When the
O(1)-problem has two complex valued eigenvalues (namely,
λ_{1} and
λ_{2}), we restrict ourselves to two cases for
r: 0<
r<1, or
r=1, where
r is the absolute value of the complex eigenvalues. It follows from (
14) that
where
\(c_{1}=\frac{\lambda_{1}-a_{22}}{a_{21}}\) and
\(c_{2}=\frac{\lambda _{2}-a_{22}}{a_{21}}\) are complex conjugates, and where
g_{0}(
εn) and
h_{0}(
εn) are still arbitrary functions which can be used to avoid unbounded behavior in
x_{1}(
n,
εn) and
y_{1}(
n,
εn) on the
\(O(\frac {1}{\varepsilon})\) iteration scale. Then, by substituting (
16) into the
O(
ε)-problem (
15), and after rearranging terms, one finally obtains as
O(
ε)-problem
where
M_{ij} (for
i=1,2, and
j=0,…,9) are given by
$$\lefteqn{\begin{aligned}[c]&\begin{pmatrix}M_{10}\\M_{20}\end{pmatrix}= \varepsilon \begin{pmatrix}b_{10}\\b_{20}\end{pmatrix},\\&\begin{pmatrix}M_{11}\\M_{21}\end{pmatrix}=\begin{pmatrix}\varepsilon(b_{11}c_1+b_{12})g_0-c_1\lambda_1\Delta_\varepsilon g_0(\varepsilon n)\\\varepsilon(b_{21}c_1+b_{22})g_0-\lambda_1\Delta_\varepsilon g_0(\varepsilon n)\end{pmatrix},\\&\begin{pmatrix}M_{12}\\M_{22}\end{pmatrix}=\begin{pmatrix}\varepsilon(b_{11}c_2+b_{12})h_0-c_2\lambda_2\Delta_\varepsilon h_0(\varepsilon n)\\\varepsilon(b_{21}c_2+b_{22})h_0-\lambda_2\Delta_\varepsilon h_0(\varepsilon n)\end{pmatrix},\\&\begin{pmatrix}M_{13}\\M_{23}\end{pmatrix}= \varepsilon \begin{pmatrix}b_{13}c_1^2+b_{14}c_1+b_{15}\\b_{23}c_1^2+b_{24}c_1+b_{25}\end{pmatrix}g_0^2,\\&\begin{pmatrix}M_{14}\\M_{24}\end{pmatrix}= \varepsilon \begin{pmatrix}2b_{13}c_1c_2+b_{14}(c_1+c_2)+2b_{15}\\2b_{23}c_1c_2+b_{24}(c_1+c_2)+2b_{25}\end{pmatrix}g_0h_0,\\&\begin{pmatrix}M_{15}\\M_{25}\end{pmatrix}= \varepsilon \begin{pmatrix}b_{13}c_2^2+b_{14}c_2+b_{15}\\b_{23}c_2^2+b_{24}c_2+b_{25}\end{pmatrix}h_0^2,\\&\begin{pmatrix}M_{16}\\M_{26}\end{pmatrix}= \varepsilon \begin{pmatrix}b_{16}c_1^3+b_{17}c_1^2+b_{18}c_1+b_{19}\\b_{26}c_1^3+b_{27}c_1^2+b_{28}c_1+b_{29}\end{pmatrix}g_0^3,\\&\begin{pmatrix}M_{17}\\M_{27}\end{pmatrix}= \varepsilon \begin{pmatrix}3b_{16}c_1^2c_2+b_{17}c_1(c_1+2c_2)+b_{18}(2c_1+c_2)+3b_{19}\\3b_{26}c_1^2c_2+b_{27}c_1(c_1+2c_2)+b_{28}(2c_1+c_2)+3b_{29}\end{pmatrix}g_0^2h_0,\\&\begin{pmatrix}M_{18}\\M_{28}\end{pmatrix}= \varepsilon \begin{pmatrix}3b_{16}c_1c_2^2+b_{17}c_2(2c_1+c_2)+b_{18}(c_1+2c_2)+3b_{19}\\3b_{26}c_1c_2^2+b_{27}c_2(2c_1+c_2)+b_{28}(c_1+2c_2)+3b_{29}\end{pmatrix}g_0h_0^2,\\&\begin{pmatrix}M_{19}\\M_{29}\end{pmatrix}= \varepsilon \begin{pmatrix}b_{16}c_2^3+b_{17}c_2^2+b_{18}c_2+b_{19}\\b_{26}c_2^3+b_{27}c_2^2+b_{28}c_2+b_{29}\end{pmatrix}h_0^3,\end{aligned}} $$
(18)
where
g_{0}=
g_{0}(
εn) and
h_{0}=
h_{0}(
εn). In system (
17) for
x_{1}(
n,
εn) and
y_{1}(
n,
εn), it is obvious that the right-hand side contains terms (i.e., multiples of
\((c_{1},1)^{T}\lambda_{1}^{n}\) and of
\((c_{2},1)^{T}\lambda_{2}^{n}\), where
T refers to the transpose of the matrix), which are solutions of the homogeneous system. For the case 0<
r<1, it follows from (
17) that only two vectors
\((M_{11},M_{21})^{T}\lambda_{1}^{n}\) and
\((M_{12},M_{22})^{T}\lambda_{2}^{n}\) contain secular, and also nonsecular terms. Therefore, we decompose the sum of these vectors into two linearly independent directions to separate secular and nonsecular terms,
where
M_{k} (for
k=1,…,4) are obtained based on the relationship between the eigenvalues
λ_{1} and
λ_{2}, see Table
1. From (
19), we obtain
$$\begin{cases}m_1=\frac{c_2M_2-M_1}{c_2-c_1},\\m_2=\frac{M_1-c_1M_2}{c_2-c_1},\\m_3=\frac{c_2M_4-M_3}{c_2-c_1},\\m_4=\frac{M_3-c_1M_4}{c_2-c_1}.\end{cases} $$
(20)
To avoid secular behavior in
x_{1}(
n,
εn) and
y_{1}(
n,
εn), it follows that
m_{1}=0 and
m_{4}=0, that is,
$$M_1=c_2M_2,\quad\mbox{and}\quad M_3=c_1M_4. $$
(21)
If we solve system (
21) for Δ
_{ε}g_{0}(
εn) and Δ
_{ε}h_{0}(
εn), for the case 0<
r<1 and based on Table
1, it follows that
$$\begin{cases}\Delta_\varepsilon g_0(\varepsilon n)=\varepsilon k_1g_0(\varepsilon n),\\\Delta_\varepsilon h_0(\varepsilon n)=\varepsilon k_2h_0(\varepsilon n),\end{cases} $$
(22)
where
$$\lefteqn{\begin{aligned}[c]&k_1=\frac{c_1b_{11}+b_{12}-c_1c_2b_{21}-c_2b_{22}}{\lambda _1(c_1-c_2)},\quad\mbox{and}\\& k_2=-\frac{c_2b_{11}+b_{12}-c_1c_2b_{21}-c_1b_{22}}{\lambda_2(c_1-c_2)}.\end{aligned}} $$
(23)
Since, for the complex eigenvalues,
c_{1} and
c_{2} are complex conjugates, and so
k_{1} and
k_{2} are, and since the solutions
x_{0}(
n,
εn) and
y_{0}(
εn) need to be real, it then follows that
\(h_{0}(\varepsilon n)=\overline {g_{0}(\varepsilon n)}\), where the overline refers to complex conjugates. Therefore, system (
22) can be reduced to a single equation
$$\Delta_\varepsilon g_0(\varepsilon n)=\varepsilon k_1g_0(\varepsilon n), $$
(24)
which has as a solution
$$g_0(\varepsilon n)=g_0(0) (1+\varepsilon k_1)^{n}. $$
(25)
From (
24), it then follows that if |1+
εk_{1}| is less than 1, then the equilibrium point (
x(
n,
εn),
y(
n,
εn))=(0 +
O(
ε),0+
O(
ε)) of system (
1) is an asymptotically stable focus, if |1+
εk_{1}| is bigger than 1, then the equilibrium point (
x(
n,
εn),
y(
n,
εn))=(0+
O(
ε),0+
O(
ε)) of system (
1) is an unstable focus, and if
k_{1} is zero, then the equilibrium point (
x(
n,
εn),
y(
n,
εn))=(0+
O(
ε),0+
O(
ε)) is a higher singularity and the
O(
ε^{2})-problem for
x_{2}(
n,
εn) and
x_{2}(
n,
εn) has to be studied (which is outside the scope of this paper).
Table 1Values of M_{k} (for k=1,…,4) in (19) for some representations of the absolute value of the eigenvalues λ_{1} and λ_{2}
For the case
r=1 and when the eigenvalues of matrix
A in (
3) are complex with nonzero real and imaginary parts, it follows from (
17) that only four vectors
\((M_{11},M_{21})^{T}\lambda_{1}^{n}\),
\((M_{12},M_{22})^{T}\lambda_{2}^{n}\),
\((M_{17},M_{27})^{T}\lambda_{1}^{2n}\lambda_{2}^{n}\), and
\((M_{18},M_{28})^{T}\lambda_{1}^{n}\lambda_{2}^{2n}\) contain secular, and also nonsecular terms. Therefore, we decompose the sum of these vectors into two linearly independent directions to separate secular and nonsecular terms which follow from the same equation as (
19), for the case
r=1, where
M_{k} (for
k=1,…,4) are obtained based on the relationship
λ_{1}λ_{2}=1, see Table
1. If we solve system (
21) for Δ
_{ε}g_{0}(
εn) and Δ
_{ε}h_{0}(
εn), where
\(h_{0}(\varepsilon n)=\overline{g_{0}(\varepsilon n)}\), for the case
r=1, and based on Table
1, we have again a single equation
$$\Delta_\varepsilon g_0(\varepsilon n)=\varepsilon g_0(\varepsilon n) \bigl(k_1+k_3g_0(\varepsilon n)\overline{g_0(\varepsilon n)}\, \bigr), $$
(26)
where
Consider
$$\lefteqn{\begin{aligned}[c]&g_0(\varepsilon n)=g_{0,1}(\varepsilon n)+ig_{0,2}(\varepsilon n),\\&k_1=k_{11}+ik_{12},\quad\mbox{and}\\& k_3=k_{31}+ik_{32},\end{aligned}} $$
(28)
where
g_{0,1}(
εn) and
g_{0,2}(
εn) are real functions, and where
k_{11},
k_{12},
k_{31,} and
k_{32} are real constants. Then (
26) becomes
$$\begin{cases}\Delta_\varepsilon g_{0,1}=\varepsilon\{[k_{11}+k_{31}(g_{0,1}^2+g_{0,2}^2)]g_{0,1}\\\hspace*{40pt}-[k_{12}+\,k_{32}(g_{0,1}^2+g_{0,2}^2)]g_{0,2}\},\\\Delta_\varepsilon g_{0,2}=\varepsilon\{[k_{11}+k_{31}(g_{0,1}^2+g_{0,2}^2)]g_{0,2}\\\hspace*{40pt}+\,[k_{12}+k_{32}(g_{0,1}^2+g_{0,2}^2)]g_{0,1}\},\end{cases} $$
(29)
where
g_{0,1}=
g_{0,1}(
εn) and
g_{0,2}=
g_{0,2}(
εn). As far as we know, there are no exact solutions available for system (
29). However system (
29) has always an equilibrium point in (
g_{0,1},
g_{0,2})=(0,0), and an equilibrium “circle”
$$g_{0,1}^2+g_{0,2}^2=-\frac{k_{11}k_{31}+k_{12}k_{32}}{k_{31}^2+k_{32}^2}, $$
(30)
when
k_{31}k_{12}=
k_{32}k_{11}.
For an additional analysis, we define a new variable
R, where
\(R^{2}=g_{0,1}^{2}+g_{0,2}^{2}\). After doing some computations, it follows from (
29) that
Since the dynamics of the system is mostly influenced by the
O(
ε)-problem rather than
O(
ε^{2})-problem we consider
O(
ε) terms in (
31) for the analysis. Then it follows that if
k_{11} and
k_{31} have different signs, and if
k_{11} is positive, there is a stable limit cycle, and if
k_{11} is negative then the limit cycle is unstable. To compute the radius of this limit cycle, the right-hand side of (
31) is set equal to zero, that is, Δ
_{ε}(
R^{∗}^{2})=0. To construct an approximation for
R^{∗}^{2}, one now has to substitute into the right-hand side of (
31) a formal power series (in
ε) for
R^{∗}^{2}, that is,
$${R^*}^2=R_0^*+\varepsilon R_1^*+\varepsilon^2 R_2^*+O \bigl(\varepsilon^3\bigr), $$
(32)
One solution of Δ
_{ε}(
R^{∗}^{2})=0 is the origin, that is
R^{∗}=0. To find a nontrivial approximation, by taking together those terms of equal powers in
ε, one obtains as O(1)-problem
$$k_{11}+k_{31}R_0^*=0, $$
(33)
which has as a nontrivial solution
$$R_0^*=-\frac{k_{11}}{k_{31}}, $$
(34)
when
\(-\frac{k_{11}}{k_{31}}>0\), and, as O(
ε)-problem
which has as a solution
$$R_1^*=-\frac{(k_{31}k_{12}-k_{11}k_{32})^2}{2k_{31}^3}, $$
(36)
and, as O(
ε^{2})-problem
which has as a solution
$$R_2^*=\frac{k_{32}(k_{31}k_{12}-k_{11}k_{32})^3}{2k_{31}^5}, $$
(38)
and so on. So, nontrivial equilibrium points follow from
$$\lefteqn{\begin{cases}g_{0,1}^{*2}+g_{0,2}^{*2}=R^{*2},\\(k_{11}+k_{31}{R^*}^2)g_{0,1}^*-(k_{12}+k_{32}{R^*}^2)g_{0,2}^*=0,\\(k_{11}+k_{31}{R^*}^2)g_{0,2}^*+(k_{12}+k_{32}{R^*}^2)g_{0,1}^*=0,\end{cases}} $$
(39)
where
R^{∗}^{2} is defined in (
32). Now, we expand a formal power series (in
ε) for
\(g_{0,1}^{*}\) and
\(g_{0,2}^{*}\), that is,
$$\begin{cases}g_{0,1}^*=a_0+\varepsilon a_1+\varepsilon^2a_2+O(\varepsilon^3),\\g_{0,2}^*=b_0+\varepsilon b_1+\varepsilon^2b_2+O(\varepsilon^3).\end{cases} $$
(40)
Then, we substitute (
40) into (
39) and take together those terms of equal powers in
ε to find
a_{0},
b_{0},
a_{1},
b_{1},…. One obtains as O(1)-problem
$$\begin{cases}a_0^2+b_0^2=R_0^*,\\a_0(k_{11}+k_{31}R_0^*)-b_0(k_{12}+k_{32}R_0^*)=0,\\b_0(k_{11}+k_{31}R_0^*)+a_0(k_{12}+k_{32}R_0^*)=0,\end{cases} $$
(41)
where
\(R_{0}^{*}\) is defined in (
34). It follows from (
34) that (
41) has nontrivial solutions for
a_{0} and
b_{0} if and only if
k_{11}k_{32}=
k_{12}k_{31} and
\(a_{0}^{2}+b_{0}^{2}=-\frac{k_{11}}{k_{31}}\). And, as O(
ε)-problem
$$\begin{cases}2a_0a_1+2b_0b_1=R_1^*,\\(a_0k_{31}-b_0k_{32})R_1^*+a_1(k_{11}+k_{31}R_0^*)\\\quad-\,b_1(k_{12}+k_{32}R_0^*)=0,\\(b_0k_{31}+a_0k_{32})R_1^*+b_1(k_{11}+k_{31}R_0^*)\\\quad+\,a_1(k_{12}+k_{32}R_0^*)=0,\end{cases} $$
(42)
where
\(R_{1}^{*}\) is defined in (
36). Then it follows that, if
k_{11}k_{32}=
k_{12}k_{31}, (
42) has nontrivial solution for
a_{1} and
b_{1} if and only if
a_{0}a_{1}+
b_{0}b_{1}=0.
If we define
g_{0,1}=
Rcos(
ϕ) and
g_{0,2}=
Rsin(
ϕ), after doing some computations, it follows from (
29) that
$$\Delta_\varepsilon\biggl(\frac{g_{0,2}}{g_{0,1}} \biggr)=\frac {\varepsilon(k_{12}+k_{32}R^2)R^2}{g_{0,1}(\varepsilon n)g_{0,1}(\varepsilon(n+1))}. $$
(43)
or, after expanding the right-hand side with respect to
ε, it follows that
$$\Delta_\varepsilon\biggl(\frac{g_{0,2}}{g_{0,1}} \biggr)=\frac {\varepsilon(k_{12}+k_{32}R^2)R^2}{g_{0,1}^2}\Biggl[1-\frac{\varepsilon(k_{11}g_{0,1}-k_{12}g_{0,2}+(g_{0,1}k_{31}-g_{0,2}k_{32})R^2)}{g_{0,1}}\sum_{m=0}^{\infty}\biggl(\frac{-\varepsilon}{g_{0,1}} \biggr)^m \Biggr], $$
(44)
where
g_{0,1}=
g_{0,1}(
εn) and
g_{0,2}=
g_{0,2}(
εn). It follows from (
31) and (
44) that if
k_{11}k_{32}=
k_{12}k_{31} there are infinitely many equilibrium points and as a result singularity exists, therefore, for this case we need higher order terms when we compute secular terms. But if
k_{11}k_{32}≠
k_{12}k_{31,} then the system has only one equilibrium point (which is located in the origin). And, by linearization of (
29) around origin, it follows that if |1+
εk_{1}| is less than 1, then the equilibrium point (
x(
n,
εn),
y(
n,
εn))=(0+
O(
ε),0+
O(
ε)) of system (
1) is an asymptotically stable focus, if |1+
εk_{1}| is bigger than 1, then the equilibrium point (
x(
n,
εn),
y(
n,
εn))=(0+
O(
ε),0+
O(
ε)) of system (
1) is an unstable focus, and if
k_{1} is zero, then the equilibrium point (
x(
n,
εn),
y(
n,
εn))=(0+
O(
ε),0+
O(
ε)) is a higher singularity and the
O(
ε^{2})-problem for
x_{2}(
n,
εn) and
x_{2}(
n,
εn) has to be studied (which is outside the scope of this paper). Then it also follows from (
16) that the real solutions for
x_{0}(
n,
εn) and
y_{0}(
n,
εn), for the case when
r=1, are as follows:
$$\begin{pmatrix}x_0(n,\varepsilon n)\\y_0(n,\varepsilon n)\end{pmatrix}=2R(\varepsilon n)\begin{pmatrix}\frac{1}{a_{21}}[\cos(\phi(\varepsilon n)+(n+1)\theta)-a_{22}\cos (\phi (\varepsilon n)+n\theta)]\\\cos(\phi(\varepsilon n)+n\theta)\end{pmatrix}, $$
(45)
where λ_{1}=cos(θ)+isin(θ) and \(\lambda _{2}=\overline {\lambda_{1}}\).