Bifurcation
Definition
- the qualitative structure of the flow in a 1D system can change as parameters are varied
- in particular fixed points can be created or destroyed and their stability can change
- these qualitative changes are called bifurcations
Bifurcation Points
Definition
- parameter values at which bifurcations occur
- at bifurcation points f(x) = x’ = 0
Saddle Node Bifurcation
Definition
- basic mechanism by which fixed points are created and destroyed
- as a parameter is varied two fixed points move towards each other, collide and mutually annhilate
Saddle Node Bifurcation
Normal Form
x’ = r + x²
Saddle Node Bifurcation
Phase Space
- for r<0, a +x² shape crossing the x axis twice, stable point on the left, unstable on the right
- for r=0, a +x² shape touching the x axis at the origin, one semi stable fixed point
- for r>0 a +x² shape above the x axis so no fixed points
Saddle Node Bifurcation
Description
- as r tends to 0 fixed points move closer together
- at r=0, fixed points coalesce into a single semi stable fixed point
- as soon as r>0, the semi stable fixed point vanishes and there are no fixed points
Saddle Node Bifurcation
Bifurcation Diagram
-bifurcation points at
x² = -r
-so solutions only exist for r<0
-stable for x<0, unstable for x>0
Normal Forms / Prototypical
- representative of all bifurcations of that type
- i.e close to the bifurcation point dynamics all functions typically behave like normal forms
Taylor Expansion
-examine the behaviour of f(x,r) = x’ near a bifurcation point at x=xo, r=ro
-Taylor expansion:
f(x,r) = f(xo,ro) + (x-xo) ∂f/∂x|xo + (r-ro) ∂f/∂r|ro + 1/2 (x-xo)² ∂²f/∂x²|xo + …..
-truncate after (x-xo)² term
-and f(xo,ro) = 0 since xo is a fixed point and f=x’
-*********
Transcritical Bifurcation
Definition
- there exist certain situations where a fixed point must exist for all values of a parameter and can never be destroyed
- however stability of the fixed point can change as the parameter is varied
- the standard mechanism for this is transcritical bifurcation
Transcritical Bifurcation
Normal Form
x’ = rx - x²
Transcritical Bifurcation
Phase Space
- for r<0, a -x² shape crossing at x<0, an unstable fixed point and x=0, a stable fixed point
- for r=0, a -x² shape touching the x axis only at the origin at a semistable fixed point
- for r>0, a -x² shape crossing the x axis at the origin with an unstable fixed point and at x>0 with a stable fixed point
Transcritical Bifurcation
Description
- for r<0 there is an unstable fixed point at x=r and a stable fixed point at x=0
- as r increases, the unstable fixed point approaches the origin and coalesces with it when r=0
- when r>0 the origin has become unstable and x=r is stable, we can say that an exchange of stabilities has taken place between the two fixed points
Difference between saddle-point and transcritical bifurcations
-in transcritical bifurcation the two fixed points don’t disappear after the bifurcation, they just switch their stability
Transcritical Bifurcation
Bifurcation Diagram
- one bifurcation point at x=0, stable for r<0 and unstable for r>0
- another bifurcation point at x=r which is stable for r>0 and unstable for r<0
Pitchfork Bifuration
- many problems have spatial symmetry between left and right
- in such cases, fixed points tend to appear and disappear in symmetrical pairs
- there are two types, supercritical and subcritical
- e.g. a weight balanced on a beam, as the weight is increased the beam will eventually buckle either to the left or to the right
Supercritical Pitchfork Bifurcation
Normal Form
x’ = rx - x³
Supercritical Pitchfork Bifurcation
Phase Space
- for r<0 a -x³ shape crossing through the origin and linear through the origin, stable fixed point at x=0
- for r=0 -x³ through the origin but not linear through the origin, still stable
- for r<0, wavy -x³ shape crosses at x<0 stable fixed point, an unstable fixed point at the origin and another stable fixed point symmetrically at x>0
Supercritical Pitchfork Bifurcation
Description
- for r<0, the origin is the only fixed point and is stable
- for r=0, the origin is stable but more weakly stable, solutions no longer decay towards it exponentially, instead they decay as a much slower algebraic function of time, a critical slowing down
- for r>0, the origin has become unstable, two new stable points appear located symmetrically as x=±√r
Supercritical Pitchfork Bifurcation
Bifurcation Diagram
- one bifurcation point at x=0, stable for r<0 but unstable for r>0
- another bifurcation point at x=√r, stable
Difference between supercritical and subcritical pitchfork bifurcation
-in the supercritical case, the cubic term is stabilising, in the subcritical case, the cubic term is destabilising
Subcritical Pitchfork Bifurcation
Normal Form
x’ = rx + x³
Subcritical Pitchfork Bifurcation
Bifurcation Diagram
- one bifurcation point at x=0, stable for r<0 and unstable for r>0
- another bifurcation point at x=-√r , unstable
Subcritical Pitchfork Bifurcation
Description
- non-zero fixed points x=±√r are unstable and only exist below the bifurcation, r<0, hence “sub”
- origin stable for r<0 and unstable for r>0, the same as for supercritical
- BUT instability for r>0 is not opposed by cubic term, in fact the cubic term drives trajectories out to infinity
