{4}{π^{2} − 4} − 4) = −0.5824.

We calculate ψ for the eigenvalues ±iπ and use e_{3} := (0, 0, 1, 0)^{T} and e_{4} := (0, 0, 0, 1)^{T} to define the right eigenvector e_{3} + ie_{4} and the left eigenvector e_{3}^{T} − ie_{4}^{T} for the eigenvalue iπ of A(0). L1 is satisfied for these vectors.

The result of the calculation is 1. φ_{111}^{1} + φ_{122}^{1} + φ_{112}^{2} + φ_{222}^{2} = 6.

2. −φ_{11}^{1}φ_{12}^{1} + φ_{11}^{1}φ_{11}^{2} − φ_{12}^{1}φ_{22}^{1} − φ_{22}^{1}φ_{22}^{2} + φ_{11}^{2}φ_{12}^{2} + φ_{12}^{2}φ_{22}^{2} = 4.

3. ζ_{11} = −\frac{4π^{2}−2}{4π^{2}−1}(0, 1, 0, 0)^{T}, ζ_{22} = −\frac{4π^{2}}{4π^{2}−1}(0, 1, 0, 0)^{T}, ζ_{12} = −\frac{2π}{4π^{2}−1}(1, 0, 0, 0)^{T} is the solution of the linear system (2.1) and

φ_{13}^{1}(3ζ_{11} + ζ_{22}) + 2φ_{23}^{1}ζ_{12} + 2φ_{13}^{2}ζ_{12} + φ_{23}^{2}(ζ_{11} + 3ζ_{22}) = −\frac{16π^{2} − 6}{4π^{2} − 1}.

Alltogether we have

ψ := \frac{1}{16}(6 − \frac{16π^{2} − 6}{4π^{2} − 1} + \frac{4}{π}) = 0.2078.

At ν = 0 we have a Hopf–Hopf bifurcation and two pairs of eigenvalues cross the imaginary axis with positive speed from the left to the right half of the complex plane. The stability coefficient is positive for one and negative for the other pair of eigenvalues. Therefore we have a supercritical and a subcritical bifurcation at the same time.

References

[1] H. Amann, Ordinary Differential Equations (De Gruyter, Berlin, 1990).

[2] M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. MatMech. Anal. 67 (1977) 53–72.

[3] E. J. Doedel and J. P. Kernévez, AUTO: Software for continuation and bifurcation problems in ordinary differential equations, Appl. Math. Technical Report, Caltech (1986).

[4] T. Göke, Konjunkturzyklen in einem makroökonomischen Modell, Numerik und Analysis der Hopf–Bifurkation, Diploma Thesis, University of Bielefeld (1995).

[5] B. Hassard, N. D. Kazarinov and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, (London Mathematical Society lecture note series 41, Cambridge University Press, 1981).

[6] Y. Kuznetsov, Elements of Applied Bifurcation Theory, (Applied Mathematical Sciences 112, Springer, New York, 1995).

[7] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, (Applied mathematical sciences 19, Springer, New York, 1976).

[8] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, (Springer, New York, 1990).