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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 f ( x, y ) and ˙ = g ( x, y

= f ( x, µ

), the combination of signs of both f

), the sign of and g f gives you the

= 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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