Abstract

CHAPTER 3. BIFURCATION ANALYSIS 47
constant function, the plot shows what this constant value is as α changes. The
Steady−State Solution (w)
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
Unstable
Stable
Turning Point Bifurcation
−1
−1 −0.8 −0.6 −0.4 −0.2 0
Parameter
0.2 0.4 0.6 0.8 1
Figure 3.1: Steady-State Solution Branch of dw
dt = h(w, α) =w2 − α
solution curve has a lower branch that is stable until the parameter α =0. The
curve then reverses direction at the turning point (α, w) =(0, 0) and continues in the
direction of increasing parameter α. Note that the stability of the solutions change
along the curve at the turning point as well.
The second example is of a pitchfork bifurcation. Consider the differential equation
dw
dt = h(w, α) =w3 − αw (3.2)