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

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-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

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##
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

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## Normal Forms / Prototypical

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-representative of all bifurcations of that type

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

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## Taylor Expansion

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-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

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## 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

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