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L. Zhao et al.

The characteristic equation of the system (4) is
系统(4)的特征方程为
Equation (5) is equivalent to
等式(5)等价于
The following two cases are discussed according to Eq. (6):
根据式(6)讨论以下两种情况:
(i) . (一) .
(ii) . (二) .

4.1. Bifurcation caused by
4.1. 分岔引起的

For (i), it is only necessary to consider , suppose that is one of its roots. Separating the real and imaginary parts, then it follows that
对于(i),只需要考虑 ,假设这是 它的根之一。将真实部分和虚部分分开,然后得出
Combined with the expression of
结合表达
Accordingly 因此
and
.
It is found from Eq. (8) that
从式(8)中可以看出,
where 哪里
If the values of in system (3) are given, substitute Eq. (10) of in Eq. (8) and the value of can be calculated from by Maple software. Finally, the value of can be obtained by Eq. (11).
如果给出了系统(3)中的值 ,则将方程(10) 代入方程(8),则可以通过Maple软件计算 出的值 。最后,可以通过方程(11)得到的 值。
Define 定义
For (ii), take into account 1) . Let be one of its roots, thus
对于(ii),考虑 1)。 让我们 成为它的根源之一,因此
In view of Eq. (12)
根据式(12)
Hence 因此
and
From Eq. (13) 由式 (13)
where 哪里
If the values of in system (3) are given, use the same method as (i), the value of can be obtained by Eq. (13).
如果给出了系统(3)中的值 ,则使用与(i)相同的方法,可以通过方程(13)获得的 值。
Let
Next, define the first bifurcation point as
接下来,将第一个分岔点定义为
The following assumption is given hereinafter:
以下给出以下假设:
(H1) or .
(H1) .
Lemma 1. Suppose that is the root of Eq. (6) near meeting , , then the following transversality condition holds
引理 1.假设这是 方程(6)的根 以下横向条件成立
Proof. For (i), differentiating both sides of with respect to , and bring in the expression of , it can be deduced that
证明。对于(i),将 的两侧 差,并引入 的 表达式,可以推导出
where 哪里
From Eq. (23), after direct calculation, the following result can be obtained
从式(23)直接计算后,可得到以下结果
where 哪里