Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 923809, 29 pages
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

Received 12 August 2009; Accepted 2 November 2009

Academic Editor: Xue-Zhong He

Copyright © 2009 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.

Abstract

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.

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 [2] introduced the idea that there is a time delay for investment before a business decision. Krawiec and Szydłowski [3–5] incorporated the idea of Kalecki into the model of Kaldor by proposing the following Kaldor-Kalecki model of business cycles:

𝑑 𝑌 ( 𝑡 ) [ ] , 𝑑 𝑡 = 𝛼 𝐼 ( 𝑌 ( 𝑡 ) , 𝐾 ( 𝑡 ) ) − 𝑆 ( 𝑌 ( 𝑡 ) , 𝐾 ( 𝑡 ) ) 𝑑 𝐾 ( 𝑡 ) 𝑑 𝑡 = 𝐼 ( 𝑌 ( 𝑡 − 𝜏 ) , 𝐾 ( 𝑡 ) ) − 𝑞 𝐾 ( 𝑡 ) , ( 1 . 1 ) where 𝑌 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, 𝐼 ( 𝑌 , 𝐾 ) and 𝑆 ( 𝑌 , 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 decisions [6] also influence the change in the capital stock, Kaddar and Talibi Alaoui [15] extended the model (1.1) by imposing delays in both the gross product and capital stock. Thus adding the same delay to the capital stock 𝐾 in the investment function 𝐼 ( 𝑌 , 𝐾 ) of the second equation of Sys. (1.1) leads to the following Kaldor-Kalecki model of business cycles:

𝑑 𝑌 ( 𝑡 ) [ ] , 𝑑 𝑡 = 𝛼 𝐼 ( 𝑌 ( 𝑡 ) , 𝐾 ( 𝑡 ) ) − 𝑆 ( 𝑌 ( 𝑡 ) , 𝐾 ( 𝑡 ) ) 𝑑 𝐾 ( 𝑡 ) 𝑑 𝑡 = 𝐼 ( 𝑌 ( 𝑡 − 𝜏 ) , 𝐾 ( 𝑡 − 𝜏 ) ) − 𝑞 𝐾 ( 𝑡 ) . ( 1 . 2 )

As in [3]; also see [10, 16, 17], using the following saving and investment functions 𝑆 and 𝐼 , respectively, 𝑆 ( 𝑌 , 𝐾 ) = 𝛾 𝑌 , 𝐼 ( 𝑌 , 𝐾 ) = 𝐼 ( 𝑌 ) − 𝛽 𝐾 , ( 1 . 3 ) where 𝛽 > 0 and 𝛾 ∈ ( 0 , 1 ) are constants, we obtain the following system:

𝑑 𝑌 ( 𝑡 ) [ ] , 𝑑 𝑡 = 𝛼 𝐼 ( 𝑌 ( 𝑡 ) ) − 𝛽 𝐾 ( 𝑡 ) − 𝛾 𝑌 ( 𝑡 ) 𝑑 𝐾 ( 𝑡 ) 𝑑 𝑡 = 𝐼 ( 𝑌 ( 𝑡 − 𝜏 ) ) − 𝛽 𝐾 ( 𝑡 − 𝜏 ) − 𝑞 𝐾 ( 𝑡 ) . ( 1 . 4 ) Kaddar and Talibi Alaoui [15] studied the characteristic equation of the linear part of Sys. (1.4) at an equilibrium point and used the delay 𝜏 as a bifurcation parameter to show that the Hopf bifurcation may occur under some conditions as 𝜏 passes some critical values. However, they did not obtain the stability of the bifurcating limit cycles and the direction of the Hopf bifurcation. Wang and Wu [18] further studied Sys. (1.4) and 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. (1.4) to give the direction of the Hopf bifurcation and the stability of the bifurcating limit cycles for some critical values of 𝜏 .

However, under certain conditions, the characteristic equation of the linear part of Sys. (1.4) may 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-zero (fold) and double-zero (Bogdanov-Takens) singularities for Sys. (1.4) and their corresponding dynamical behaviors are investigated by using 𝑘 and 𝜏 as bifurcation parameters (where 𝑘 is defined in Section 2). The discussion of zero-Hopf singularity will be addressed in a coming paper.

The rest of this manuscript is organized as follows. In Section 2, a detailed presentation is given for the distribution of eigenvalues of the linear part of Sys. (1.4) at an equilibrium point in the ( 𝑘 , 𝜏 ) -parameter space. In Section 3, the theory of center manifold reduction for general delayed differential equations (DDEs) is briefly introduced. In Sections 4 and 5, 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. In Section 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. of (1.4). 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 ) , a n d 𝐼 ( 𝑠 ) i s a n o n l i n e a r 𝐶 4 f u n c t i o n , ( 2 . 1 ) and that ( 𝑌 ∗ , 𝐾 ∗ ) is an equilibrium point of Sys. (1.4). Let 𝐼 ∗ = 𝐼 ( 𝑌 ∗ ) , 𝑢 1 = 𝑌 − 𝑌 ∗ , 𝑢 2 = 𝐾 − 𝐾 ∗ , and 𝑖 ( 𝑠 ) = 𝐼 ( 𝑠 + 𝑌 ∗ ) − 𝐼 ∗ . Then Sys. (1.4) can be transformed as

𝑑 𝑢 1 ( 𝑡 ) 𝑖 𝑢 𝑑 𝑡 = 𝛼 1 ( 𝑡 ) − 𝛽 𝑢 2 ( 𝑡 ) − 𝛾 𝑢 1 , ( 𝑡 ) 𝑑 𝑢 2 ( 𝑡 ) 𝑢 𝑑 𝑡 = 𝑖 1 ( 𝑡 − 𝜏 ) − 𝛽 𝑢 2 ( 𝑡 − 𝜏 ) − 𝑞 𝑢 2 ( 𝑡 ) . ( 2 . 2 ) Let the Taylor expansion of 𝑖 at 0 be

𝑖 ( 𝑢 ) = 𝑘 𝑢 + 𝑖 ( 2 ) 𝑢 2 + 𝑖 ( 3 ) 𝑢 3 + 𝒪 | 𝑢 | 4 , ( 2 . 3 ) where

𝑘 = 𝑖 ( 0 ) = 𝐼 𝑌 ∗ , 𝑖 ( 2 ) = 1 2 𝑖 1 ( 0 ) = 2 𝐼 𝑌 ∗ , 𝑖 ( 3 ) = 1 𝑖 3 ! 1 ( 0 ) = 𝐼 3 ! 𝑌 ∗ . ( 2 . 4 ) The linear part of Sys. (2.2) at ( 0 , 0 ) is

𝑑 𝑢 1 ( 𝑡 ) 𝑑 𝑡 = 𝛼 ( 𝑘 − 𝛾 ) 𝑢 1 ( 𝑡 ) − 𝛽 𝑢 2 , ( 𝑡 ) 𝑑 𝑢 2 ( 𝑡 ) 𝑑 𝑡 = 𝑘 𝑢 1 ( 𝑡 − 𝜏 ) − 𝛽 𝑢 2 ( 𝑡 − 𝜏 ) − 𝑞 𝑢 2 ( 𝑡 ) , ( 2 . 5 ) and the corresponding characteristic equation is

Δ ( 𝜆 ) ≡ 𝜆 2 + 𝐴 𝜆 + 𝐵 + ( 𝛽 𝜆 + 𝐶 ) 𝑒 − 𝜆 𝜏 = 0 , ( 2 . 6 ) where

𝐴 = 𝑞 − 𝛼 ( 𝑘 − 𝛾 ) , 𝐵 = − 𝛼 𝑞 ( 𝑘 − 𝛾 ) , 𝐶 = 𝛼 𝛽 𝛾 . ( 2 . 7 ) For 𝜏 = 0 , (2.6) becomes

𝜆 2 + ( 𝐴 + 𝛽 ) 𝜆 + 𝐵 + 𝐶 = 0 . ( 2 . 8 ) Define

𝑘 ∗ = 𝛽 𝛾 𝑞 + 𝛾 , 𝑘 ∗ ∗ = 𝑞 + 𝛽 𝛼 + 𝛾 . ( 2 . 9 )

Theorem 2.1. Let 𝜏 = 0 . If 𝑘 < m i n { 𝑘 ∗ , 𝑘 ∗ ∗ } , then all roots of (2.8) have negative real parts, and hence ( 𝑌 ∗ , 𝐾 ∗ ) is asymptotically stable. If 𝑘 > m i n { 𝑘 ∗ , 𝑘 ∗ ∗ } , then (2.8) has a positive root and a negative root, and hence, ( 𝑌 ∗ , 𝐾 ∗ ) is unstable.

Now assume 𝜏 > 0 . Clearly Δ ( 0 ) = 0 if and only if 𝑘 = 𝑘 ∗ . Next we always assume that 𝑘 = 𝑘 ∗ . It is easy to attain

Δ ( 𝜆 ) = 2 𝜆 + 𝑞 − 𝛼 𝛽 𝛾 𝑞 + 𝛽 𝑒 − 𝜆 𝜏 − ( 𝛽 𝜆 + 𝐶 ) 𝜏 𝑒 − 𝜆 𝜏 , Δ ( 𝜆 ) = 2 − 2 𝛽 𝜏 𝑒 − 𝜆 𝜏 + 𝛽 𝜏 2 𝜆 𝑒 − 𝜆 𝜏 + 𝐶 𝜏 2 𝑒 − 𝜆 𝜏 . ( 2 . 1 0 ) Define 𝜏 ∗ = ( 𝑞 2 + 𝑞 𝛽 − 𝛼 𝛽 𝛾 ) / 𝛼 𝛽 𝛾 𝑞 . Then we have that,

Δ ( 0 ) = 𝛼 𝛽 𝛾 𝑞 𝜏 ∗ − 𝜏 , Δ | | ( 0 ) 𝜏 = 𝜏 ∗ = 𝑞 4 − 𝛽 2 𝑞 2 + 𝛼 2 𝛽 2 𝛾 2 𝛼 𝛽 𝛾 𝑞 2 . ( 2 . 1 1 ) Define

𝑓 ( 𝑥 ) = 𝑥 2 + 𝛽 𝑥 − 𝛼 𝛽 𝛾 , 𝑔 ( 𝑥 ) = 𝑥 2 − 𝛽 2 𝑥 + 𝛼 2 𝛽 2 𝛾 2 . ( 2 . 1 2 ) Hence if 𝑓 ( 𝑞 ) ≤ 0 , 𝜏 ∗ ≤ 0 , and hence Δ ( 0 ) < 0 , and if 𝑓 ( 𝑞 ) > 0 , 𝜏 ∗ > 0 , and hence Δ ( 0 ) = 0 if and only if 𝜏 = 𝜏 ∗ . Also Δ ( 0 ) | 𝜏 = 𝜏 ∗ ≠ 0 if and only if 𝑔 ( 𝑞 2 ) ≠ 0 . Thus we obtain the following result.

Lemma 2.2. Suppose that 𝑘 = 𝑘 ∗ . Then the following are considered. (i)If 𝜏 ∗ ≤ 0 , then (2.6) has a simple root 0 for all 𝜏 > 0 . (ii)Let 𝜏 ∗ > 0 . Then the following are given. (a)Equation (2.6) has a simple root 0 if and only if 𝜏 ≠ 𝜏 ∗ , (b)Equation (2.6) has a double root 0 if and only if 𝜏 = 𝜏 ∗ and 𝑔 ( 𝑞 2 ) ≠ 0 .

Let 𝜔 𝑖 ( 𝜔 > 0 ) be a purely imaginary root of (2.6). After plugging it into (2.6) and separating the real and imaginary parts, we have that

𝜔 2 𝑞 + 𝛼 𝛽 𝛾 = 𝛼 𝛽 𝛾 c o s ( 𝜔 𝜏 ) + 𝛽 𝜔 s i n ( 𝜔 𝜏 ) , 2 − 𝛼 𝛽 𝛾 𝑞 𝜔 = 𝛼 𝛽 𝛾 s i n ( 𝜔 𝜏 ) − 𝛽 𝜔 c o s ( 𝜔 𝜏 ) . ( 2 . 1 3 ) Adding squares of two equations yields 𝜔 2 + 𝑔 𝑞 2 𝑞 2 = 0 . ( 2 . 1 4 ) Then (2.14) has a nonzero solution if and only if 𝑔 ( 𝑞 2 ) < 0 and does not have a nonzero solution if and only if 𝑔 ( 𝑞 2 ) ≥ 0 . If 𝑔 ( 𝑞 2 ) < 0 , from (2.14), we solve 𝜔 as follows:

𝜔 = 𝜔 0 ≡ 1 𝑞 𝑞 − 𝑔 2 , ( 2 . 1 5 ) and from (2.13), we solve c o s ( 𝜔 0 𝜏 ) , s i n ( 𝜔 0 𝜏 ) as:

𝜔 c o s 0 𝜏 = − 𝑞 2 𝜔 2 0 + 𝛼 𝛽 𝛾 𝜔 2 0 + 𝑞 𝛼 𝛾 𝛼 𝛽 𝛾 + 𝜔 2 0 𝛼 𝑞 𝛽 2 𝛾 2 + 𝜔 2 0 𝜔 ≡ 𝑎 , s i n 0 𝜏 = 𝑞 2 𝛼 𝛾 𝜔 0 − 𝛼 2 𝛽 𝛾 2 𝜔 0 + 𝑞 𝛼 𝛽 𝛾 𝜔 0 + 𝑞 𝜔 3 0 𝛼 𝑞 𝛽 2 𝛾 2 + 𝜔 2 0 ≡ 𝑏 . ( 2 . 1 6 ) Define

𝛿 = a r c c o s 𝑎 , i f 𝑏 ≥ 0 , 2 𝜋 − a r c c o s 𝑎 , i f 𝑏 < 0 . ( 2 . 1 7 ) From (2.16), we obtain

𝜏 = 𝜏 𝑗 ≡ 1 𝜔 0 ( 𝛿 + 2 𝑗 𝜋 ) , 𝑗 = 0 , 1 , 2 , … . ( 2 . 1 8 )

Clearly if 𝛽 > 2 𝛼 𝛾 , then 𝑔 ( 𝑥 ) = 0 has two positive roots, and if 𝛽 ≤ 2 𝛼 𝛾 , then 𝑔 ( 𝑥 ) ≥ 0 . Now, under 𝑘 = 𝑘 ∗ , we impose the following conditions:

(H1) 𝛽 ≤ 2 𝛼 𝛾 , 𝜏 ∗ ≤ 0 , (H2) 𝛽 ≤ 2 𝛼 𝛾 , 𝜏 ∗ > 0 , 𝜏 ≠ 𝜏 ∗ , (H3) 𝛽 ≤ 2 𝛼 𝛾 , 𝜏 ∗ > 0 , 𝜏 = 𝜏 ∗ , (H4) 𝛽 > 2 𝛼 𝛾 , 𝜏 ∗ > 0 , 𝜏 ≠ 𝜏 ∗ , 𝑔 ( 𝑞 2 ) ≥ 0 ,(H5) 𝛽 > 2 𝛼 𝛾 , 𝜏 ∗ > 0 , 𝜏 ≠ 𝜏 ∗ , 𝑔 ( 𝑞 2 ) < 0 ,(H6) 𝛽 > 2 𝛼 𝛾 , 𝜏 ∗ > 0 , 𝜏 = 𝜏 ∗ , 𝑔 ( 𝑞 2 ) ≥ 0 ,(H7) 𝛽 > 2 𝛼 𝛾 , 𝜏 ∗ > 0 , 𝜏 = 𝜏 ∗ , 𝑔 ( 𝑞 2 ) < 0 .

Based on Lemma 2.2, we have the following result.

Lemma 2.3. Suppose that 𝑘 = 𝑘 ∗ and 0 < 𝑞 < 1 . Then the following are obtained. (i)Under one of the conditions (H1), (H2), and (H4), (2.6) has a simple zero root and does not have other roots in the imaginary axis.(ii)Under the condition (H5), (2.6) has a simple zero root and a pair of purely imaginary roots ± 𝜔 0 𝑖 in the imaginary axis if 𝜏 = 𝜏 𝑗 , 𝑗 = 0 , 1 , 2 , … . (iii)Under one of the conditions (H3) and (H6), then (2.6) has a double root 0 and does not have other roots in the imaginary axis. (iv)Under the condition (H7), (2.6) has a double zero root and a pair of purely imaginary roots ± 𝜔 0 𝑖 in the imaginary axis if 𝜏 ∗ = 𝜏 𝑗 for some 𝑗 .

Now we use the roots of 𝑓 ( 𝑥 ) = 0 , 𝑔 ( 𝑥 ) = 0 to give a more detailed discussion for the roots of (2.6). Define

𝑞 0 = 1 2 − 𝛽 + 𝛽 2 , 𝑞 + 4 𝛼 𝛽 𝛾 1 = 1 2 𝛽 2 − 𝛽 4 − 4 𝛼 2 𝛽 2 𝛾 2 , 𝑞 2 = 1 2 𝛽 2 + 𝛽 4 − 4 𝛼 2 𝛽 2 𝛾 2 . ( 2 . 1 9 ) Clearly 𝑞 0 is the positive root of 𝑓 ( 𝑥 ) = 0 and 𝑞 1 , 𝑞 2 are two positive roots of 𝑔 ( 𝑥 ) = 0 if 𝛽 > 2 𝛼 𝛾 . Note that 𝑓 ( 𝑥 ) ≤ 0 if 0 < 𝑥 ≤ 𝑞 0 , a n d 𝑓 ( 𝑥 ) > 0 if 𝑥 > 𝑞 0 , 𝑔 ( 𝑥 ) ≥ 0 if 0 < 𝑥 ≤ 𝑞 1 , or 𝑥 ≥ 𝑞 2 , then 𝑔 ( 𝑥 ) < 0 if 𝑞 1 < 𝑥 < 𝑞 2 . Also note that as well as if 𝛽 > 2 𝛼 𝛾 , 𝑞 2 0 < 𝑞 1 . In fact it is based on the following calculation:

𝑞 1 − 𝑞 2 0 = 1 2 𝛽 2 − 𝛽 4 − 4 𝛼 2 𝛽 2 𝛾 2 − 1 4 − 𝛽 + 𝛽 2 + 4 𝛼 𝛽 𝛾 2 = 𝛽 2 𝛽 2 + 4 𝛼 𝛽 𝛾 − 𝛽 2 − 4 𝛼 2 𝛾 2 = √ − 2 𝛼 𝛾 2 𝛼 𝛽 𝛾 𝛽 − 𝛽 2 − 4 𝛼 2 𝛾 2 √ 𝛽 2 √ + 4 𝛼 𝛽 𝛾 + 𝛽 2 − 4 𝛼 2 𝛾 2 + 2 𝛼 𝛾 > 0 . ( 2 . 2 0 ) Thus for 𝛽 > 2 𝛼 𝛾 , we always have 𝑞 0 < √ 𝑞 1 < √ 𝑞 2 . Noting that 𝑞 ∈ ( 0 , 1 ) , we have the following result.

Lemma 2.4. Let 𝛽 > 2 𝛼 𝛾 . Then the following are given. (i)Suppose that 𝑞 0 ≥ 1 . Then for 0 < 𝑞 < 1 , then (2.6) has a simple zero root and does not have roots in the imaginary axis. (ii)Suppose that 𝑞 0 √ < 1 ≤ 𝑞 1 < √ 𝑞 2 . If 0 < 𝑞 ≤ 𝑞 0 , then (2.6) has a simple zero root and does not have roots in the imaginary axis. And if 𝑞 0 < 𝑞 < 1 , (2.6) has a double zero root and does not have roots in the imaginary axis. (iii)Suppose that 𝑞 0 < √ 𝑞 1 √ < 1 < 𝑞 2 . If 0 < 𝑞 ≤ 𝑞 0 , then (2.6) has a simple zero root and does not have roots in the imaginary axis. If 𝑞 0 √ < 𝑞 ≤ 𝑞 1 , then (2.6) has a double zero root and does not have roots in the imaginary axis. And if √ 𝑞 1 < 𝑞 < 1 , then (2.6) has a double zero root and has a pair of purely imaginary roots. (iv)Suppose that √ 𝑞 2 ≥ 1 . Then if 0 < 𝑞 ≤ 𝑞 0 , then (2.6) has a simple zero root and does not have roots in the imaginary axis. If 𝑞 0 √ < 𝑞 ≤ 𝑞 1 , then (2.6) has a double zero root and does not have roots in the imaginary axis. If √ 𝑞 1 √ < 𝑞 < 𝑞 2 , then (2.6) has a double zero root and has a pair of purely imaginary roots when 𝜏 ∗ = 𝜏 𝑗 for some 𝑗 . And if √ 𝑞 2 ≤ 𝑞 < 1 , (2.6) has a double zero root and does not have a pair of purely imaginary roots.

Define 𝜆 ( 𝜏 ) = 𝜎 ( 𝜏 ) + 𝑖 𝜔 ( 𝜏 ) to be the root of (2.6) such that 𝜎 ( 𝜏 𝑗 ) = 0 and 𝜔 ( 𝜏 𝑗 ) = 𝜔 0 . Then we have the following result.

Lemma 2.5. Suppose that 𝑘 = 𝑘 ∗ and 𝑔 ( 𝑞 2 ) < 0 . Then 𝜎 ( 𝜏 𝑗 ) > 0 .

Proof. Differentiating (2.6) with respect to 𝜏 yields 𝑑 𝜆 𝑑 𝜏 − 1 = [ ] 𝑒 2 𝜆 + 𝑞 − 𝛼 ( 𝑘 − 𝛾 ) 𝜆 𝜏 + 𝛽 − 𝜏 𝜆 𝛽 ( 𝜆 + 𝛼 𝛾 ) 𝜆 , ( 2 . 2 1 ) and a simple calculation gives R e 𝑑 𝜆 𝑑 𝜏 − 1 | | | | 𝜏 = 𝜏 𝑗 = 𝛼 2 𝛽 2 𝛾 2 + 𝑞 2 − 𝛽 2 + 𝑞 2 + 2 𝜔 2 0 𝛽 2 𝑞 2 𝛼 2 𝛾 2 + 𝜔 2 0 = 𝑞 2 𝛽 2 − 𝛼 2 𝛽 2 𝛾 2 − 𝑞 4 𝛽 2 𝑞 2 𝛼 2 𝛾 2 + 𝜔 2 0 , ( 2 . 2 2 ) which gives S i g n R e 𝑑 𝜆 𝑑 𝜏 − 1 | | | | 𝜏 = 𝜏 𝑗 𝑞 = S i g n − 𝑔 2 = 1 , ( 2 . 2 3 ) thus completing the proof.

Next we discuss the distribution of other roots of (2.6). We need the following lemma due to Ruan and Wei [19].

Lemma 2.6. Consider the exponential polynomial 𝑃 𝜆 , 𝑒 − 𝜆 𝜏 = 𝑝 ( 𝜆 ) + 𝑞 ( 𝜆 ) 𝑒 − 𝜆 𝜏 , ( 2 . 2 4 ) where 𝑝 , 𝑞 are real polynomials such that d e g ( 𝑞 ) < d e g ( 𝑝 ) and 𝜏 ≥ 0 . As 𝜏 varies, the sum of the order of zeros of 𝑃 ( 𝜆 , 𝑒 − 𝜆 𝜏 ) on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.

Lemma 2.7. Let 𝑘 = 𝑘 ∗ and 𝜏 > 0 . Then, the following are obtained. (i)If 𝑞 > 𝑞 0 , then all roots of (2.6) except 0 and purely imaginary roots have negative real parts,(ii)If 0 < 𝑞 ≤ 𝑞 0 , then (2.6) has at least one positive root.

Proof. Note that, for 𝜏 = 0 , if 𝑞 > 𝑞 0 or 𝑞 2 + 𝑞 𝛽 > 𝛼 𝛽 𝛾 , Δ ( 𝜆 ) = 0 has a zero root and a negative root. Using Lemmas 2.2 and 2.6, we obtain claim (i). For 𝜏 = 0 , Δ ( 𝜆 ) = 0 has a zero root and a positive root if 0 < 𝑞 ≤ 𝑞 0 or 𝑞 2 + 𝑞 𝛽 ≤ 𝛼 𝛽 𝛾 . For 𝜏 > 0 , let 𝑓 ( 𝜆 ) = Δ ( 𝜆 ) 𝜆 = 𝜆 + 𝐴 + 𝛽 𝑒 − 𝜆 𝜏 + 𝐵 + 𝐶 𝑒 − 𝜆 𝜏 𝜆 . ( 2 . 2 5 ) Also noting that 𝐵 + 𝐶 = 0 when 𝑘 = 𝑘 ∗ , we have that l i m 𝜆 → 0 + 1 𝑓 ( 𝜆 ) = 𝐴 + 𝛽 − 𝐶 𝜏 = 𝑞 𝑞 2 + 𝑞 𝛽 − 𝛼 𝛽 𝛾 − 𝛼 𝛽 𝛾 𝜏 < 0 , ( 2 . 2 6 ) and l i m 𝜆 → ∞ 𝑓 ( 𝜆 ) = ∞ . 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 from [20, 21]. Consider the following DDE:

𝑑 𝑥 𝑑 𝑡 = 𝐿 ( 𝜇 ) 𝑥 𝑡 𝑥 + 𝐺 𝑡 , , 𝜇 ( 3 . 1 ) where 𝑥 ∈ 𝐶 ( [ − 𝜏 , 0 ] , ℝ 𝑛 ) , 𝜇 ∈ ℝ 𝑝 . This equation is equivalent to

𝑑 𝑥 𝑑 𝑡 = 𝐿 ( 𝜇 ) 𝑥 𝑡 𝑥 + 𝐺 𝑡 , , 𝜇 𝑑 𝜇 𝑑 𝑡 = 0 , ( 3 . 2 ) which can be written as

𝑑 𝑋 𝑑 𝑡 = ℒ 𝑋 𝑡 𝑋 + 𝐹 𝑡 , ( 3 . 3 ) where 𝑋 = ( 𝑥 , 𝜇 ) 𝑇 , 𝐹 ( 𝑋 𝑡 ) = ( 𝐺 ( 𝑥 𝑡 ) , 0 ) 𝑇 , and ℒ = d i a g ( 𝐿 , 0 ) . Define 𝑋 ∈ 𝐶 ∶ = 𝐶 ( [ − 𝜏 , 0 ] , ℝ 𝑛 + p ) with supreme norm and 𝑋 𝑡 ∈ 𝐶 is defined by 𝑋 𝑡 ( 𝜃 ) = 𝑋 ( 𝑡 + 𝜃 ) , − 𝜏 ≤ 𝜃 ≤ 0 ; ℒ ∶ 𝐶 → 𝐿 ( ℝ 𝑛 + 𝑝 ) is a bounded linear operator; and 𝐹 ∶ 𝐶 → 𝐶 is a 𝐶 𝑘 ( 𝑘 ≥ 2 ) function with 𝐹 ( 0 ) = 0 , 𝐷 𝐹 ( 0 ) = 0 . Consider the following linear system:

̇ 𝑋 ( 𝑡 ) = ℒ 𝑋 𝑡 . ( 3 . 4 ) Since ℒ is a bounded linear operator, then ℒ can be represented by a Riemann-Stieltjes integral

ℒ 𝜑 = 0 − 𝜏 𝑑 𝜂 ( 𝜃 ) 𝜑 ( 𝜃 ) , ∀ 𝜑 ∈ 𝐶 , ( 3 . 5 ) by the Riesz representation theorem, where 𝜂 ( 𝜃 ) ( 𝜃 ∈ [ − 𝜏 , 0 ] ) is an ( 𝑛 + 𝑝 ) × ( 𝑛 + 𝑝 ) matrix function of bounded variation. Let 𝒜 0 be the infinitesimal generator for the solution semigroup defined by Sys. (3.4) such that

𝒜 0 𝒜 𝜑 = ̇ 𝜑 , 𝐷 0 = 𝜑 ∈ 𝐶 1 [ ] − 𝜏 , 0 , ℝ 𝑛 + 𝑝 ∶ ̇ 𝜑 ( 0 ) = 0 − 𝜏 . 𝑑 𝜂 ( 𝜃 ) 𝜑 ( 𝜃 ) ( 3 . 6 ) Define the bilinear form between 𝐶 and 𝐶 ∗ = 𝐶 ( [ 0 , 𝜏 ] , ℝ ( 𝑛 + 𝑝 ) ∗ ) (where ℝ ( 𝑛 + 𝑝 ) ∗ is the space of all row ( 𝑛 + 𝑝 ) -vectors) by

⟨ 𝜓 , 𝜑 ⟩ = 𝜓 ( 0 ) 𝜑 ( 0 ) − 0 − 𝜏 𝜃 0 𝜓 ( 𝜉 − 𝜃 ) 𝑑 𝜂 ( 𝜃 ) 𝜑 ( 𝜉 ) 𝑑 𝜉 , ∀ 𝜓 ∈ 𝐶 ∗ , ∀ 𝜑 ∈ 𝐶 . ( 3 . 7 ) The adjoint of 𝒜 0 is defined by 𝒜 ∗ 0 as

𝒜 ∗ 0 𝒜 𝜓 = − ̇ 𝜓 , 𝐷 ∗ 0 = 𝜑 ∈ 𝐶 1 [ ] 0 , 𝜏 , ℝ ( 𝑛 + 𝑝 ) ∗ ∶ ̇ 𝜓 ( 0 ) = − 0 − 𝜏 . 𝜓 ( − 𝜃 ) 𝑑 𝜂 ( 𝜃 ) ( 3 . 8 ) In our setting, (3.3) has 𝑝 trivial components. Assume that the characteristic equation of (3.3) has eigenvalue zero with multiplicity 2 𝑝 and all other eigenvalues have negative real parts. Then ℒ has a generalized eigenspace 𝑃 which is invariant under the flow (3.4). Let 𝑃 ∗ be the space adjoint with 𝑃 in 𝐶 ∗ . Then 𝐶 can be decomposed as 𝐶 = 𝑃 ⊕ 𝑄 where 𝑄 = { 𝜑 ∈ 𝐶 ∶ ⟨ 𝜓 , 𝜑 ⟩ = 0 , ∀ 𝜓 ∈ 𝑃 ∗ } . Choose the bases Φ and Ψ for 𝑃 and 𝑃 ∗ , respectively, such that

̇ ̇ ⟨ Ψ , Φ ⟩ = 𝐼 , Φ = Φ 𝐽 , Ψ = − 𝐽 Ψ , ( 3 . 9 ) where 𝐽 is Jordan matrix associated with the eigenvalue 0.

To consider Sys. (3.3), we need to enlarge the space 𝐶 to the following 𝐵 𝐶 :

[ ] 𝐵 𝐶 = 𝜑 ∶ − 𝜏 , 0 → ℝ 𝑛 + 𝑝 [ ∶ 𝜑 i s c o n t i n u o u s o n − 𝜏 , 0 ) , ∃ l i m 𝜃 → 0 − 𝜑 ( 𝜃 ) ∈ ℝ 𝑛 + 𝑝 . ( 3 . 1 0 ) The elements of 𝐵 𝐶 can be expressed as 𝜓 = 𝜑 + 𝑋 0 𝛼 with 𝜑 ∈ 𝐶 , 𝛼 ∈ ℝ 𝑛 + 𝑝 , and

𝑋 0 ( 𝜃 ) = 0 , − 𝜏 ≤ 𝜃 < 0 , 𝐼 , 𝜃 = 0 , ( 3 . 1 1 ) where 𝐼 is the 𝑛 × 𝑛 identity matrix. Define the projection 𝜋 ∶ 𝐵 𝐶 → 𝑃 by

𝜋 𝜑 + 𝑋 0 𝛼 [ ] = Φ ( Ψ , 𝜑 ) + Ψ ( 0 ) 𝛼 . ( 3 . 1 2 ) Then the enlarged phase space 𝐵 𝐶 can be decomposed as 𝐵 𝐶 = 𝑃 ⊕ k e r 𝜋 . Let 𝑋 = Φ 𝑥 + 𝑦 with 𝑥 ∈ ℝ 2 𝑝 and 𝑦 ∈ 𝑄 1 = { 𝜑 ∈ 𝑄 ∶ ̇ 𝜑 ∈ 𝐶 } . Then (3.3) can be decomposed as

̇ 𝑥 = 𝐽 𝑥 + Ψ ( 0 ) 𝐹 ( Φ 𝑥 + 𝑦 ) , ̇ 𝑦 = 𝒜 𝑄 1 𝑦 + ( 𝐼 − 𝜋 ) 𝑋 0 𝐹 ( Ψ 𝑥 + 𝑦 ) , ( 3 . 1 3 ) where 𝒜 is an extension of the infinitesimal generator 𝒜 0 from 𝐶 1 to 𝐵 𝐶 , defined by

𝒜 0 𝜑 = ̇ 𝜑 + 𝑋 0 [ ] = ⎧ ⎪ ⎨ ⎪ ⎩ 𝐿 𝜑 − ̇ 𝜑 ( 0 ) ̇ 𝜑 , − 1 ≤ 𝜃 < 0 , 0 − 𝜏 𝑑 𝜂 ( 𝑡 ) 𝜑 ( 𝑡 ) , 𝜃 = 0 , ( 3 . 1 4 ) for 𝜑 ∈ 𝐶 1 and its adjoint by 𝒜 ∗ is defined by

𝒜 ∗ ⎧ ⎪ ⎨ ⎪ ⎩ 𝜓 = − ̇ 𝜓 , 0 < 𝑠 ≤ 𝜃 , 0 − 𝜏 𝜓 ( − 𝜃 ) 𝑑 𝜂 ( 𝜃 ) , 𝑠 = 0 , ( 3 . 1 5 ) for 𝜓 ∈ 𝐶 1 ∗ . Let ∑ 𝐹 ( 𝑣 ) = 𝑗 ≥ 2 ( 1 / 𝑗 ! ) 𝐹 𝑗 ( 𝑣 ) . Then Sys. (3.13) becomes

̇ 𝑥 = 𝐽 𝑥 + 𝑗 ≥ 2 1 𝑓 𝑗 ! 1 𝑗 ( 𝑥 , 𝑦 ) , ̇ 𝑦 = 𝒜 𝑄 1 𝑦 + 𝑗 ≥ 2 1 𝑓 𝑗 ! 2 𝑗 ( 𝑥 , 𝑦 ) , ( 3 . 1 6 ) where

𝑓 1 𝑗 ( 𝑥 , 𝑦 ) = Ψ ( 0 ) 𝐹 𝑗 ( Φ 𝑥 + 𝑦 ) , 𝑓 2 𝑗 ( 𝑥 , 𝑦 ) = ( 𝐼 − 𝜋 ) 𝑋 0 𝐹 𝑗 ( Φ 𝑥 + 𝑦 ) . ( 3 . 1 7 ) On the center manifold, (3.16) can be approximated as

̇ 𝑥 = 𝐽 𝑥 + 𝑗 ≥ 2 1 𝑓 𝑗 ! 1 𝑗 ( 𝑥 , 0 ) . ( 3 . 1 8 )

4. Simple-Zero Singularity

In this section, we assume that the condition (H2) holds. From the definition of 𝜏 ∗ , we know that 𝜏 ∗ > 0 if and only if 𝑞 > 𝑞 0 . Therefore (H2) is equivalent to

𝑘 = 𝑘 ∗ , 𝑞 > 𝑞 0 , 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 treat 𝑘 as a bifurcation parameter near 𝑘 ∗ .

Set 𝐶 ∶ = 𝐶 ( [ − 𝜏 , 0 ] , ℝ 3 ) , 𝐶 ∗ ∶ = 𝐶 ( [ 0 , 𝜏 ] , ℝ 3 ∗ ) . Let 𝜇 = 𝑘 − 𝑘 ∗ . Then Sys. (2.5) can be rewritten as

𝑑 𝑢 1 𝑑 𝑡 = 𝛼 𝛽 𝛾 𝑞 𝑢 1 ( 0 ) − 𝛽 𝑢 2 ( 0 ) + 𝜇 𝑢 1 ( 0 ) + 𝑖 ( 2 ) 𝑢 2 1 ( 0 ) + 𝑖 ( 0 ) 𝑢 3 1 | | 𝜇 | | ( 𝑡 ) + 𝒪 | 𝑢 | 2 + | 𝑢 | 4 , 𝑑 𝑢 2 𝑑 𝑡 = 𝑘 ∗ 𝑢 1 ( − 𝜏 ) − 𝑞 𝑢 2 ( 𝑡 ) + 𝜇 𝑢 1 ( − 𝜏 ) − 𝛽 𝑢 2 ( − 𝜏 ) + 𝑖 ( 2 ) 𝑢 2 1 ( − 𝜏 ) + 𝑖 ( 3 ) 𝑢 3 1 | | 𝜇 | | ( − 𝜏 ) + 𝒪 | 𝑢 | 2 + | 𝑢 | 4 , 𝑑 𝜇 𝑑 𝑡 = 0 . ( 4 . 2 ) The linearization of Sys. (4.2) at ( 0 , 0 , 0 ) is

𝑑 𝑢 1 = 𝑑 𝑡 𝛼 𝛽 𝛾 𝑞 𝑢 1 ( 0 ) − 𝛼 𝛽 𝑢 2 ( 0 ) , 𝑑 𝑢 2 𝑑 𝑡 = 𝑘 ∗ 𝑢 1 ( − 𝜏 ) − 𝑞 𝑢 2 ( 0 ) − 𝛽 𝑢 2 ( − 𝜏 ) , 𝑑 𝜇 𝑑 𝑡 = 0 . ( 4 . 3 ) Let 𝜂 ( 𝜃 ) = 𝔸 𝛿 ( 𝜃 ) + 𝔹 𝛿 ( 𝜃 + 𝜏 ) where

⎛ ⎜ ⎜ ⎜ ⎝ 𝔸 = 𝛼 𝛽 𝛾 𝑞 ⎞ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎝ 𝑘 − 𝛼 𝛽 0 0 − 𝑞 0 0 0 0 , 𝔹 = 0 0 0 ∗ ⎞ ⎟ ⎟ ⎟ ⎠ − 𝛽 0 0 0 0 . ( 4 . 4 ) Let 𝑋 = ( 𝑢 1 , 𝑢 2 , 𝜇 ) 𝑇 and

𝐹 𝑋 𝑡 = ⎛ ⎜ ⎜ ⎜ ⎝ 𝛼 𝜇 𝑢 1 ( 0 ) + 𝛼 𝑖 ( 2 ) 𝑢 2 1 ( 0 ) + 𝛼 𝑖 ( 3 ) 𝑢 3 1 | | 𝜇 | | ( 0 ) + 𝒪 | 𝑢 | 2 + | 𝑢 | 4 𝜇 𝑢 1 ( − 𝜏 ) + 𝑖 ( 2 ) 𝑢 2 1 ( − 𝜏 ) + 𝑖 ( 3 ) 𝑢 3 1 | | 𝜇 | | ( − 𝜏 ) + 𝒪 | 𝑢 | 2 + | 𝑢 | 4 0 ⎞ ⎟ ⎟ ⎟ ⎠ . ( 4 . 5 ) Define

𝐿 𝜑 = 0 − 𝜏 𝑑 𝜂 ( 𝜃 ) 𝜑 ( 𝜃 ) , ∀ 𝜑 ∈ 𝐶 . ( 4 . 6 ) Then Sys. (4.2) becomes

̇ 𝑋 ( 𝑡 ) = 𝐿 𝑋 𝑡 𝑋 + 𝐹 𝑡 . ( 4 . 7 ) From (3.7), the bilinear form can be expressed as

⟨ 𝜓 , 𝜑 ⟩ = 𝜓 ( 0 ) 𝜑 ( 0 ) + 0 − 𝜏 𝜓 ( 𝜉 + 𝜏 ) 𝔹 𝜑 ( 𝜉 ) 𝑑 𝜉 . ( 4 . 8 ) It is not hard to see that the infinitesimal generator 𝒜 ∶ 𝐶 1 → 𝐵 𝐶 is given by

𝒜 𝜑 = ̇ 𝜑 + 𝑋 0 [ ] = 𝐿 𝜑 − ̇ 𝜑 ( 0 ) ̇ 𝜑 , − 𝜏 ≤ 𝜃 < 0 , 𝔸 𝜑 ( 0 ) + 𝔹 𝜑 ( − 𝜏 ) , 𝜃 = 0 , ( 4 . 9 ) for 𝜑 ∈ 𝐶 1 and its adjoint 𝒜 ∗ by

𝒜 ∗ 𝜓 = − ̇ 𝜓 , 0 < 𝑠