Bifurcation

Question 1

Bifurcation

Definition

Answer 1

• 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

Question 2

Bifurcation Points

Definition

Answer 2

• parameter values at which bifurcations occur

- at bifurcation points f(x) = x’ = 0

Question 3

Saddle Node Bifurcation

Definition

Answer 3

• 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

Question 4

Saddle Node Bifurcation

Normal Form

Answer 4

x’ = r + x²

Question 5

Saddle Node Bifurcation

Phase Space

Answer 5

• 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

Question 6

Saddle Node Bifurcation

Description

Answer 6

• 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

Question 7

Saddle Node Bifurcation

Bifurcation Diagram

Answer 7

-bifurcation points at
x² = -r
-so solutions only exist for r<0
-stable for x<0, unstable for x>0

Question 8

Normal Forms / Prototypical

Answer 8

• representative of all bifurcations of that type

- i.e close to the bifurcation point dynamics all functions typically behave like normal forms

Question 9

Taylor Expansion

Answer 9

-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’
-*********

Question 10

Transcritical Bifurcation

Definition

Answer 10

• 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

Question 11

Transcritical Bifurcation

Normal Form

Answer 11

x’ = rx - x²

Question 12

Transcritical Bifurcation

Phase Space

Answer 12

• 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

Question 13

Transcritical Bifurcation

Description

Answer 13

• 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

Question 14

Difference between saddle-point and transcritical bifurcations

Answer 14

-in transcritical bifurcation the two fixed points don’t disappear after the bifurcation, they just switch their stability

Question 15

Transcritical Bifurcation

Bifurcation Diagram

Answer 15

• 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

Question 16

Pitchfork Bifuration

Answer 16

• 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

Question 17

Supercritical Pitchfork Bifurcation

Normal Form

Answer 17

x’ = rx - x³

Question 18

Supercritical Pitchfork Bifurcation

Phase Space

Answer 18

• 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

Question 19

Supercritical Pitchfork Bifurcation

Description

Answer 19

• 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

Question 20

Supercritical Pitchfork Bifurcation

Bifurcation Diagram

Answer 20

• one bifurcation point at x=0, stable for r<0 but unstable for r>0
• another bifurcation point at x=√r, stable

Question 21

Difference between supercritical and subcritical pitchfork bifurcation

Answer 21

-in the supercritical case, the cubic term is stabilising, in the subcritical case, the cubic term is destabilising

Question 22

Subcritical Pitchfork Bifurcation

Normal Form

Answer 22

x’ = rx + x³

Question 23

Subcritical Pitchfork Bifurcation

Bifurcation Diagram

Answer 23

• one bifurcation point at x=0, stable for r<0 and unstable for r>0
• another bifurcation point at x=-√r , unstable

Question 24

Subcritical Pitchfork Bifurcation

Description

Answer 24

• 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