4.6 Bifurcations

If the differential equations modeling a particular system depend on a real parameter, ?, then it is expected that the solutions will also have a dependence on the parameter, i.e.,

A value ?* of the real parameter ? such that the qualitative properties of the trajectories in phase space change their fundamental character, as ? passes through ?*, is called a bifurcation point.

Figures 4.6.1, 4.6.2, and 4.6.3 illustrate this phenomenon. First, Figure 4.6.1, shows the behavior of solutions to the differential equation

Figure 4.6.1: Solution behaviors for the differential equations .

Figure 4.6.2: Solution behaviors for the differential equation corresponds to the fixed-points of the equation.

Figure 4.6.3: Solution behaviors for the damped linear oscillator, .

for various values of the parameter ?. For all three cases studied, the fixedpoint is x( ?) = 0. While the fixed-point does not depend on ?, its stability properties do. The three cases are:

1. ? < 0: All solutions monotonically decrease in value to zero. The fixed-point is stable.

2. ? = 0: All solutions are constant, i.e.,

   

   These solutions have neutral stability; this means that changing the initial condition changes the solution, but only by a finite amount if the change in the initial condition was finite.

3. ? > 0: All solutions are monotonically increasing and become unbounded, i.e.,

   

For this equation, it is clear that ?* = 0.

The second equation presented is

There...