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MATH2391 Non-Linear Differential Equations > Bifurcation > Flashcards

Flashcards in Bifurcation Deck (24)
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1

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

2

Bifurcation Points
Definition

-parameter values at which bifurcations occur
-at bifurcation points f(x) = x' = 0

3

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

4

Saddle Node Bifurcation
Normal Form

x' = r + x²

5

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

6

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

7

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

8

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

9

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'
-*******************

10

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

11

Transcritical Bifurcation
Normal Form

x' = rx - x²

12

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

13

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

14

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

15

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

16

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

17

Supercritical Pitchfork Bifurcation
Normal Form

x' = rx - x³

18

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

19

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

20

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

21

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

22

Subcritical Pitchfork Bifurcation
Normal Form

x' = rx + x³

23

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

24

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