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 1
Values 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 \(\lambda _{2}=\overline {\lambda_{1}}\).