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.