4.3.1 Examples A

Perhaps the simples, nontrivial first-order differential equation is

Inspection shows that f( x) = ? x and

1. x = 0 is the only fixed-point;

2. the derivative has the property

   

3. 
   

   

All of these results imply that the fixed-point, x = 0, is both linearly and globally stable and that all nontrivial solutions approach zero monotonically.

Consider the differential equation

For this case, f( x) = x and

1. x = 0 is the unique fixed-point;

2. the derivative has the property

   

3. 
   

   

We conclude that the fixed-point, x = 0, is both linearly and globally unstable and all nontrivial solutions become unbounded.

The fact that the respective solutions to Eq. (4.3.1) and (4.3.4) are exp( ? x) and exp( x) shows the correctness of this analysis.

4.3.2 Example B

The logistic differential equation is

For this case, f( x) = x(1 ? x) and the two fixed-points are

Likewise, we have df / dx = 1 ? 2 x and

Also, the derivative has the following properties

Using this information, we can draw typical solutions; these are represented in Figure 4.3.1.

Figure 4.3.1: The logistic equation . (a) Domains of a constant sign for the derivative. (b) Typical solutions.

Note that if x(0) > 0, then all solutions approach the stable fixed-point x = 1, i.e.,

For x(0) < 0, we can only state that,...