Fixed Points and Bifurcations

Type 1

Stable fixed point

Equivalent

$f(x)$ decreases at

Filled circle. Arrows point towards it from both sides

Type 2

Unstable fixed point

Equivalent

$f(x)$ increases at

Empty circle. Arrows move away from both sides

Type 3

Semi-stable fixed point

Equivalent

$f(x)$ neither increases nor decreases at

Half-filled circle. Arrows go in the same direction on both sides

What is a bifurcation?

A bifurcation is a qualitative change in the behaviour of a system $\dot{x} = f (x, r)$ when the parameter $r$ crosses a critical value $r_{c}$ . At this point, the number or stability of the fixed points changes abruptly. Fixed points are found where $f (x^{*}, r) = 0$ , and their stability is determined by the sign of $f^{'} (x^{*})$ .

Legend

Stable fixed point ·

f^{'} (x^{*}) < 0

Unstable fixed point ·

f^{'} (x^{*}) > 0

Semi-stable fixed point ·

f^{'} (x^{*}) = 0

Option 1 · Analytical solution

When you can isolate

Find fixed points: values

such that

. May depend on

Determine the stability of each fixed point: sign of

as a function of

Identify bifurcation: analyse the number of fixed points and stability on each side of

(see the 4 types below)

How to draw the diagram

Horiz. axisrepresents the parameter

Vert. axisrepresents the fixed points

Curvesexpress each

as a function of

(or constant value) and draw it

Stabilitysolid line = stable · dashed line = unstable

Option 2 · No analytical solution

When you cannot isolate

Bifurcation conditions: solve the system

and

Obtain the critical values

Pitchfork if: symmetry

Super/Subcritical: do the new fixed points appear for

(super) or

(sub)?

Transcritical if:

is a fixed point for all

, i.e.

for any value of

Saddle-node if: none of the above +

≠ 0

How to draw the diagram

Horiz. axisrepresents the parameter

Vert. axisrepresents the fixed points

Curveisolate

and draw the parametric curve

Trivialif there are fixed points independent of

, draw them as horizontal lines

Stabilitysolid line = stable · dashed line = unstable

Type 1

Saddle-node

Two fixed points (one stable and one unstable) collide and disappear.

Behaviour

r < r_{c}

2 fixed points

stable

unstable

r = r_{c}

1 fixed point

semi-stable

r > r_{c}

0 fixed points

no fixed points

Type 2

Transcritical

There are always two fixed points. When crossing $r_{c}$ they collide and exchange stability.

Behaviour

r < r_{c}

2 fixed points

x_{0}^{*}

stable

x_{1}^{*}

unstable

r = r_{c}

1 fixed point

semi-stable

r > r_{c}

2 fixed points

x_{0}^{*}

unstable

x_{1}^{*}

stable

Type 3

Supercritical Pitchfork

A stable fixed point loses stability and two new symmetric ones appear.

Behaviour

r < r_{c}

1 fixed point

x_{0}^{*}

stable

r = r_{c}

1 fixed point

x_{0}^{*}

semi-stable

r > r_{c}

3 fixed points

x_{-}^{*}

stable

x_{0}^{*}

unstable

x_{+}^{*}

stable

Type 4

Subcritical Pitchfork

Two symmetric unstable fixed points collapse with the stable fixed point.

Behaviour

r < r_{c}

3 fixed points

x_{-}^{*}

unstable

x_{0}^{*}

stable

x_{+}^{*}

unstable

r = r_{c}

1 fixed point

x_{0}^{*}

semi-stable

r > r_{c}

1 fixed point

x_{0}^{*}

unstable