Simple-Zero and Double-Zero

Volume 2009, Article ID 923809,29pages doi:10.1155/2009/923809

Research Article

Simple-Zero and Double-Zero

Singularities of a Kaldor-Kalecki Model of Business Cycles with Delay

Xiaoqin P. Wu

Department of Mathematics, Computer & Information Sciences, Mississippi Valley State University, Itta Bena, MS 38941, USA

Correspondence should be addressed to Xiaoqin P. Wu,xpaul wu@yahoo.com Received 12 August 2009; Accepted 2 November 2009

Recommended by Xue-Zhong He

We study the Kaldor-Kalecki model of business cycles with delay in both the gross product and the capital stock. Simple-zero and double-zero singularities are investigated when bifurcation parameters change near certain critical values. By performing center manifold reduction, the normal forms on the center manifold are derived to obtain the bifurcation diagrams of the model such as Hopf, homoclinic and double limit cycle bifurcations. Some examples are given to confirm the theoretical results.

Copyrightq2009 Xiaoqin P. Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In the last decade, the study of delayed differential equations that arose in business cycles has received much attention. The first model of business cycles can be traced back to Kaldor 1 who used a system of ordinary differential equations to study business cycles in 1940 by proposing nonlinear investment and saving functions so that the system may have cyclic behaviors or limit cycles, which are important from the point of view of economics. Kalecki 2introduced the idea that there is a time delay for investment before a business decision.

Krawiec and Szydłowski3–5incorporated the idea of Kalecki into the model of Kaldor by proposing the following Kaldor-Kalecki model of business cycles:

dYt

dt αIYt, Kt−SYt, Kt, dKt

dt IYt−τ, Kt−qKt,

1.1

whereY is the gross product, K is the capital stock,α > 0 is the adjustment coefficient in the goods market, q ∈ 0,1 is the depreciation rate of capital stock, IY, Kand SY, K

are investment and saving functions, and τ ≥ 0 is a time lag representing delay for the investment due to the past investment decision. This model has been studied extensively by many authors; see 6–11. Several authors also discussed similar models 12–14 and established the existence of limit cycles.

Considering that past investment decisions6also influence the change in the capital stock, Kaddar and Talibi Alaoui15extended the model1.1by imposing delays in both the gross product and capital stock. Thus adding the same delay to the capital stock K in the investment function IY, K of the second equation of Sys.1.1leads to the following Kaldor-Kalecki model of business cycles:

dYt

dt αIYt, Kt−SYt, Kt, dKt

dt IYt−τ, Kt−τ−qKt.

1.2

As in3; also see10,16,17, using the following saving and investment functionsS andI, respectively,

SY, K γY, IY, K IY−βK, 1.3

whereβ >0 andγ∈0,1are constants, we obtain the following system:

dYt dt α

IYt−βKt−γYt , dKt

dt IYt−τ−βKt−τ−qKt.

1.4

Kaddar and Talibi Alaoui15studied the characteristic equation of the linear part of Sys.

1.4at an equilibrium point and used the delayτas a bifurcation parameter to show that the Hopf bifurcation may occur under some conditions asτpasses some critical values. However, they did not obtain the stability of the bifurcating limit cycles and the direction of the Hopf bifurcation. Wang and Wu18further studied Sys.1.4and gave a more detailed discussion of the distribution of the eigenvalues of the characteristic equation which has a pair of purely imaginary roots. They derived the normal forms on the center manifold for sys.1.4to give the direction of the Hopf bifurcation and the stability of the bifurcating limit cycles for some critical values ofτ.

However, under certain conditions, the characteristic equation of the linear part of Sys.1.4may have a simple-zero root, a double-zero root, or a simple zero root and a pair of purely imaginary roots. In this paper, simple-zerofoldand double-zeroBogdanov-Takens singularities for Sys.1.4and their corresponding dynamical behaviors are investigated by usingkandτ as bifurcation parameterswherekis defined inSection 2. The discussion of zero-Hopf singularity will be addressed in a coming paper.

The rest of this manuscript is organized as follows. InSection 2, a detailed presentation is given for the distribution of eigenvalues of the linear part of Sys.1.4at an equilibrium point in thek, τ-parameter space. InSection 3, the theory of center manifold reduction for general delayed differential equationsDDEsis briefly introduced. In Sections4and5, center

manifold reduction is performed for Sys.1.4; and hence, the normal forms for simple-zero and double-zero singularities are obtained on the center manifold, respectively. InSection 6, the normal forms for the double-zero singularity are used to predict the bifurcation diagrams such as Hopf, homoclinic, and double limit cycle bifurcations for the original Sys. of1.4.

Finally in Section 7, some numerical simulations are presented to confirm the theoretical results.

2. Distribution of Eigenvalues

Throughout the rest of this paper, we assume that

α, β >0, q, γ∈0,1, andIsis a nonlinearC⁴ function, 2.1 and thatY^∗, K^∗is an equilibrium point of Sys.1.4. LetI^∗IY^∗,u1Y−Y^∗,u2K−K^∗, andis Is Y^∗−I^∗. Then Sys.1.4can be transformed as

du₁t dt α

iu1t−βu₂t−γu₁t , du₂t

dt iu1t−τ−βu₂t−τ−qu₂t.

2.2

Let the Taylor expansion ofiat 0 be

iu ku i²u² i³u³ O

|u|⁴

, 2.3

where

ki0 IY^∗, i² 1

2i0 1

2IY^∗, i³ 1

3!i0 1

3!IY^∗. 2.4 The linear part of Sys.2.2at0,0is

du1t

dt α

k−γ

u1t−βu2t , du2t

dt ku1t−τ−βu2t−τ−qu2t,

2.5

and the corresponding characteristic equation is Δλ≡λ² Aλ B

βλ C

e^−λτ0, 2.6

where

Aq−α k−γ

, B−αq

k−γ

, Cαβγ. 2.7

Forτ 0,2.6becomes

λ² A β

λ B C0. 2.8

Define

k^∗ βγ

q γ, k^∗∗ q β

α γ. 2.9

Theorem 2.1. Letτ 0. Ifk <min{k^∗, k^∗∗}, then all roots of 2.8have negative real parts, and hence Y^∗, K^∗ is asymptotically stable. Ifk > min{k^∗, k^∗∗}, then2.8has a positive root and a negative root, and hence,Y^∗, K^∗is unstable.

Now assumeτ >0. ClearlyΔ0 0 if and only ifkk^∗. Next we always assume that kk^∗. It is easy to attain

Δλ 2λ q−αβγ

q βe^−λτ−

βλ C τe^−λτ, Δλ 2−2βτe^−λτ βτ²λe^−λτ Cτ²e^−λτ.

2.10

Defineτ^∗ q² qβ−αβγ/αβγq. Then we have that,

Δ0 αβγ

q τ^∗−τ, Δ0

ττ^∗ q⁴−β²q² α²β²γ²

αβγq² . 2.11

Define

fx x² βx−αβγ, gx x²−β²x α²β²γ². 2.12

Hence iffq ≤0,τ^∗ ≤0, and henceΔ0< 0, and iffq> 0,τ^∗ > 0, and henceΔ0 0 if and only ifτ τ^∗. AlsoΔ0|ττ^∗/0 if and only ifgq²/0. Thus we obtain the following result.

Lemma 2.2. Suppose thatkk^∗. Then the following are considered.

iIfτ^∗≤0, then2.6has a simple root 0 for allτ >0.

iiLetτ^∗>0. Then the following are given.

aEquation2.6has a simple root 0 if and only ifτ /τ^∗,

bEquation2.6has a double root 0 if and only ifττ^∗andgq²/0.

Letωi ω > 0be a purely imaginary root of2.6. After plugging it into 2.6and separating the real and imaginary parts, we have that

ω² αβγ αβγcosωτ βωsinωτ, q²−αβγ

q ωαβγsinωτ−βωcosωτ. 2.13

Adding squares of two equations yields

ω² g q²

q² 0. 2.14

Then2.14 has a nonzero solution if and only if gq² < 0 and does not have a nonzero solution if and only ifgq²≥0. Ifgq²<0, from2.14, we solveωas follows:

ωω0 ≡ 1 q −g

q²

, 2.15

and from2.13, we solve cosω0τ, sinω0τas:

cosω0τ −q²ω²₀ αβγω²₀ qαγ

αβγ ω²₀ qβ

α²γ² ω²₀ ≡a, sinω0τ q²αγω0−α²βγ²ω0 qαβγω0 qω³₀

qβ

α²γ² ω²₀ ≡b.

2.16

Define

δ

⎧⎨

⎩

arccosa, if b≥0,

2π−arccosa, if b <0. 2.17

From2.16, we obtain

τ τj≡ 1 ω₀

δ 2jπ

, j 0,1,2, . . . . 2.18

Clearly ifβ >2αγ, thengx 0 has two positive roots, and ifβ≤2αγ, thengx≥0.

Now, underkk^∗, we impose the following conditions:

H1β≤2αγ,τ^∗≤0, H2β≤2αγ,τ^∗>0,τ /τ^∗, H3β≤2αγ,τ^∗>0,ττ^∗,

H4β >2αγ,τ^∗>0,τ /τ^∗, gq²≥0,

H5β >2αγ,τ^∗>0,τ /τ^∗, gq²<0, H6β >2αγ,τ^∗>0,ττ^∗, gq²≥0, H7β >2αγ,τ^∗>0,ττ^∗, gq²<0.

Based onLemma 2.2, we have the following result.

Lemma 2.3. Suppose thatkk^∗and 0< q <1. Then the following are obtained.

iUnder one of the conditions (H1), (H2), and (H4),2.6has a simple zero root and does not have other roots in the imaginary axis.

iiUnder the condition (H5),2.6has a simple zero root and a pair of purely imaginary roots

±ω0iin the imaginary axis ifτ τj,j 0,1,2, . . . .

iiiUnder one of the conditions (H3) and (H6), then2.6has a double root 0 and does not have other roots in the imaginary axis.

ivUnder the condition (H7),2.6has a double zero root and a pair of purely imaginary roots

±ω0iin the imaginary axis ifτ^∗τ_jfor somej.

Now we use the roots offx 0,gx 0 to give a more detailed discussion for the roots of2.6. Define

q₀ 1 2

−β β² 4αβγ

, q1 1

2

β²− β⁴−4α²β²γ²

, q₂ 1

2

β² β⁴−4α²β²γ²

.

2.19

Clearly q0 is the positive root offx 0 and q1,q2 are two positive roots of gx 0 if β > 2αγ. Note thatfx ≤0 if 0 < x ≤ q₀,andfx > 0 ifx > q₀,gx≥ 0 if 0 < x ≤ q₁, or x≥ q₂,thengx< 0 ifq₁ < x < q₂. Also note that as well as ifβ >2αγ,q₀² < q₁. In fact it is based on the following calculation:

q₁−q₀² 1 2

β²− β⁴−4α²β²γ²

−1 4

−β β² 4αβγ ₂

β

2 β² 4αβγ − β²−4α²γ²−2αγ

2αβγ

β− β²−4α²γ² β² 4αβγ β²−4α²γ² 2αγ

>0.

2.20

Thus for β > 2αγ, we always have q0 < √q1 < √q2. Noting that q ∈ 0,1, we have the following result.

Lemma 2.4. Letβ >2αγ. Then the following are given.

iSuppose thatq0 ≥1. Then for 0< q <1, then2.6has a simple zero root and does not have roots in the imaginary axis.

iiSuppose thatq0<1≤ √q1<√q2. If 0< q≤q0, then2.6has a simple zero root and does not have roots in the imaginary axis. And ifq₀ < q <1,2.6has a double zero root and does not have roots in the imaginary axis.

iiiSuppose thatq₀<√q₁<1<√q₂. If 0< q≤q₀, then2.6has a simple zero root and does not have roots in the imaginary axis. Ifq₀< q≤ √q1, then2.6has a double zero root and does not have roots in the imaginary axis. And if√q1< q <1, then2.6has a double zero root and has a pair of purely imaginary roots.

ivSuppose that√q2 ≥1. Then if 0< q ≤q0, then2.6has a simple zero root and does not have roots in the imaginary axis. Ifq0 < q ≤ √q1, then2.6has a double zero root and does not have roots in the imaginary axis. If√q₁ < q <√q₂, then2.6has a double zero root and has a pair of purely imaginary roots whenτ^∗τjfor somej. And if√q2≤q <1, 2.6has a double zero root and does not have a pair of purely imaginary roots.

Defineλτ στ iωτto be the root of2.6such thatστj 0 andωτj ω₀. Then we have the following result.

Lemma 2.5. Suppose thatkk^∗andgq²<0. Thenστj>0.

Proof. Differentiating2.6with respect toτ yields dλ

dτ ₋₁

2λ q−α k−γ

e^λτ β λβ

λ αγ −τ

λ, 2.21

and a simple calculation gives

Re dλ

dτ ₋₁

ττj

α²β²γ² q²

−β² q² 2ω²₀ β²q²

α²γ² ω²₀ q²β²−α²β²γ²−q⁴ β²q²

α²γ² ω²₀ , 2.22

which gives

Sign Re dλ

dτ ₋₁

ττj

Sign

−g q²

1, 2.23

thus completing the proof.

Next we discuss the distribution of other roots of2.6. We need the following lemma due to Ruan and Wei19.

Lemma 2.6. Consider the exponential polynomial P

λ, e^−λτ

pλ qλe^−λτ, 2.24

wherep,qare real polynomials such that degq < degpandτ ≥ 0. As τ varies, the sum of the order of zeros ofPλ, e^−λτon the open right half-plane can change only if a zero appears on or crosses the imaginary axis.

Lemma 2.7. Letkk^∗andτ >0. Then, the following are obtained.

iIfq > q₀, then all roots of 2.6except 0 and purely imaginary roots have negative real parts,

iiIf 0< q≤q0, then2.6has at least one positive root.

Proof. Note that, forτ 0, ifq > q₀orq² qβ > αβγ,Δλ 0 has a zero root and a negative root. Using Lemmas2.2and2.6, we obtain claimi. Forτ0,Δλ 0 has a zero root and a positive root if 0< q≤q0orq² qβ≤αβγ. Forτ >0, let

fλ Δλ

λ λ A βe^−λτ B Ce^−λτ

λ . 2.25

Also noting thatB C0 whenkk^∗, we have that

λ→lim0fλ A β−Cτ 1 q

q² qβ−αβγ

−αβγτ

<0, 2.26

and lim_{λ→ ∞}fλ ∞. This proves the second part of the lemma and completes the proof of the lemma.

3. Center Manifold Reduction

In this section, we briefly summarize the theory of center manifold reduction for general DDEs. The material is mainly taken from20,21. Consider the following DDE:

dx dt L

μ x_t G

x_t, μ

, 3.1

wherex∈C−τ,0,Rⁿ,μ∈R^p. This equation is equivalent to dx

dt L μ

x_t G x_t, μ

, dμ

dt 0, 3.2

which can be written as

dX

dt LXt FXt, 3.3

whereX x, μ^T,FXt Gxt,0^T, andLdiagL,0. DefineX ∈C:C−τ,0,R^{n p} with supreme norm andX_t∈Cis defined byX_tθ Xt θ,−τ≤θ≤0;L:C → LR^{n p}is

a bounded linear operator; andF:C → Cis aC^k k≥2function withF0 0,DF0 0.

Consider the following linear system:

Xt ˙ LXt. 3.4

Since L is a bounded linear operator, then L can be represented by a Riemann-Stieltjes integral

Lϕ ₀

−τdηθϕθ, ∀ϕ∈C, 3.5

by the Riesz representation theorem, where ηθ θ ∈ −τ,0 is an n p × n p matrix function of bounded variation. LetA0 be the infinitesimal generator for the solution semigroup defined by Sys.3.4such that

A0ϕϕ,˙ DA0

ϕ∈C¹−τ,0,R^{n p}: ˙ϕ0 ₀

−τdηθϕθ

. 3.6

Define the bilinear form betweenCandC^∗ C0, τ,R^{n p∗} whereR^{n p∗} is the space of all rown p-vectorsby

ψ, ϕ

ψ0ϕ0− ₀

−τ

_θ

0

ψξ−θdηθϕξdξ, ∀ψ∈C^∗, ∀ϕ∈C. 3.7

The adjoint ofA0is defined byA^∗₀as

A₀^∗ψ −ψ,˙ D A₀^∗

ϕ∈C¹

0, τ,R^{n p∗}

: ˙ψ0 − ₀

−τψ−θdηθ

. 3.8

In our setting,3.3hasptrivial components. Assume that the characteristic equation of3.3 has eigenvalue zero with multiplicity 2pand all other eigenvalues have negative real parts.

ThenLhas a generalized eigenspaceP which is invariant under the flow3.4. LetP^∗be the space adjoint withP inC^∗. ThenCcan be decomposed asC P ⊕QwhereQ {ϕ ∈ C : ψ, ϕ0,∀ψ∈P^∗}. Choose the basesΦandΨforP andP^∗, respectively, such that

Ψ,ΦI, Φ ΦJ,˙ Ψ ˙ −JΨ, 3.9

whereJis Jordan matrix associated with the eigenvalue 0.

To consider Sys.3.3, we need to enlarge the spaceCto the followingBC:

BC

ϕ:−τ,0 → R^{n p}:ϕis continuous on −τ,0, ∃lim

θ→0⁻ϕθ∈R^{n p}

. 3.10

The elements ofBCcan be expressed asψϕ X0αwithϕ∈C, α∈R^{n p},and

X0θ

⎧⎨

⎩

0, −τ ≤θ <0,

I, θ0, 3.11

whereIis then×nidentity matrix. Define the projectionπ:BC → Pby π

ϕ X₀α Φ

Ψ, ϕ

Ψ0α

. 3.12

Then the enlarged phase spaceBCcan be decomposed asBC P ⊕kerπ.LetX Φx y withx∈R^2pandy∈Q¹{ϕ∈Q: ˙ϕ∈C}. Then3.3can be decomposed as

˙

xJx Ψ0F

Φx y ,

˙

yA_Q¹y I−πX0F

Ψx y

, 3.13

whereAis an extension of the infinitesimal generatorA0fromC¹toBC, defined by

A0ϕϕ˙ X₀

Lϕ−ϕ0˙

⎧⎪

⎪⎨

⎪⎪

⎩

˙

ϕ, −1≤θ <0, ₀

−τdηtϕt, θ0, 3.14

forϕ∈C¹and its adjoint byA^∗is defined by

A^∗ψ

⎧⎪

⎪⎨

⎪⎪

⎩

−ψ,˙ 0< s≤θ, ₀

−τψ−θdηθ, s0,

3.15

forψ∈C^1∗. LetFv

j≥21/j!Fjv. Then Sys.3.13becomes

˙

xJx

j≥2

1 j!f_j¹

x, y ,

˙

yAQ¹y

j≥2

1 j!f_j²

x, y ,

3.16

where

f_j¹ x, y

Ψ0Fj

Φx y

, f_j² x, y

I−πX0F_j

Φx y

. 3.17

On the center manifold,3.16can be approximated as

˙

xJx

j≥2

1

j!f_j¹x,0. 3.18

4. Simple-Zero Singularity

In this section, we assume that the conditionH2holds. From the definition ofτ^∗, we know thatτ^∗>0 if and only ifq > q0. ThereforeH2is equivalent to

kk^∗, q > q0, τ >0, τ /τ^∗. 4.1 From ii of Lemma 2.4 and ii of Lemma 2.7, we know that, at 0,0, the characteristic equation of the linear part of Sys. 2.5 has a simple zero root and the rest of roots have negative parts. We treatkas a bifurcation parameter neark^∗.

SetC : C−τ,0,R³,C^∗ : C0, τ,R^3∗. Letμ k−k^∗. Then Sys.2.5 can be rewritten as

du₁ dt α

βγ

q u10−βu20 μu10 i²u²₁0 i⁰u³₁t

Oμ|u|² |u|⁴ , du2

dt k^∗u1−τ−qu2t μu1−τ−βu2−τ i²u²₁−τ i³u³₁−τ Oμ|u|² |u|⁴ , dμ

dt 0.

4.2

The linearization of Sys.4.2at0,0,0is du₁

dt αβγ

q u₁0−αβu₂0, du2

dt k^∗u1−τ−qu20−βu2−τ, dμ

dt 0.

4.3

Letηθ Aδθ Bδθ τwhere

A

⎛

⎜⎜

⎜⎝ αβγ

q −αβ 0

0 −q 0

0 0 0

⎞

⎟⎟

⎟⎠, B

⎛

⎜⎜

⎝

0 0 0

k^∗ −β 0

0 0 0

⎞

⎟⎟

⎠. 4.4

LetX u1, u2, μ^Tand

FXt

⎛

⎜⎜

⎜⎝

αμu10 αi²u²₁0 αi³u³₁0 Oμ|u|² |u|⁴ μu₁−τ i²u²₁−τ i³u³₁−τ Oμ|u|² |u|⁴

0

⎞

⎟⎟

⎟⎠. 4.5

Define

Lϕ ₀

−τdηθϕθ, ∀ϕ∈C. 4.6

Then Sys.4.2becomes

Xt ˙ LXt FXt. 4.7

From3.7, the bilinear form can be expressed as ψ, ϕ

ψ0ϕ0

₀

−τψξ τBϕξdξ. 4.8

It is not hard to see that the infinitesimal generatorA:C¹ → BCis given by

Aϕϕ˙ X₀

Lϕ−ϕ0˙

⎧⎨

⎩

˙

ϕ, −τ≤θ <0,

Aϕ0 Bϕ−τ, θ0, 4.9

forϕ∈C¹and its adjointA^∗by

A^∗ψ

⎧⎨

⎩

−ψ,˙ 0< s≤θ,

ψ0A ψτB, s0, 4.10

forψ∈C^1∗.

Next we obtain the bases for the center spaceP and its adjoint spaceP^∗, respectively.

LetAϕ0 forϕ∈C¹, that is,

ϕθ ˙ 0 for −τ ≤θ <0, Aϕ0 Bϕ−τ 0 for θ0. 4.11

then we know thatϕis a constant vectora1, a2, a3^T ∈R³\ {0}such that

A Ba1, a2, a3^T 0. 4.12

Then we have two linearly independent solutions ϕ₁ q, γ,0^T,ϕ₂ 0,0,1^T which are bases for the center spaceP. LetΦ ϕ1, ϕ2.

Similarly, letA^∗ψ 0 forψ∈C^1∗, that is,

−ψs ˙ 0 for 0< s≤τ, ψ0A ψτB0 fors0, 4.13

then we know thatψis a constant vectorb1, b2, b3∈R^3∗\ {0}such that

b1, b₂, b₃A B 0. 4.14

From this we have two linearly independent solutionsψ₁ −q β, αβ,0andψ₂ 0,0,1 which are bases for the center spaceP^∗. LetΨ rψ1, ψ2^Twithrbeing determined such that ψ1, ϕ₁1. In fact

r 1

qαβγτ−τ^∗. 4.15

Clearlyr is well defined sinceτ−τ^∗/0. It is not hard to check that ˙Φ ΦJ, ˙Ψ −JΨand Ψ,ΦI, whereJ_{0 0}

0 0

.

Letu Φx y. Then Sys.4.2can be decomposed as

˙ x ΨF

Φx y ,

˙

yA_Q¹y I−πX0F

Φx y

. 4.16

Writex x1, μ. Note that

Ψ0FΦx

#−rαq²

μx1 qi²x₁² q²i³x³₁ 0

$

h.o.t.. 4.17

Here h.o.t. represents higher-order terms. Thus, for sufficiently small μ, on the center manifold, ifi²/0, then Sys.4.2becomes

˙

x1−rαq²μx1−rαq³i²x₁² h.o.t.,

˙

μ0. 4.18

Ifi²0 andi³/0, then Sys.4.2can be transformed into the following form:

˙

x1−rαq²μx1−rαq⁴i³x₁³ h.o.t.,

˙

μ0. 4.19

Thus we have the following results.

Theorem 4.1. Letμbe small. Then consider what follows.

iSuppose thatμ 0. Then ifi²/0, the equilibriumY^∗, K^∗is unstable, and ifi² 0 andi³/0, then the equilibriumY^∗, K^∗is asymptotically stable forτ−τ^∗i³>0 and unstable ifτ−τ^∗i³<0.

iiThe equilibriumY^∗, K^∗is asymptotically stable ifτ−τ^∗μ >0 and unstable ifτ−τ^∗μ <

0.

iiiAtY^∗, K^∗, k^∗, Sys.1.4undergoes a transcritical bifurcation ifi²/0 and a pitchfork bifurcation ifi²0 andi³/0.

5. Double-Zero Singularity

In this section, we assume that one of the conditionsH3andH6holds andgq²>0, or equivalently, as

kk^∗, ττ^∗, q > q₀, g q²

>0. 5.1

FromSection 2, we can see that, at0,0, the characteristic equation of Sys.2.5has a double root 0 and all other roots have negative real parts ifk k^∗ andτ τ^∗. We treatk, τas a bifurcation parameter neark^∗, τ^∗.

By scalingt → t/τ, Sys.2.2can be written as du1t

dt ατ k−γ

u₁t−αβτu₂t ατi²u²₁t τi³u³₁t O

|u1|⁴ , du2t

dt τku1t−1− q β

τu2t τi²u²₁t−1 τi³u³₁t−1 O

|u1|⁴ .

5.2

LetC:C−1,0,R⁴, C^∗:C0,1,R^4∗. Letμ1k−k^∗,μ2τ−τ^∗. Then onCwe have du1

dt α βγ

qτ^∗u10−βτ^∗u20 τ^∗μ1u10 βγ

qμ₂u₁0−βμ₂u₂0 τ^∗i²u²₁0 i²μ₂u²₁0 i³τ^∗u³₁0 i³μ₂u³₁0

Oμ²|u| μ|u|⁴ , du2

dt βγ

q τ^∗u1t−1−qτ^∗u2t−βτ^∗u2−1 τ^∗μ₁u₁−1 βγ

qμ₂u₁−1−qμ₂u₂0−βμ₂u₂−1 i²τ^∗u²₁−1 i²μ₂u²₁−1

i³τ^∗u³₁−1 i³μ2u³₁−1 Oμ²|u| μ|u|⁴ , dμ₁

dt 0, dμ₂ dt 0.

5.3

The linearization of Sys.5.3at0,0,0,0is du1t

dt αβγ

q τ^∗u₁0−αβτ^∗u₂0, du2t

dt k^∗τ^∗u1−1−qτ^∗u20−βτ^∗u2−1, dμ1

dt 0, dμ2

dt 0.

5.4

Let

ηθ Aδθ Bδθ 1, 5.5

where

A

⎛

⎜⎜

⎜⎜

⎜⎜

⎝ αβγ

q τ^∗ −αβτ^∗ 0 0 0 −qτ^∗ 0 0

0 0 0 0

0 0 0 0

⎞

⎟⎟

⎟⎟

⎟⎟

⎠

, B

⎛

⎜⎜

⎜⎜

⎜⎝

0 0 0 0

k^∗τ^∗ −βτ^∗ 0 0

0 0 0 0

0 0 0 0

⎞

⎟⎟

⎟⎟

⎟⎠. 5.6

Define

Lϕ ₀

−1dηθϕθ, ∀ϕ∈C. 5.7

LetC¹C¹−1,0,R⁴. LetX u1, u2, μ1, μ2^TandFXt F¹, F²,0,0^Twhere F¹α

τ^∗μ₁u₁0 βγ

q μ₂u₁0−βμ₂u₂0 i²τ^∗u²₁0 i³τ^∗u³₁0

Oμ²|u| μ|u|⁴ , F²τ^∗μ1u1−1 βγ

q μ2u1−1−qμ2u20−βμ2u2−1 i²τ^∗u²₁0 i³τ^∗u³₁−1 Oμ²|u| μ|u|⁴ .

5.8

Then Sys.5.3can be transformed into

Xt ˙ LXt FXt. 5.9

Let C^∗ C0,1,R^4∗. From 3.7, the bilinear inner product between C and C^∗ can be expressed by

ψ, ϕ

ψ0ϕ0

₀

−1ψξ 1Bϕξdξ, 5.10

forϕ∈ Candψ ∈C^∗. As inSection 4, the infinitesimal generatorA :C¹ → BCassociated withLis given by

Aϕϕ˙ X₀

Lϕ−ϕ0˙

⎧⎨

⎩

˙

ϕ, −1≤θ <0,

Aϕ0 Bϕ−1, θ0, 5.11 forϕ∈C¹and its adjoint by

A^∗ψ

⎧⎨

⎩

−ψ,˙ 0< s≤1,

ψ0A ψ1B, s0, 5.12 forψ ∈C^1∗. FromSection 2, we know that 0 is an eigenvalue ofAandA^∗with multiplicity 4. Now we compute eigenvectors ofAandA^∗associated with 0, respectively.

Next we obtain the bases for the center spaceP and its adjoint spaceP^∗, respectively.

LetAϕ0 forϕ∈C¹. This means that

ϕθ ˙ 0 for −1≤θ <0, Aϕ0 Bϕ−1 0 for θ0. 5.13

From this we obtain thatϕθ ϕ⁰is a constant vector inR⁴satisfying

A Bϕ⁰ 0. 5.14

This equation has three linearly independent solutions:a1 q, γ,0,0^T, a3 0,0,1,0^T, a₄ 0,0,0,1^T. Letϕ⁰₁be one of those. Suppose thatAa2ϕ⁰₁fora₂ ∈C¹, namely,

˙

a₂θ ϕ⁰₁ for −1≤θ <0, Aa20 Ba2−1 ϕ⁰₁ forθ0. 5.15

This implies that there is a constant vectorϕ⁰₂inR⁴such thata₂θ ϕ⁰₁θ ϕ⁰₂and L

ϕ⁰₁θ ϕ⁰₂

ϕ⁰₁. 5.16

Since

L

ϕ⁰₁θ ϕ⁰₂ L

ϕ⁰₁θ L

ϕ⁰₂

−Bϕ⁰₁ A Bϕ⁰₂, 5.17

we have that

A Bϕ⁰₂ I Bϕ⁰₁. 5.18

It is easy to see that5.18has no solution ifϕ⁰₁ is eithera₃ ora₄. Forϕ⁰₁ a₁, settingϕ⁰₂ 0, l,0,0^Tin5.18, we obtain

l− q²γ

q² qβ−αβγ, 5.19

and hencea2θ θq, l γθ,0,0^T. Thus we obtain basesa1,a2,a3,a4for the center spaceP.

LetΦ a1, a₂, a₃, a₄. Then we have that ˙Φ ΦJwhereJ

⎛

⎜⎝

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

⎞

⎟⎠.

Similarly, letA^∗ψ20 forψ2∈C^1∗, that is,

−ψ˙2s 0 for 0< s≤1, ψ20A ψ2−1B0 for s0, 5.20

which means thatψ2s ψ₂⁰is a constant vectorψ₂⁰∈R^4∗\ {0}satisfying

ψ⁰₂A B 0. 5.21

This equation has three linearly independent solutions: b2 mq β,−mαβ,0,0,b3 0,0,1,0,b₄ 0,0,0,1. Asserting thatb2, a₂1 gives

m 2

q² qβ−αβγ

q⁴−q²β² α²β²γ². 5.22

Letψ₂⁰be one ofb₂,b₃,b₄. SupposeA^∗b₁ψ₂⁰, that is,

−b˙₁s ψ₂⁰ for 0< s≤1, b₁0A b₁1Bψ₂⁰ fors0, 5.23

which implies that there isψ₁⁰∈R^4∗such thatb₁s −ψ₂⁰s ψ₁⁰satisfying L^∗

−ψ₂⁰s ψ₁⁰

ψ₂⁰. 5.24

Since

L^∗

−ψ₂⁰s ψ₁⁰

−L^∗ ψ₂⁰s

L^∗ ψ₁⁰

−ψ⁰₂B ψ₁⁰A B, 5.25

we have

ψ₁⁰A B ψ₂⁰I B. 5.26

It is not hard to check that5.26has no solution ifψ⁰₂ b₃ orb₄. Letting ψ₂⁰ b₂, setting ψ₁⁰ n1, n2,0,0in5.26and usingb1, a20, we can getn1andn2:

n₁ 2

q β

q⁶−3q⁵β−3q⁴β² q³β³ 3q²α²β²γ²−3qα²β³γ² 2α³β³γ³ 3

q⁴−q²β² α²β²γ²₂ , n2 2αβ

−q² 2qβ αβγ

q² qβ−αβγ2

3

q⁴−q²β² α²β²γ²₂ .

5.27

Hence

b₁s ψ₁⁰−sψ₂⁰

−m q β

s n₁, mαβs n₂,0,0

. 5.28

Thenb1,b2,b3,b4are bases of the center spaceP^∗. LetΨ b1, b2, b3, b4^T. ThenΨ,Φ I, Φ ΦJ˙ and ˙Ψ −JΨ.

Letu Φx y, namely,

u₁θ qx₁ qθx₂ y₁θ, u2θ γx1

l γθ

x2 y2θ. 5.29

Then Sys.5.9can be decomposed as

˙

xJx Ψ0F

Φx y ,

˙

yA_Q¹y I−πX0F

Φx y

. 5.30

Writex x1, x2, μ1, μ2. Then, on the center manifold, Sys.5.30becomes

˙ x₁

a₁₁μ₁ a₂₁μ₂ x₁

a₂₁μ₁ a₂₂μ₂ x₂ αn₁τ^∗

i²q²x²₁ i³q³τ^∗x³₁ n₂τ^∗

×

i²q²x1−x₂² i³q³τ^∗x1−x₂³

h.o.t.,

˙ x2

b11μ1 b21μ2

x1

b12μ1 b22μ2

x2

mα q β

τ^∗

i²q²x₁² i³q³τ^∗x³₁

mαβτ^∗

×

i²q²x1−x₂² i³q³τ^∗x1−x₂³

h.o.t.,

˙

μ₁0, μ˙₂0,

5.31

where

a11qτ^∗αn1 n2, a21 −α q β

γ−qk^∗ n1

− 2q β

γ qk^∗ n2, a₁₂−qn2τ^∗, a₂₂−

lαβn₁

q β

l−γ qk^∗

n₂ , b11 mq²ατ^∗, b21mqαβγ, b12 mqαβτ^∗, b22 mαβ

− q β

γ qk^∗ .

5.32

Next we use techniques of nonlinear transformations in22to transform Sys.5.31 into normal forms. Ifi²/0, then up to the second order, Sys.5.31can be written as

˙ x1

a11μ1 a21μ2

x1

a21μ1 a22μ2

x2

αn1τ^∗i²q²x²₁ n2τ^∗i²q²x1−x2² h.o.t.,

˙ x₂

b₁₁μ₁ b₂₁μ₂ x₁

b₁₂μ₁ b₂₂μ₂ x₂ mα

q β

τ^∗i²q²x²₁ mαβτ^∗i²q²x1−x₂² h.o.t.,

˙

μ₁0, μ˙₂0.

5.33

This system can be transformed into the following normal form:

˙

x1x2 h.o.t.,

˙

x2ρ1x1 ρ2x2 a1x²₁ b1x1x2 h.o.t., 5.34

where

ρ1 b11μ1 b21μ2, ρ2 a11 b12μ1 a21 b22μ2, a1mq³ατ^∗i², b12q²

mαβ αn1 n2

τ^∗i².

5.35

Since

∂ρ

∂μ det

⎛

⎜⎜

⎝

∂ρ₁

∂μ₁

∂ρ₁

∂μ₂

∂ρ₂

∂μ1

∂ρ₂

∂μ2

⎞

⎟⎟

⎠

−mq²αγ

mαβ² αβn1

q β n2

τ^∗ −4q³α³β²γ²τ^∗

q² qβ−αβγ q⁴−q²β² α²β²γ²₂ /0,

5.36

we have thatμ1, μ2 → ρ1, ρ2is regular and hence the transversality condition holds.

Ifi²0 andi³/0, then up to the third order, Sys.5.31becomes

˙ x1

a11μ1 a21μ2

x1

a21μ1 a22μ2

x2

αn1τ^∗i³q³τ^∗x³₁ n2τ^∗i³q³τ^∗x1−x2³ h.o.t.,

˙ x₂

b₁₁μ₁ b₂₁μ₂ x₁

b₁₂μ₁ b₂₂μ₂ x₂ mα

q β

τ^∗i³q³τ^∗x₁³ mαβτ^∗i³q³τ^∗x1−x₂³ h.o.t.,

˙

μ10, μ˙20.

5.37

This system can be transformed into the following normal form:

˙

x1x2 h.o.t.,

˙

x₂ρ₁x₁ ρ₂x₂ a₂x₁³ b₂x²₁x₂ h.o.t., 5.38 wherea₂mq⁴ατ^∗i³,b₂3q³mαβ αn₁ n₂τ^∗i³.

6. Bifurcation Diagrams

In this section, we will use the truncated systems 5.34 and 5.38 to obtain bifurcation diagrams of Sys.5.3.

First, we consider the truncated system of5.34:

˙ x1x2,

˙

x2 ρ1x1 ρ2x2 a1x²₁ b1x1x2, 6.1