$S^{4}_{in}=\left\\{(n_{1},n_{2})|n_{1}\leq
l^{2}_{in}(n_{2}),~{}n_{2}<\frac{n_{2}+2s_{2}}{2}\right\\};~{}S^{4}_{out}=\left\\{(n_{1},n_{2})|n_{1}\leq
l^{2}_{out}(n_{2}),~{}n_{2}<\frac{n_{2}+2s_{2}}{2}\right\\}.$

  3. (3)

Under Condition ($\hat{\mathbb{K}}^{4}_{1}\wedge\mathbb{H}_{3}$),

$S^{4}_{in}=\left\\{(n_{1},n_{2})|n_{1}<\frac{n_{1}+2s_{1}}{2},~{}n_{2}\leq
l^{1}_{in}(n_{1})\right\\};~{}S^{4}_{out}=\left\\{(n_{1},n_{2})|n_{1}<\frac{n_{1}+2s_{1}}{2},~{}n_{2}\leq
l^{1}_{out}(n_{1})\right\\}.$

  4. (4)

Under Condition ($\hat{\mathbb{K}}^{4}_{1}\wedge\mathbb{H}_{4}$),
$S^{4}_{in}=S^{4}_{out}=\left\\{(n_{1},n_{2})|n_{1}<\frac{n_{1}+2s_{1}}{2},~{}n_{2}<\frac{n_{2}+2s_{2}}{2}\right\\}.$

Note that $n_{i}\geq 0$ ($i=1,2$) holds all the time.

Clearly, according to Theorem 7.2, the inner and outer estimations of
attraction regions for ${P}^{4}_{1}$ under
($\hat{\mathbb{K}}^{4}_{1}\wedge\mathbb{H}_{i}$) with $i=1,2,3,4$ can be
obtained, as shown in the Figs. 9(a)–9(d).

Moreover, we will derive the boundaries of the attraction region of
${P}^{4}_{1}$ under Condition
($\hat{\mathbb{K}}^{4}_{1}\wedge\mathbb{H}_{1}$), by finding certain
corresponding separatrices located in
$\bigcup\limits_{i=1}^{3}\hat{V}^{4,i}=\hat{S}^{4}_{out}\setminus\hat{S}^{4}_{in}$
of equivalent systems (24), as shown in Fig. 17(a), where

$\displaystyle\hat{V}^{4,1}$ $\displaystyle=$
$\displaystyle\left\\{(x_{1},x_{2})|x_{1}<0,~{}\hat{l}^{1}_{in}(x_{1})<x_{2}<\hat{l}^{1}_{out}(x_{1})\right\\},$
$\displaystyle\hat{V}^{4,3}$ $\displaystyle=$
$\displaystyle\left\\{(x_{1},x_{2})|\hat{l}^{2}_{in}(x_{2})<x_{1}<\hat{l}^{2}_{out}(x_{2}),~{}x_{2}<0\right\\},$

and $\hat{V}^{4,2}=\hat{V}^{4,2}_{1}\vee\hat{V}^{4,2}_{2}$, where

$\displaystyle\hat{V}^{4,2}_{1}$ $\displaystyle=$
$\displaystyle\left\\{(x_{1},x_{2})|0<x_{1}<2s_{1},~{}\hat{l}^{1}_{in}(x_{1})<x_{2}<\hat{l}^{1}_{out}(x_{1})\right\\},$
$\displaystyle\hat{V}^{4,2}_{2}$ $\displaystyle=$
$\displaystyle\left\\{(x_{1},x_{2})|\hat{l}^{2}_{in}(x_{2})<x_{1}<\hat{l}^{2}_{out}(x_{2}),~{}0<x_{2}<2s_{2}\right\\}.$

Obviously, we can obtain a stable separatrix
$\hat{\Phi}^{1}(\hat{P}^{4}_{3})\in\hat{V}^{4,1}$ for the saddle point
$\hat{P}^{4}_{3}$ and a stable separatrix
$\hat{\Phi}^{1}(\hat{P}^{4}_{2})\in\hat{V}^{4,3}$ for the saddle point
$\hat{P}^{4}_{2}$. Moreover, we can further obtain another stable separatrix
$\hat{\Phi}^{2}(\hat{P}^{4}_{3})$ of $\hat{P}^{4}_{3}$, which satisfy
$\hat{\Phi}^{2}(\hat{P}^{4}_{3})\in\hat{V}^{4,2}_{1}$ and
$\lim_{t\to-\infty}\hat{\Phi}^{2}(\hat{P}^{4}_{3})=\hat{P}^{4}_{4}$, by
following two steps. Firstly, we can verify that the region
$\hat{V}^{4,2}_{1}$ is a negative invariant set, otherwise there will exist
contradiction with the trajectory direction at the boundary line of region
$\hat{V}^{4,2}_{1}$. Secondly, the trajectory starting from any point in
bounded region $\hat{V}^{4,2}_{1}$ will go to $\hat{P}^{4}_{4}$ as
$t\rightarrow-\infty$, since there is no closed orbit, and there exists only
one unstable node $\hat{P}^{4}_{4}$ in the bounded region (Zhang (2006)).
Similarly, we can obtain another stable separatrix
$\hat{\Phi}^{2}(\hat{P}^{4}_{2})$ for $\hat{P}^{4}_{2}$, which satisfies
$\hat{\Phi}^{2}(\hat{P}^{4}_{2})\in\hat{V}^{4,2}_{2}$ and
$\lim_{t\to-\infty}\hat{\Phi}^{2}(\hat{P}^{4}_{2})=\hat{P}^{4}_{4}$. Clearly,
these four separatrices are exactly the boundaries of the attraction region
for $\hat{P}^{4}_{1}$.

Further, letting $n_{1}(t)=x_{1}(t)+\frac{p_{1}-2s_{1}}{2}$ and
$n_{2}(t)=x_{2}(t)+\frac{p_{2}-2s_{2}}{2}$, the attraction region boundary for
$P^{4}_{1}$ under Condition ($\hat{\mathbb{K}}^{4}_{1}\wedge\mathbb{H}_{1}$)
can be obtained. Denoting $\Phi^{i}(P^{4}_{2})$ ($i=1,2$) to be the two stable
separatrices of $P^{4}_{2}$, and $\Phi^{j}(P^{4}_{3})$ ($j=1,2$) to be the two
stable separatrices of $P^{4}_{3}$, the below theorem can be arrived.

###### Theorem 7.3

Under Condition ($\hat{\mathbb{K}}^{4}_{1}\wedge\mathbb{H}_{1}$), for the
local stable point $P^{4}_{1}$, we have $S^{4}_{A}=\mathcal{R}(P^{4}_{1})$,
where

$S^{4}_{A}:=\left\\{(n_{1},n_{2})|n_{i}<g^{4}_{j}(n_{j}),\ i,j=1,2,~{}j\neq
i\right\\}$

satisfying $l^{j}_{in}(n_{j})<g^{4}_{j}(n_{j})<l^{j}_{out}(n_{j})$, and
$g^{4}_{j}(n_{j}),\ j=1,2$ are defined as:

$g^{4}_{1}(n_{1})=\begin{cases}\Phi^{1}(P^{4}_{3})&\text{$n_{1}<\frac{p_{1}-2s_{1}}{2}$}\\\
\Phi^{2}(P^{4}_{3})&\text{$\frac{p_{1}-2s_{1}}{2}<n_{1}<\frac{p_{1}+2s_{1}}{2}$}\end{cases}$

and

$g^{4}_{2}(n_{2})=\begin{cases}\Phi^{1}(P^{4}_{2})&\text{$n_{2}<\frac{p_{2}-2s_{2}}{2}$}\\\
\Phi^{2}(P^{4}_{2})&\text{$\frac{p_{2}-2s_{2}}{2}<n_{2}<\frac{p_{2}+2s_{2}}{2}$}\end{cases}$

with $\lim_{t\to-\infty}\Phi^{2}(P^{4}_{2})=P^{4}_{4}$ and
$\lim_{t\to-\infty}\Phi^{2}(P^{4}_{3})=P^{4}_{4}$. Note that $n_{i}\geq 0$
($i=1,2$) holds all the time.

In summary, the attraction region of $P^{4}_{1}$ under corresponding
conditions are plotted as light pink zones in the Figs. 9(a)-9(d).

Secondly, we consider Condition ($\hat{\mathbb{K}}^{4}_{2}\wedge\mathbb{H}$),
which can be partitioned into two sub-Conditions:
($\hat{\mathbb{K}}^{4}_{2}\wedge\mathbb{H}_{1}$) and
($\hat{\mathbb{K}}^{4}_{2}\wedge\mathbb{H}_{2}$). Note that
$\hat{\mathbb{K}}^{4}_{2}\wedge\mathbb{H}_{3}=\varnothing$ and
$\hat{\mathbb{K}}^{4}_{2}\wedge\mathbb{H}_{4}=\varnothing$. By repeating the
analysis as above, the corresponding phase portraits of (24) under these two
sub-conditions can be obtained; and further prove that there is no close orbit
for system (24) under above two sub-conditions. In addition, we can obtain the
inner and outer estimations of attraction regions for $\hat{P}^{4}_{1}$, and
deduce the boundary of attraction region for $\hat{P}^{4}_{1}$, as shown in
the Figs. 18(a) and 18(b). Further, letting
$n_{1}(t)=x_{1}(t)+\frac{p_{1}-2s_{1}}{2}$ and
$n_{2}(t)=x_{2}(t)+\frac{p_{2}-2s_{2}}{2}$, we can obtain the inner and outer
estimations of attraction regions for ${P}^{4}_{1}$ which are defined in
Theorem 7.2 (1), and the attraction region boundary for $P^{4}_{1}$ under
Conditions ($\hat{\mathbb{K}}^{4}_{2}\wedge\mathbb{H}_{1}$) and
($\hat{\mathbb{K}}^{4}_{2}\wedge\mathbb{H}_{2}$), as shown in Figs. 9(e) and
9(f).

(a) $\hat{\mathbb{K}}^{4}_{2}\wedge\mathbb{H}_{1}$

(b) $\hat{\mathbb{K}}^{4}_{2}\wedge\mathbb{H}_{2}$

(c) $\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}_{1}$

(d) $\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}_{3}$

Figure 18: Qualitative characteristics of system (24) under corresponding
conditions.

Thirdly, we analyze the phase portrait and the attraction region of
$P^{4}_{1}$ under Condition ($\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}$),
which can be partitioned into two sub-Conditions:
($\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}_{1}$) and
($\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}_{3}$). Note that
$\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}_{2}=\varnothing$ and
$\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}_{4}=\varnothing$. Notice that
Condition ($\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}$) is symmetric to
Condition ($\hat{\mathbb{K}}^{4}_{2}\wedge\mathbb{H}$), so the analysis
process and results are similar. Thus, we can obtain the inner and outer
estimations of attraction regions for $\hat{P}^{4}_{1}$ in a similar way, as
shown in Figs. 18(c) and 18(d). Further, letting
$n_{1}(t)=x_{1}(t)+\frac{p_{1}-2s_{1}}{2}$ and
$n_{2}(t)=x_{2}(t)+\frac{p_{2}-2s_{2}}{2}$, the attraction region of
$P^{4}_{1}$ under Conditions ($\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}_{1}$)
and ($\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}_{3}$) can be obtained, as shown
in Figs. 9(g) and 9(h).

Fourthly, we consider Condition ($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}$),
which can be partitioned into two sub-Conditions: Condition
($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1}$) and
($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{2}$). Note that
$\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{3}=\varnothing$ and
$\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{4}=\varnothing$.

We first consider Condition ($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1}$).
For system (24) under Condition
($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1}$), following the steps as
Condition ($\hat{\mathbb{K}}^{4}_{1}\wedge\mathbb{H}_{1}$), the global phase
portrait can be obtained, as shown in Fig. 19(a); further prove that there
exist no close orbits; and subsequently derive the inner estimation of
attraction region $\hat{S}^{4,1}_{in}=\left\\{(x_{1},x_{2})|x_{1}\leq
2s_{1},~{}x_{2}\leq\hat{l}^{1,1}_{in}(x_{1})\right\\}$ and the outer
estimation of attraction region
$\hat{S}^{4,1}_{out}=\mathbb{R}^{2}\setminus\hat{U}^{4,1}=\left\\{(x_{1},x_{2})|x_{1}\in\mathbb{R},~{}x_{2}<\hat{l}^{1,1}_{out}(x_{1})\right\\}$,
where

$\hat{l}^{1,1}_{in}(x_{1})=\begin{cases}-\sqrt{\frac{M_{1}}{u_{2}a_{2}}}+s_{2}&\text{$x_{1}<s_{1}$}\\\
-\sqrt{\frac{a_{1}}{u_{2}a_{2}}\left(x_{1}-s_{1}\right)^{2}+\frac{M_{1}}{u_{2}a_{2}}}+s_{2}&\text{$s_{1}<x_{1}<2s_{1}$}\end{cases}$

and

$\hat{l}^{1,1}_{out}(x_{1})=\begin{cases}\sqrt{\frac{u_{1}a_{1}}{a_{2}}\left(x_{1}-s_{1}\right)^{2}-\frac{M_{2}}{a_{2}}}+s_{2}&\text{$x_{1}<0$}\\\
2s_{2}&\text{$0<x_{1}<2s_{1}$}\\\
-\sqrt{\frac{a_{1}}{u_{2}a_{2}}\left(x_{1}-s_{1}\right)^{2}+\frac{M_{1}}{u_{2}a_{2}}}+s_{2}&\text{$x_{1}>2s_{1}$}\end{cases}$

(a) $\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}_{1}$

(b) $\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}_{1}$

Figure 19: Phase diagram (a) and qualitative characteristics (b) of system
(24) under Condition ($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1}$).

Thus, we will next deduce the boundaries of the attraction region of
${P}^{4}_{1}$ by finding certain corresponding separatrices located in
$\bigcup\limits_{i=3}^{4}\hat{V}^{4,i}=\hat{S}^{4,1}_{out}\setminus\hat{S}^{4,1}_{in}$
of equivalent systems (24), as shown in Fig. 19(b), where
$\hat{V}^{4,4}=\left\\{(x_{1},x_{2})|x_{1}<2s_{1},~{}\hat{l}^{1,1}_{in}(x_{1})<x_{2}<\hat{l}^{1,1}_{out}(x_{1})\right\\}$
and
$\hat{V}^{4,3}=\left\\{(x_{1},x_{2})|x_{1}>2s_{1},~{}x_{2}<\hat{l}^{1,1}_{out}(x_{1})\right\\}$.
By similar analysis as above, we can obtain a stable separatrix
$\hat{\Phi}^{1}(\hat{P}^{4}_{3})\in\hat{V}^{4,1}$ for the saddle point
$\hat{P}^{4}_{3}$; a stable separatrix
$\hat{\Phi}^{1}(\hat{P}^{4}_{2})\in\hat{V}^{4,3}$ for the saddle point
$\hat{P}^{4}_{2}$; and another stable separatrix
$\hat{\Phi}^{2}(\hat{P}^{4}_{3})$ of $\hat{P}^{4}_{3}$, which satisfy
$\hat{\Phi}^{2}(\hat{P}^{4}_{3})\in\hat{V}^{4,2}_{1}$ and
$\lim_{t\to-\infty}\hat{\Phi}^{2}(\hat{P}^{4}_{3})=\hat{P}^{4}_{4}$. Note that
$\hat{V}^{4,1}$ and $\hat{V}^{4,2}_{1}$ are defined above. As for another
stable separatrix for $\hat{P}^{4}_{2}$ in $\hat{V}^{4,4}$, there are three
types, where in type 1,
$\lim_{t\to-\infty}\hat{\Phi}^{2}(\hat{P}^{4}_{2})=-\infty$; in type 2,
$\lim_{t\to-\infty}\hat{\Phi}^{2}(\hat{P}^{4}_{2})=\hat{P}^{4}_{3}$; and in
type 3, $\lim_{t\to-\infty}\hat{\Phi}^{2}(\hat{P}^{4}_{2})=\hat{P}^{4}_{4}$;
as shown in Figs. 20(a), 20(b) and 20(c), respectively.

(a) $\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1}$, type 1

(b) $\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1}$, type 2

(c) $\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1}$, type 3

Figure 20: Three types of qualitative characteristics of system (24) ((a)-(c))
and system (1) ((d)-(f)) under Condition
($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1}$).

Further, we can refine the inner and outer estimations of attraction regions
under each type. Specifically, In type 1, the outer estimation of attraction
region can be refined as
$\hat{S}^{4,2}_{out}=\left\\{(x_{1},x_{2})|x_{1}\in\mathbb{R},~{}x_{2}<\hat{l}^{1,2}_{out}(x_{1})\right\\}$,
where

$\hat{l}^{1,2}_{out}(x_{1})=\begin{cases}\sqrt{\frac{u_{1}a_{1}}{a_{2}}\left(x_{1}-s_{1}\right)^{2}-\frac{M_{2}}{a_{2}}}+s_{2}&\text{$x_{1}<0$}\\\
\sqrt{\frac{a_{1}}{u_{2}a_{2}}\left(x_{1}-s_{1}\right)^{2}+\frac{M_{1}}{u_{2}a_{2}}}+s_{2}&\text{$0<x_{1}<s_{1}$}\\\
\sqrt{\frac{M_{1}}{u_{2}a_{2}}}+s_{2}&\text{$s_{1}<x_{1}<2s_{1}$}\\\
-\sqrt{\frac{a_{1}}{u_{2}a_{2}}\left(x_{1}-s_{1}\right)^{2}+\frac{M_{1}}{u_{2}a_{2}}}+s_{2}&\text{$x_{1}>2s_{1}$}\end{cases}$

since the region
$\hat{V}^{4,4}_{1}=\hat{S}^{4,2}_{out}\setminus\hat{S}^{4,1}_{out}$ is a
negative invariant set and the trajectory starting from any point in bounded
region $\hat{V}^{4,4}_{1}$ will go to $\hat{P}^{4}_{4}$ as
$t\rightarrow-\infty$. In type 2, the outer estimation of attraction region
can also be refined as $\hat{S}^{4,2}_{out}$ for the same reason. Moreover,
the inner estimation of attraction region can be refined as
$\hat{S}^{4,2}_{in}=\left\\{(x_{1},x_{2})|x_{1}\in\mathbb{R},~{}x_{2}<\hat{l}^{1,2}_{in}(x_{1})\right\\}$,
where

$\hat{l}^{1,2}_{in}(x_{1})=\begin{cases}2s_{2}&\text{$x_{1}<0$}\\\
-\sqrt{\frac{M_{1}}{u_{2}a_{2}}}+s_{2}&\text{$0<x_{1}<s_{1}$}\\\
-\sqrt{\frac{a_{1}}{u_{2}a_{2}}\left(x_{1}-s_{1}\right)^{2}+\frac{M_{1}}{u_{2}a_{2}}}+s_{2}&\text{$s_{1}<x_{1}<2s_{1}$}\end{cases}$

since any trajectory starting from the region
$\hat{V}^{4,4}_{2}=\hat{S}^{4,1}_{in}\setminus\hat{S}^{4,2}_{in}$ will also go
to $\hat{P}^{4}_{1}$ as $t\rightarrow+\infty$. In type 3, the inner estimation
of attraction region can be refined as $\hat{S}^{4,2}_{in}$ for the same
reason.

In summary, the refined inner and outer estimations of attraction regions in
each type are obtained, as shown in Figs. 20(a), 20(b) and 20(c). Then,
letting $n_{1}(t)=x_{1}(t)+\frac{p_{1}-2s_{1}}{2}$ and
$n_{2}(t)=x_{2}(t)+\frac{p_{2}-2s_{2}}{2}$, we can obtain the attraction
region, inner and outer estimations of attraction regions in each type of
system (1) under Conditions ($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1}$),
which are shown in Fig. 20. Moreover, denoting
$l^{1,i}_{out}(n_{1})=\hat{l}^{1,i}_{out}(n_{1}-\frac{p_{1}-2s_{1}}{2})+\frac{p_{2}-2s_{2}}{2}$
and
$l^{1,i}_{in}(n_{1})=\hat{l}^{1,i}_{in}(n_{1}-\frac{p_{1}-2s_{1}}{2})+\frac{p_{2}-2s_{2}}{2}$,
where $i=1,2$, the below theorem can be arrived.

###### Theorem 7.4

Under Condition ($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1}$), for the
local stable point $P^{4}_{1}$, we have $S^{4}_{in}\subset
S^{4}_{A}=\mathcal{R}(P^{4}_{1})\subset S^{4}_{out}$, where $S^{4}_{A}$, the
inner estimation of attraction region $S^{4}_{in}$ and outer estimation of
attraction region $S^{4}_{out}$ have three types, which are defined as
follows:

  1. (1)

In type 1: $S^{4}_{A}=\left\\{(n_{1},n_{2})|n_{1}\leq{p_{1}},\
~{}n_{2}<g^{4}_{3}(n_{1})\right\\}$, where $g^{4}_{3}(n_{1})$ represents
$\Phi^{1}(P^{4}_{2})$ when $n_{1}<\frac{p_{1}+2s_{1}}{2}$ and
$\Phi^{2}(P^{4}_{2})$ when $\frac{p_{1}+2s_{1}}{2}<n_{1}\leq{p_{1}}$;

${S}^{4}_{in}=\left\\{(n_{1},n_{2})|n_{1}\leq{p_{1}},\
~{}n_{2}<{l}^{1,1}_{in}(n_{1})\right\\};~{}{S}^{4}_{out}=\left\\{(n_{1},n_{2})|n_{1}\leq{p_{1}},\
~{}n_{2}<{l}^{1,2}_{out}(n_{1})\right\\}.$

  2. (2)

In type 2: $S^{4}_{A}=\left\\{(n_{1},n_{2})|n_{1}\leq{p_{1}},\
~{}n_{2}<g^{4}_{4}(n_{1})\right\\}$, where $g^{4}_{4}(n_{1})$ represents
$\Phi^{1}(P^{4}_{3})$ when $n_{1}<\frac{p_{1}-2s_{1}}{2}$, represents
$\Phi^{2}(P^{4}_{2})$ when
$\frac{p_{1}-2s_{1}}{2}<n_{1}<\frac{p_{1}+2s_{1}}{2}$, and represents
$\Phi^{1}(P^{4}_{2})$ when $\frac{p_{1}+2s_{1}}{2}<n_{1}\leq{p_{1}}$;

${S}^{4}_{in}=\left\\{(n_{1},n_{2})|n_{1}\leq{p_{1}},\
~{}n_{2}<{l}^{1,2}_{in}(n_{1})\right\\};~{}{S}^{4}_{out}=\left\\{(n_{1},n_{2})|n_{1}\leq{p_{1}},\
~{}n_{2}<{l}^{1,2}_{out}(n_{1})\right\\}.$

  3. (3)

In type 3: $S^{4}_{A}$ is defined in Theorem 7.3;

${S}^{4}_{in}=\left\\{(n_{1},n_{2})|n_{1}\leq{p_{1}},\
~{}n_{2}<{l}^{1,2}_{in}(n_{1})\right\\};~{}{S}^{4}_{out}=\left\\{(n_{1},n_{2})|n_{1}\leq{p_{1}},\
~{}n_{2}<{l}^{1,1}_{out}(n_{1})\right\\}.$

Note that $n_{i}\geq 0$ ($i=1,2$) holds all the time.

Then, for Conditions ($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{2}$), we can
also simplify the system (24) and then perform the above analysis. In a
similar way, we can obtain the phase portrait for system (24) and further
verify that there are no close orbits under Conditions
($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{2}$). Thus, we can obtain three
types of attraction regions, and further obtain three types of refined the
inner and outer estimations of attraction regions under each type, as shown in
Figs. 21(a), 21(b) and 21(c). Then letting
$n_{1}(t)=x_{1}(t)+\frac{p_{1}-2s_{1}}{2}$ and
$n_{2}(t)=x_{2}(t)+\frac{p_{2}-2s_{2}}{2}$, we can obtain three types of
attraction regions of (1) under Conditions
($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{2}$), as shown in Figs. 9(j), 9(n)
and 9(r).

(a) $\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{2}$, type 1

(b) $\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{2}$, type 2

(c) $\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{2}$, type 3

(d) $\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{1}$, type 1

(e) $\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{1}$, type 2

(f) $\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{1}$, type 3

(g) $\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{3}$, type 1

(h) $\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{3}$, type 2

(i) $\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{3}$, type 3

Figure 21: Three types of qualitative characteristics of system (24) under
corresponding condition.

Finally, we analyze the phase portrait and the attraction region of
$P^{4}_{1}$ under Condition ($\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}$),
which can be partitioned into two sub-Conditions:
($\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{1}$) and
($\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{3}$). Note that
$\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{2}=\varnothing$ and
$\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{4}=\varnothing$. Notice that
Condition ($\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}$) is symmetric to
Condition ($\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}$), so the analysis
process and results are similar. By repeating the analysis as above, we can
get the refined results under Conditions
($\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{1}$) and
($\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{3}$), as shown in Figs. 21 and 9.

##  8 Numerical verification of the Global phase portrait and attraction
region

(a) $\mathbb{K}^{2a}\wedge\mathbb{H}_{1}$

(b) $\mathbb{K}^{2a}\wedge\mathbb{H}_{2}$

(c) $\mathbb{K}^{2a}\wedge\mathbb{H}_{3}$

(d) $\mathbb{K}^{2a}\wedge\mathbb{H}_{4}$

(e) $\mathbb{K}^{2b}\wedge\mathbb{H}_{1}$

(f) $\mathbb{K}^{2b}\wedge\mathbb{H}_{2}$

(g) $\mathbb{K}^{2b}\wedge\mathbb{H}_{3}$

(h) $\mathbb{K}^{2b}\wedge\mathbb{H}_{4}$

Figure 22:  Numerical simulation phase portraits for system (1) under
Condition under Conditions $(\mathbb{K}^{2a})$ and $(\mathbb{K}^{2b})$. Red
circle: undesired saddle-node equilibrium. Blue circle: desired uncongested
saddle-node equilibrium. Green line with arrows: the trajectories. Purple line
and light blue line represent the inner and outer estimations of attraction
regions, respectively. The yellow lines represent the numerical attraction
region boundaries, solid yellow lines indicating the boundary line belongs to
attraction region, while dashed yellow lines indicating the boundary line does
not belong to attraction region. The agreement of this figure with Figs. 7 and
Fig. 8 validate the theoretical analysis.

(a) $\mathbb{K}^{4}_{1}\wedge\mathbb{H}_{1}$

(b) $\mathbb{K}^{4}_{1}\wedge\mathbb{H}_{2}$

(c) $\mathbb{K}^{4}_{1}\wedge\mathbb{H}_{3}$

(d) $\mathbb{K}^{4}_{1}\wedge\mathbb{H}_{4}$

(e) $\mathbb{K}^{4}_{2}\wedge\mathbb{H}_{1}$

(f) $\mathbb{K}^{4}_{2}\wedge\mathbb{H}_{2}$

(g) $\mathbb{K}^{4}_{3}\wedge\mathbb{H}_{1}$

(h) $\mathbb{K}^{4}_{3}\wedge\mathbb{H}_{3}$

(i) $\mathbb{K}^{4}_{4}\wedge\mathbb{H}_{1}$, type 1

(j) $\mathbb{K}^{4}_{4}\wedge\mathbb{H}_{2}$, type 1

(k) $\mathbb{K}^{4}_{5}\wedge\mathbb{H}_{1}$, type 1

(l) $\mathbb{K}^{4}_{5}\wedge\mathbb{H}_{3}$, type 1

(m) $\mathbb{K}^{4}_{4}\wedge\mathbb{H}_{1}$, type 2

(n) $\mathbb{K}^{4}_{4}\wedge\mathbb{H}_{2}$, type 2

(o) $\mathbb{K}^{4}_{5}\wedge\mathbb{H}_{1}$, type 2

(p) $\mathbb{K}^{4}_{5}\wedge\mathbb{H}_{3}$, type 2

(q) $\mathbb{K}^{4}_{4}\wedge\mathbb{H}_{1}$, type 3

(r) $\mathbb{K}^{4}_{4}\wedge\mathbb{H}_{2}$, type 3

(s) $\mathbb{K}^{4}_{5}\wedge\mathbb{H}_{1}$, type 3

(t) $\mathbb{K}^{4}_{5}\wedge\mathbb{H}_{3}$, type 3

Figure 23: Numerical simulation phase portraits for system (1) under Condition
$(\mathbb{K}^{4})$. Green circle: desired uncongested stable equilibrium.
Yellow circle: saddle equilibrium. Yellow-brown circle: unstable equilibrium.
Green line with arrows: the trajectories. Purple line and light blue line
represent the inner and outer estimations of attraction regions, respectively.
The yellow lines represent the numerical attraction region boundaries, solid
yellow lines indicating the boundary line belongs to attraction region, while
dashed yellow lines indicating the boundary line does not belong to attraction
region. The agreement of this figure with Fig. 9 validate the theoretical
analysis. Table 4: Parameter values for different scenarios

Scenarios | Flow rates | Network-outside demands $[veh/h]$ | Satisfying Condition | Corresponding phase portrait
---|---|---|---|---
Scenario 1 | $u_{1}=0.3$ | $u_{2}=0.4$ | $d_{1}=3\times 10^{4}$ | $d_{2}=5\times 10^{4}$ | Condition $(\mathbb{K}^{2a}\wedge\mathbb{H}_{1})$ | Fig. 22(a)
Scenario 2 | $u_{1}=0$ | $u_{2}=0.8$ | $d_{1}=3\times 10^{4}$ | $d_{2}=5\times 10^{4}$ | Condition $(\mathbb{K}^{2a}\wedge\mathbb{H}_{2})$ | Fig. 22(b)
Scenario 3 | $u_{1}=0.4$ | $u_{2}=0$ | $d_{1}=7\times 10^{4}$ | $d_{2}=5\times 10^{4}$ | Condition $(\mathbb{K}^{2a}\wedge\mathbb{H}_{3})$ | Fig. 22(c)
Scenario 4 | $u_{1}=0$ | $u_{2}=0$ | $d_{1}=7\times 10^{4}$ | $d_{2}=5\times 10^{4}$ | Condition $(\mathbb{K}^{2a}\wedge\mathbb{H}_{4})$ | Fig. 22(d)
Scenario 5 | $u_{1}=0.4386$ | $u_{2}=0.48$ | $d_{1}=3\times 10^{4}$ | $d_{2}=5\times 10^{4}$ | Condition $(\mathbb{K}^{2b}\wedge\mathbb{H}_{1})$ | Fig. 22(e)
Scenario 6 | $u_{1}=0$ | $u_{2}=0.4$ | $d_{1}=3\times 10^{4}$ | $d_{2}=8\times 10^{4}$ | Condition $(\mathbb{K}^{2b}\wedge\mathbb{H}_{2})$ | Fig. 22(f)
Scenario 7 | $u_{1}=0.8$ | $u_{2}=0$ | $d_{1}=0.5\times 10^{4}$ | $d_{2}=8\times 10^{4}$ | Condition $(\mathbb{K}^{2b}\wedge\mathbb{H}_{3})$ | Fig. 22(g)
Scenario 8 | $u_{1}=0$ | $u_{2}=0.8$ | $d_{1}=6\times 10^{4}$ | $d_{2}=8\times 10^{4}$ | Condition $(\mathbb{K}^{2b}\wedge\mathbb{H}_{4})$ | Fig. 22(h)
Scenario 9 | $u_{1}=0.4$ | $u_{2}=0.4$ | $d_{1}=3\times 10^{4}$ | $d_{2}=5\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{1}\wedge\mathbb{H}_{1})$ | Fig. 23(a)
Scenario 10 | $u_{1}=0$ | $u_{2}=0.8$ | $d_{1}=5\times 10^{4}$ | $d_{2}=1\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{1}\wedge\mathbb{H}_{2})$ | Fig. 23(b)
Scenario 11 | $u_{1}=0.8$ | $u_{2}=0$ | $d_{1}=0.2\times 10^{4}$ | $d_{2}=5\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{1}\wedge\mathbb{H}_{3})$ | Fig. 23(c)
Scenario 12 | $u_{1}=0$ | $u_{2}=0$ | $d_{1}=4\times 10^{4}$ | $d_{2}=3\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{1}\wedge\mathbb{H}_{4})$ | Fig. 23(d)
Scenario 13 | $u_{1}=0.8$ | $u_{2}=0.8$ | $d_{1}=0.6\times 10^{4}$ | $d_{2}=2\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{2}\wedge\mathbb{H}_{1})$ | Fig. 23(e)
Scenario 14 | $u_{1}=0$ | $u_{2}=0.8$ | $d_{1}=0.6\times 10^{4}$ | $d_{2}=3\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{2}\wedge\mathbb{H}_{2})$ | Fig. 23(f)
Scenario 15 | $u_{1}=0.8$ | $u_{2}=0.8$ | $d_{1}=0.3\times 10^{4}$ | $d_{2}=2.4\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{3}\wedge\mathbb{H}_{1})$ | Fig. 23(g)
Scenario 16 | $u_{1}=0.8$ | $u_{2}=0$ | $d_{1}=4\times 10^{4}$ | $d_{2}=2.4\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{1}\wedge\mathbb{H}_{3})$ | Fig. 23(h)
Scenario 17 | $u_{1}=0.8$ | $u_{2}=0.8$ | $d_{1}=2\times 10^{4}$ | $d_{2}=0.5\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1})$ | Fig. 23(i)
Scenario 18 | $u_{1}=0.8$ | $u_{2}=0.8$ | $d_{1}=0.78\times 10^{4}$ | $d_{2}=1.99\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1})$ | Fig. 23(m)
Scenario 19 | $u_{1}=0.8$ | $u_{2}=0.8$ | $d_{1}=0.72\times 10^{4}$ | $d_{2}=2\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{1})$ | Fig. 23(q)
Scenario 20 | $u_{1}=0$ | $u_{2}=0.8$ | $d_{1}=1.5\times 10^{4}$ | $d_{2}=6\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{2})$ | Fig. 23(j)
Scenario 21 | $u_{1}=0$ | $u_{2}=0.8$ | $d_{1}=1.5\times 10^{4}$ | $d_{2}=5.6\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{2})$ | Fig. 23(n)
Scenario 22 | $u_{1}=0$ | $u_{2}=0.8$ | $d_{1}=1.5\times 10^{4}$ | $d_{2}=5\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{4}\wedge\mathbb{H}_{2})$ | Fig. 23(r)
Scenario 23 | $u_{1}=0.8$ | $u_{2}=0.8$ | $d_{1}=0.02\times 10^{4}$ | $d_{2}=2.85\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{1})$ | Fig. 23(k)
Scenario 24 | $u_{1}=0.8$ | $u_{2}=0.8$ | $d_{1}=0.02\times 10^{4}$ | $d_{2}=2.82\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{1})$ | Fig. 23(o)
Scenario 25 | $u_{1}=0$ | $u_{2}=0.8$ | $d_{1}=0.02\times 10^{4}$ | $d_{2}=2.6\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{1})$ | Fig. 23(s)
Scenario 26 | $u_{1}=0.8$ | $u_{2}=0$ | $d_{1}=5\times 10^{4}$ | $d_{2}=3.9\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{3})$ | Fig. 23(l)
Scenario 27 | $u_{1}=0.8$ | $u_{2}=0$ | $d_{1}=5\times 10^{4}$ | $d_{2}=3.78\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{3})$ | Fig. 23(p)
Scenario 28 | $u_{1}=0.8$ | $u_{2}=0$ | $d_{1}=5\times 10^{4}$ | $d_{2}=3\times 10^{4}$ | Condition $(\hat{\mathbb{K}}^{4}_{5}\wedge\mathbb{H}_{3})$ | Fig. 23(t)

  * •

Note that these are all the scenarios used in this paper. In each scenario,
the different formula of the inner and outer estimations of attraction regions
are derived (see in Sec. 3), and we have plotted the phase portraits in each
scenario, as shown in Figs. 22 and 23.

Figure 24: Vehicle accumulations (dotted line) and completion flow rates
(solid line) of Region 1 (((a), (b) and (c))) and Region 2 ((d), (e) and (f)).
The numerical diagrams of deviation from maximum complete flow $-D_{s}$ ((g),
(h) and (i)) evolving with time $t$ under CPC, RCS-1, RCS-2 and RCS-single.
Scenario 9 is considered. Note that for (a)-(f), we employ dual y-axes. The
left y-axis represents vehicle accumulation (density), while the right y-axis
represents completion flow. Scenario 9 is considered. Here we consider case 2
(in Fig. 15 we consider case 1). Other data settings are the same as Fig. 15.
Figure 25: Vehicle accumulations (dotted line) and completion flow rates
(solid line) of Region 1 (((a), (b) and (c))) and Region 2 ((d), (e) and (f)).
The numerical diagrams of deviation from maximum complete flow $-D_{s}$ ((g),
(h) and (i)) evolving with time $t$ under CPC, RCS-1, RCS-2 and RCS-single.
Scenario 9 is considered. Note that for (a)-(f), we employ dual y-axes. The
left y-axis represents vehicle accumulation (density), while the right y-axis
represents completion flow. Scenario 9 is considered. Here we consider case 3
(in Fig. 15 we consider case 1). Other data settings are the same as Fig. 15.