(A.1)
Appendix A. Global stabilities of dynamical systems in Eqs. 1a-c and 9a and b.
I show here the global stabilities of dynamical systems in Eqs. 1a-c and 9a and b. When (4) is satisfied for (1), I define
|
|
(A.1) |
where 
and 
are
represented in (2). This function always has positive values for 
, 
and 
except
for 
where
the value is zero. Using (1) and (2), we can show
.
|
(A.2) |
The first term of the right-hand
side is negative because 
is
an increasing function of y, and the second term is also negative
from (4). Therefore, 
decreases
monotonically and therefore ( 
),
starting from any initial value, approaches 
where 
has
the minimum value, which means that 
is
globally stable. Thus 
is
a Liapunov function.
When the inequality sign is reversed in (4), we can constitute another Liapunov function:
|
(A.3) |
where 
and 
are
represented in (3). We can show that
|
(A.4) |
is negative. Therefore, ( 
) approaches 
where 
has the minimum value.
Next, I show the global stability
of dynamical system (9). When 
,
I define
|
(A.5) |
where 
and 
are
represented in (10). We can show that
|
(A.6) |
which is negative. Therefore, ( 
) approaches 
where 
has the minimum value.
When 
, I define
|
(A.7) |
where 
. We can show that
|
(A.8) |
which is negative. Therefore, ( 
) approaches 
where 
has the minimum value.