Two codimension-two bifurcations of a second-order difference equation from macroeconomics

In this paper we mainly investigate two codimension-two bifurcations of a second-order difference equation from macroeconomics. Applying the center manifold theorem and the normal form analysis, we firstly give the parameter conditions for the generalized flip bifurcation, and prove that the system does not produce a strong resonance. Then, we compute the normal forms to obtain the parameter conditions for the Neimark-Sacker bifurcation, from which we present the conditions for the Chenciner bifurcation. In order to verify the correctness of our results, we also numerically simulate a half stable invariant circle and two invariant circles, one stable and one unstable, arising from the Chenciner bifurcation.


1.
Introduction. The business cycle has been and continues to be an important source of significant analytical problems in economic dynamics. The theory of the business cycle aims to explain the observed and well documented fluctuations in employment, consumption, investment, total output, etc. In order to address these issues, the mathematical model is widely recognized as a convenient theoretical tool as it often helps reveal generic tendencies in some situation. One of the earliest discrete time models for the business cycle is Samuelson's business cycle model [12] Y n = cY n−1 + α(Y n−1 − Y n−2 ) + A 0 .
Here the constant A 0 = C 0 + I 0 + G 0 represents the sum of the minimum consumption, the autonomous investment and the fixed government spending, Y n is the output-national income or GDP in period n, α(Y n−1 − Y n−2 ) is the net investment amount in the same period, the coefficient α > 0 is the accelerator, and the constant c ∈ [0, 1) denotes Keynes' marginal propensity to consume. In 1997, Sedaghat [13] rigorously investigate them. For the first question, we consider q 3 as a small parameter and require only q 5 nonzero (K(p, q 2 , q 4 , q 5 ) in (2) might be zero). Using the center manifold theorem and the normal form analysis, we show that equation (1) undergoes a generalized flip bifurcation and give the parameter conditions under which the system possesses two, one and no 2-cycles respectively. See Theorems 2.1 and 2.2. In order to answer the second question, we also discuss the Neimark-Sacker bifurcation and give the first Lyapunov coefficient but without Assumption (H) and in particular q 2 = 0, q 3 = 0 in [10], from which we further present the conditions of the Chenciner bifurcation such that equation (1) has two invariant circles, one invariant circle and none separately.
This paper is organized as follows. In Section 2 we study the generalized flip bifurcation. In Section 3 we prove that equation (1) does not produce a strong resonance, and give the parameter conditions for the Neimark-Sacker bifurcation. Section 4 is devoted to the parameter conditions for the Chenciner bifurcation. In Section 5, the invariant circles arising from the Chenciner bifurcation are simulated numerically, which verifies the correctness of our results. Conclusions are drawn in Section 6. Some coefficients involved in the proofs are given in Appendix.
In this paper, we assume that the function f is analytic. Then it has the Taylor expansion where q n = f (n) (0)/n!, n = 2, 3, · · · . Translating E 0 to the origin with the transformation (w 1 + f (0)/(1 − p), w 2 + f (0)/(1 − p)), we change the mapping F into the mapping G : Since the translation transformation does not change the topological properties of system (3), i.e., the mapping G is equivalent to the mapping F , we focus on the dynamics of system (5) throughout this paper. The Jacobian matrix of the mapping G at the origin O, has eigenvalues If q = −(p + 1)/2, one can check that λ 1 = −1 and λ 2 = (p + 1)/2. As shown in [10, Theorem 1], if q 3 = 0, then system (3) undergoes the flip bifurcation at the fixed point E 0 when q crosses −(p + 1)/2. In this section, we consider the more degenerate case, that is, q 3 is near 0. For the convenience of discussion, let ε 1 = q 3 , ε 2 = q + (p + 1)/2 and ε = (ε 1 , ε 2 ), where |ε| is sufficiently small. Then we have the following results.
Going back to the original parameters, by (13), (15), (16) and (18), we obtain One can check that F + , F − and L correspond to the curves F + , F − and T respectively. The proof is completed.
We now focus on the case s = −1 corresponding to q 5 < 0 in (19). From (13), (15), (16) and (18), it follows that Using a similar proof to one of Theorem 2.1, we have the following result.
Theorem 2.2. Suppose that q 5 < 0 and |ε| is sufficiently small. Then system (3) undergoes a generalized flip bifurcation at the fixed point E 0 . More specifically, as shown in Figure 3 a flip bifurcation happens at the fixed point E 0 and an unstable 2-cycle appears.
Crossing the half line F + defined in Theorem 2.1 from the region D 2 to the region D 3 , where system (3) possesses two 2-cycles, one stable and one unstable. The two 2-cycles coincide as the parameter (ε 1 , ε 2 ) lies on the curve and disappear when (ε 1 , ε 2 ) enters the region D 1 .
In what follows, we will discuss the other codimension 2 bifurcation: the Chenciner bifurcation (or generalized Neimark-Sacker bifurcation). For this purpose, we need to discuss the resonant conditions and the Neimark-Sacker bifurcation of system (3).
3. Resonance and Neimark-Sacker bifurcation. In Theorem 2 in [10], if some coefficients q n of (4) are zero or nonzero, Li and Zhang investigated the Neimark-Sacker bifurcation as (p, q) crosses the segment L 0 = {(p, q) ∈ R 2 |q = 1, 0 ≤ p < 1}, and gave the expression of the n-th Lyapunov coefficient a 2n+1 . If a 2n+1 = 0, they obtained that equation (1) possesses j invariant circles for any 1 ≤ j ≤ n. In this section, we only focus on the existence of the first Lyapunov coefficient in order to study the Chenciner bifurcation.

3.1.
Resonance. For the convenience of our discussion, let which lies on the unit circle. If λ 0 = exp (i2πn/m), where m and n are coprime positive integers, then system (3) is resonant by the theory of the normal form for the mapping (see [18]). If 1 ≤ m ≤ 4, it is strong resonant, and if m ≥ 5, it is weak resonant. In the resonant case, we have the following results.
For p ∈ [0, 1), we have 1/2 ≤ (p + 1)/2 < 1. It follows from (23) that either 0 < n/m ≤ 1/6 or 5/6 < n/m ≤ 1. Since for p ∈ [0, 1), we obtain 0 < n/m ≤ 1/6. In this case the minimal option of m is 6 and then n = 1. Furthermore, assume that there are two positive integers m 1 and m 2 such that equation (23) holds for the same p. Then we can find two positive integers n 1 and n 2 , which satisfy that n i < m i , n i and m i are coprime, i = 1, 2, and cos(2πn 1 /m 1 ) = cos(2πn 2 /m 2 ). It follows that either n 1 /m 1 = n 2 /m 2 or n 1 /m 1 = 1 − n 2 /m 2 . In the case that n 1 /m 1 = n 2 /m 2 , we get m 1 = m 2 since n i and m i are coprime, i = 1, 2. In the other case, one can check that m 2 − n 2 and m 2 are also coprime. From the equality n 1 /m 1 = (m 2 − n 2 )/m 2 , we also conclude that m 1 = m 2 . Both cases indicate that m is determined uniquely by p. This completes the proof. Lemma 3.1 shows that equation (1) has not a strong resonance.
Theorem 4.1. Suppose that p ∈ (0, 1), q 3 is close to 2q 2 2 /3 and L > 0. Then system (3) undergoes the Chenciner bifurcation. Specifically, in a small neighborhood of ( 1 , 2 ) = (0, 0), if ( 1 , 2 ) ∈ Ω 1 (see Figure 4) where the fixed point E 0 is stable. As the parameter ( 1 , 2 ) crosses the curve from the region Ω 1 to the region system (3) undergoes the Neimark-Sacker bifurcation and produces a stable invariant cycle. Simultaneously, E 0 becomes unstable. As ( 1 , 2 ) crosses the curve from the region Ω 2 to the region Ω 3 , where system (3) possesses two invariant circles, a stable "outer" one Γ s and an unstable "inner" one Γ u . When the parameter ( 1 , 2 ) lies on the curve has a unique invariant circle Γ su , stable from the outside and unstable from the inside.
To verify that the mapping is regular, by (26), (29), (33) and (38), we obtain that Hence, for p ∈ (0, 1). The mapping in (40) is regular so that the condition (CH1) of Chenciner bifurcation given in [9, p. 405] is true. Therefore, system (5) undergoes the Chenciner bifurcation. In what follows, we discuss specifically the phenomenons of the bifurcation. Applying a parameter transformation we change system (37) into the mapping where is regarded as a function of (β 1 , β 2 ). We firstly consider that L > 0 so that c 5 (0) < 0. The case L < 0 is given later in Theorem 4.2. Then the complete bifurcation diagram of system (44) can be seen in Figure 5, which is given in [9, p. 407], where The bifurcations of system (44), given in [1] and [9], are as follows: (1) As the parameter (β 1 , β 2 ) is in the region 1 , i.e., the region system (44) has a stable fixed point O. As the parameter (β 1 , β 2 ) crosses the half line N − from the region 1 to the region 2 (the right side of β 2 -axis) and N + , system (44) undergoes the Neimark-Sacker bifurcation and a stable invariant circle appears (2) The Neimark-Sacker bifurcation happens again and an unstable invariant circle appears as the parameter (β 1 , β 2 ) crosses the half line N + from the region 2 to the region 3 , above the curve T c and the left side of β 2 -axis. Hence there are two invariant circles, stable "outer" one and unstable "inner" one, in the region 3 . (3) When the parameter (β 1 , β 2 ) lies on the curve T c , system (44) has a unique invariant circle, stable from the outside and unstable from the inside. Going back to the original parameter ( 1 , 2 ) and computing the second order derivative of the implicit function, from (41) and (43), we see that the curves N + , N − and T c correspond to N + , N − and T c given in the theorem respectively, and the regions 1 , 2 and 3 in (β 1 , β 2 )-plane correspond to the regions Ω 1 , Ω 2 and Ω 3 in ( 1 , 2 )-plane. The proof is completed.
For the case L < 0, we similarly obtain the following results.
Theorem 4.2. Assume that p ∈ (0, 1), q 3 is close to 2q 2 2 /3 and L < 0, then system (3) undergoes the Chenciner bifurcation at E 0 . Specifically, in a small neighborhood of ( 1 , 2 ) = (0, 0), if ( 1 , 2 ) ∈Ω 1 (see Figure 6), wherẽ the fixed point E 0 is unstable; As ( 1 , 2 ) crosses the half line N + given in Theorem 4.1 from the regionΩ 1 to the regioñ system (3) undergoes the Neimark-Sacker bifurcation and an unstable invariant circle appears. Simultaneously, the fixed point E 0 becomes stable; As ( 1 , 2 ) crosses the half line N − given in Theorem 4.1 from the regionΩ 2 to the regionΩ 3 , wherẽ system (3) possesses two invariant circles, an unstable "outer" one and a stable "inner" one; When the parameter ( 1 , 2 ) lies on the curvẽ has a unique invariant circle, unstable from the outside and stable from the inside. Remark 4.1. In [10], Li and Zhang required Assumption (H) in Section 1 in order to compute the n-th Lyapunov coefficient for any positive integer n. In particular, if n = 1, Assumption (H) implies that k = 3, q 2 = 0 and q 3 = 0. Our main purpose here is to analyze the Chenciner bifurcation of system (3) when the first Lyapunov coefficient is zero. Hence we do not need the above conditions.
As called in [9], the Chenciner bifurcation is also called the generalized Neimark-Sacker bifurcation. This bifurcation is similar to the Bautin bifurcation of a vector field. However, the Chenciner bifurcation possesses more complicated dynamical properties than the Bautin one. For instance, there are Arnold tongues near the 2axis such that system (3) has periodic orbits on the invariant circle as the parameter ( 1 , 2 ) lies in the tongues. When the parameter is near T c (resp.T c ), the system possesses more complicated dynamical properties such as homoclinic structure (see [1]).
5. Numerical simulations. In this section, we make numerical simulations to illustrate two invariant circles arising from the Chenciner bifurcation given in Section 4. For this purpose, we need to simulate system (3) near the fixed point E 0 when parameter ( 1 , 2 ) lies in T c and Ω 3 given in Theorem 4.1 for small | |. For convenience, choose Then L = 6125.25 given in (34), from which we obtain . They both require that 2 = q 3 − 2q 2 2 /3 = q 3 − 6 > 0. Thus, we choose q 3 = 6.1, that is, 2 = 0.1. In order to make ( 1 , 2 ) ∈ T c , we choose 1 = q − 1 ≈ −0.000175, which implies q ≈ 0.999825. Near the invariant circle Γ su , take two initial values (x 01 , y 01 ) = (0.15, 0) and (x 02 , y 02 ) = (0.071, 0), which are outside and inside the Γ su respectively. Using the mathematical software Matlab Version 7.11, after 200000 steps, we respectively simulate two orbits, the blue one approaching Γ su and the red one leaving Γ su and approaching E 0 (see Figure 7 (a)). In order to make ( 1 , 2 ) ∈ Ω 3 , we choose q = 0.99986, which implies 1 = q − 1 = −0.00014. Near the unstable invariant circle Γ u , take two initial values (x 01 , y 01 ) = (0.052553, 0) and (x 02 , y 02 ) = (0.052554, 0) which are outside the unstable invariant circle Γ u , sufficiently close to Γ u and inside Γ s respectively. Again using the Matlab, after 200000 steps, we respectively obtain two orbits, the red one leaving Γ u and approaching the stable fixed point E 0 and the green one leaving Γ u and approaching the stable invariant circle Γ s (see Figure 7 (b)). Take an initial value (0.15, 0) outside Γ s , using the same mathematical software and the same steps, we get a blue orbit, which approaches Γ s (see Figure 7 (b)).  6. Conclusions. In this paper, applying the center manifold theorem and the normal form theory, we prove that system (3) undergoes the generalized flip bifurcation, and give the parameter conditions such that the system possesses two, one and no 2-cycles. In our discussion, we require the coefficient q 5 = 0. However, if the coefficient q 5 = 0, the system may produce a codimension 3 flip bifurcation. Similarly, in the Chenciner bifurcation, if L = 0 in (34), the codimension 3 Neimark-Sacker bifurcation may occurs. Because much more complicated computations will be involved, we leave these to our future work. There is a very interesting and important question to compute the Arnold tongues in the case 2q 2 2 − 3q 3 = 0 such that the system has a periodic orbit on the invariant circle as the parameters are inside the tongues. Because the question involves very complicated computations, such as the calculation of the coefficient of the resonant termz q−1 in the complex normal form of (13), the calculation of the expression of the invariant circle in the resonant condition, the calculation of the higher derivatives of implicit functions, the discussion of the rotation number of the system restricted on the invariant circle and the discussion of the Taylor's expansion of the Hölder continuous implicit functions, we leave this to our next work.