Last year (and in previous years) many of the students who failed could not tell the difference between a bifurcation diagram and a phase plane; after all both of them have curves and lines and a bunch of arrows — but that does not mean they are the same thing!
Number of Dimensions
In this unit, we draw phase lines and bifurcation diagrams for a single ODE. However, we draw a phase plane for a system of two ODEs.
What goes on the axes
A bifurcation diagram has a variable on the vertical axis and a parameter on the horizontal axis. A phase plane has two variables, one for each axis.
The arrows
A bifurcation diagram has ONLY vertical arrows, the arrows point towards stable equilibria and away from unstable equilibria . A phase plane has arrows in all possible directions, the pattern of arrows provides clues about the type of equilibrium.
Direction of arrows
For a bifurcation diagram, corresponding to ˙ = f ( x, µ ), the sign of f gives you the upward or downward direction of the arrow. For a phase plane, corresponding to ˙ = f ( x, y ) and ˙ = g ( x, y ), the combination of signs of both f and g determine the direction of the arrow.
The curves or lines
The curves or lines on a phase plane represent NULLCLINES. The curves or lines on a bifurcation diagram represent all possible equilibrium solutions (stable equilibria are shown with a solid curve, and unstable equilibria or shown with a dashed curve).
The intersections
On a phase plane, equilibria occur at the intersection of different nullclines. On a bifurcation diagram, bifurcations occur when two different branches of the equilibrium solution curves cross or meet; the way they meet and/or cross determines whether it is a FOLD,
TRANSCRITICAL or PITCHFORK bifurcation.
Linear stability analysis
This is a purely algebraic (and calculus) method for classifying an equilibrium. For a phase line or a bifurcation diagram, if f
0
( x eq
) < 0 the equilibrium is stable, if f
0
( x eq
) > 0 the equilibrium is unstable. For a phase plane, the eigenvalues of the Jacobian matrix evaluated at the equilibrium determines the type.
When linear stability fails
For a phase line or a bifurcation diagram, if f
0
( x eq
) = 0 the equilibrium might be semistable, or stable or unstable; a diagram will help decide which.
For a phase plane, if the eigenvalues have zero real part (or are repeated) the type of equilibrium is harder to determine; a diagram will help sometimes and more powerful techniques and theorems might also help.
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