1. bifurcations - aalto · pitchfork bifurcation • pitchfork bifurcation is related to symmetries...

1. Bifurcations

• The qualitative structure of the flow on the vector field under investigation can change as parameters are varied: in particular, FPscan be created or destroyed, or their stability can change. These qualitative changes in the dynamics are called bifurcations, and the parameter values at which they occur are called bifurcation points.

Saddle-node bifurcation

• The prototypical example of a saddle-node bifurcation is given by

2x r x= +&

• As r<0, there are two FPs, stable at x=r and unstable at x=-r.

• As r increases the two FPs move towards each other and coalesce into a half-stable FP at x=0.

• This FP is destroyed as r increase to become positive.


All diagrams ( , ).x x&The halfThe half--stable FP above ought stable FP above ought to be designated by a halfto be designated by a half--filled filled circle.circle.

r r < 0< 0r r = 0= 0

r r > 0> 0


• The conventional way to draw a bifurcation diagram for a 1D system is to plot dx/dt = 0 in the r-x-coordinate system.

• The stable branch is plotted with a solid line.

• The unstable branch is plotted with a dashed line.

xx

rr

Saddle-node bifurcation• The prototypical examples

are representative of all saddle-node bifurcations and are hence called normal forms. Close to FPs the behaviour of other systems giving rise to saddle-node bifurcations can be mapped to normal forms. Taylor expansion close to x* and the parameter value rc at which the stable and unstable FP collide yields:

2x r x= ±&

* * *

* * * 2

( , ) ( , ) ( , )

( , )

1( , ) ( ) ( ) ( ) ...

2c c c

c cx r x r x r

x f x r

f f ff x r x x r r x x

x r x

=∂ ∂ ∂= + − + − + − +∂ ∂ ∂

&

, where the quadratic terms in r and cubic terms in x have been neglected. Now, at FP f(x*)=0, and at the saddle-node bifurcation point (x*,rc) / 0.f x∂ ∂ =

Saddle-node bifurcation2

2 2

( *, ) ( *, )

Thus, ( ) ( *)

where / and / . This is of the

normal form after a scaling of and .

c c

c

x r x r

x a r r b x x

a f r b f x

x r

= − + − += ∂ ∂ = ∂ ∂

& K

Transcritical bifurcation

• The transcritical bifurcation is the standard mechanism for a situation where a FP exists for all values of a parameter and can never be destroyed but may change its stability. The normal form for a transcritical bifurcation is

2.x rx x= −&

Notice that the logistic equation for population growth Notice that the logistic equation for population growth is of this general form.is of this general form.

Transcritical bifurcationAs the initially negative parameter As the initially negative parameter r r increases the two increases the two FPsFPs (at 0 and (at 0 and r) r) coalesce and form a halfcoalesce and form a half--stable FP stable FP when when rr = 0. When further increasing = 0. When further increasing r r to positive values the two to positive values the two FPsFPs split split again, but now they have switched again, but now they have switched stability.stability.

x&

x&

r r < 0< 0 r r > 0> 0

r r = 0= 0

xx xx

xx

dx/dtdx/dt

dx/dtdx/dt


• The bifurcation diagram corresponding to the dynamics depicted in the three diagrams above looks like this:

xx

rr

Again, the diagramAgain, the diagramdepicts the dynamicsdepicts the dynamicswhen when dx/dtdx/dt = 0, i.e. the= 0, i.e. thebehaviour of behaviour of FPsFPs..


• (Example 3.2.2 in Strogatz) Analyse the dynamics of dx/dt = r ln x + x – 1 near x = 1, and show that the system undergoes a transcritical bifurcation at a certain value of r. Then find new variables X and R such that the system reduces to approximate normal form dX/dt ~ RX – X2 near bifurcation.


• Solution: x = 1 is a FP for all r. Introduce a new variable u = x – 1, where u is small. Then

2 3

2 3

ln(1 )

1( )

2

1( 1) ( ).

2

u x

r u u

r u u O u u

r u ru O u

== + +

= − + + = + − +

& &


( )

2 3

2 3

By substituting the equation becomes

1( 1) ( ). Now we choose 2 /

2

and let 1 and to achieve the approximate

normal form

.

u a

r ra O a r

R r X

X RX X O X

ν

ν ν ν ν

ν

= = + − + =

= + =

= − +

&

&

Pitchfork bifurcation

• Pitchfork bifurcation is related to symmetries in the system. For example in systems having a spatial symmetry between left and right FPs tend to appear and disappear in symmetrical pairs (think for instance of a buckling beam).

• The normal form is 3.x rx x=& m• The minus sign gives supercritical pitchfork bifurcation.

The name implies that there can exist a FP above bifurcation.

Supercritical pitchfork bifurcation

dx/dtdx/dt dx/dtdx/dt

xx xx

rr < 0< 0

rr = 0= 0

rr > 0> 0

As As rr = 0 the solutions no longer decay exponentially fast to FP, = 0 the solutions no longer decay exponentially fast to FP, but rather algebraically. This is called but rather algebraically. This is called critical slowing down.critical slowing down.

x*x* = = ±± r r 1/1/22

Supercritical pitchfork bifurcation

• The corresponding bifurcation diagram looks like pitchfork. xx

rr

Subcritical pitchfork bifurcation

• If in the normal pf bifurcation form the x3

term is destabilising (> 0), the pitchfork in th bifurcation diagram and the stabilities of the FP are inverted xx

rr

Now both FP are below Now both FP are below the bifurcation the bifurcation (r =(r =0); hence0); hencethe name the name subcriticalsubcritical..

2. Flows on the Circle

• The vector field on the circle is of the form

( ).fθ θ=&

Here Here θ θ is a point on the circle and is a point on the circle and ddθθ/dt/dt the velocity vector the velocity vector at that point. Like the line, the circle is oneat that point. Like the line, the circle is one--dimensional, but dimensional, but now a phase point unlike on the 1D line can return to its now a phase point unlike on the 1D line can return to its starting place. Thus, vector fields on the circle provide the starting place. Thus, vector fields on the circle provide the most basic model of systems that can oscillate.most basic model of systems that can oscillate.

For example, the sketch of the vector For example, the sketch of the vector field field ddθθ/dt/dt = sin = sin θ θ on the circle would likeon the circle would likethis. this. FPsFPs are are θθ*=0 and *=0 and θθ*=*=π.π.

θ&

3. Two-Dimensional Flows3.1. Linear Systems

• A two-dimensional linear system is of the form

0

0

, or

, where and . For instance,

0for the uncoupled system with

0 1

and so ( )

y( ) .

at

at

x ax by

y cx dy

a b xA A

c d y

aA

x ax x t x e

y y t y e−

= += +

= = = = −

= == − =

x x x

&

&

&

&

&

Linear systems

• The phase portraits (x,y) for different values of a look as following (and for a set of different initial values x0, y0):

a < a < --11

Linear systems

aa==--11 --1<1<aa<0<0

aa=0=0 aa>0>0

Linear systems

• Notice that in linear systems there is always a FP at origo. The trajectories approach the stable FP tangent to the slower decaying direction as t → ∞. Looking backwards along a trajectory t → -∞, it becomes tangent to the faster decaying direction.

• If both directions decay with equal rate (a=-1) the stable node is called a symmetrical node or star.

• In the case a=0 there is a line of FPs along the x-axis. The trajectories approach these along vertical lines (please ignore the small arrow on the x-axis).

• When a>0, x* becomes unstable due to the exponential growth in the x-direction. x*=0 is a saddle point.

Linear systems

• In forward time, the trajectories are asymptotic to the x-axis; in backward time, to the y-axis. These directions intersect at the saddle point.

• The trajectory starting on the y-axis ends at x*=0. The y-axis is called the stable manifold of the saddle point: the set of initial conditions x0such that x(t) → x* as t → ∞. Likewise, the x-axis is the unstable manifold in this case: that’s where the trajectories not starting from the stable manifold end up as t → ∞.

Linear systems

• Some terminology: x*=0 is an attracting FP if all trajectories starting near x* approach it as t → ∞. If x* attracts all trajectories in the phase plane it is called globally attracting.

• A FP x* is Liapunov stable if all trajectories that start sufficiently close to x* remain close to it for all time. In the previous example the origin is Liapunov stable for all cases except for a>0.

• A FP is unstable if it is neither attracting nor Liapunov stable.

Classification of linear systems

• Also in a general case where A is non-diagonal we seek for straight-line trajectories, for which the phase point starting on such a line stays on it, and the trajectory exhibit exponential growth or decay. Now they are not necessarily the coordinate axis.

• For the general case we seek trajectories of the form x(t) = exp(λt) v, where v≠0 is some fixed vector and λ is a growth rate, both to be determined. Now, substituting the exponential form for x(t) into dx/dt = Ax yields Av = λv, which says that the desired straight-line solution exists if v is and eigenvector of A with corresponding eigenvalue λ. Then x(t) = exp(λt) v is called an eigensolution.


• Notation: Recall that the characteristic equation for the linear system dx/dt = Ax is det(A-λI)=0. For a 2x2 matrix:

2

2 2

1 2

the characteristic equation is

det 0

0, where

trace( ) , det( ) . Then

4 4, .

2 2

a bA

c d

a b

c d

A a d A ad bc

λλ

λ τλτ

τ τ τ τλ λ

= − = −

⇔ − +∆ == = + ∆ = = −

+ − ∆ − − ∆= =


• If the eigenvalues are distinct, λ1≠ λ2, then the eigenvectors v1 and v2 are linearly independent and span the entire space. Thus any initial condition can be written as x0=c1v1+c2v2, and the general solution is x(t)=c1exp(λ1t)v1+ c2exp(λ2t)v2.

• So, for the initial value problem dx/dt=x+y, dy/dt=4x-2y, subject to the initial condition (xo, y0)=(2,-3) τ=-1 and ∆=-6, λ1=2, λ2=-3, and v1=(1 1)T v2=(1 -4)T, and the linear combination of eigensolutions satisfying the initial condition is

Classification of linear systems2 3

2 3

( ) ,

( ) 4 .

t t

t t

x t e e

y t e e

−

−

= +

= −

•• In order to plot the phase portrait the system doesIn order to plot the phase portrait the system doesnot necessarily have to be solved completely. Thenot necessarily have to be solved completely. Theknowledge of knowledge of eigenvalueseigenvalues and and ––vectors is sufficient.vectors is sufficient.Because the first Because the first eigensolutioneigensolution ((λλ11=2) grows and the =2) grows and the second (second (λλ11==--3) decays exponentially the origin is a3) decays exponentially the origin is asaddle point. The stable manifold is the line spanned saddle point. The stable manifold is the line spanned by the eigenvector by the eigenvector vv22=(1,=(1,--4), corresponding to the 4), corresponding to the decaying decaying eigensolutioneigensolution, and the unstable manifold is , and the unstable manifold is the line spanned by the line spanned by vv11=(1,1). =(1,1).


yy

xx

(The rightmost curve is the one corresponding to the initial (The rightmost curve is the one corresponding to the initial conditions conditions –– so with so with cc11=c=c22=1.=1.))


• In the case of complex eigenvalues the FP is either a centre or a spiral. Centres are neutrally stable, since nearby trajectories are neither attracted to nor repelled from the FP.

centrecentre


spiral of a (repelling) FPspiral of a (repelling) FP

3.2. Nonlinear Systems on Phase Plane

• Typically, for a nonlinear system there’s no hope of finding an analytic solution for trajectories on the phase plane. So, the goal is to be able to plot a phase portrait to get an idea of the system behaviour.

• Without even solving numerically one typically starts by finding nullclines: for a 2-dimensional system defined as the curves where either dx/dt=0 or dy/dt=0. Thus, nullclines are lines on which the direction of the vector field (flow) is purely horizontal or vertical. When you’ve plotted the nullclines, then of course FPs are their intersections. (When plotting the phase portraits, if you have the time and patience, you can plot the vector field lines (in Matlab by using streamslice-function).)

Nonlinear Systems on Phase Plane

• The existence and uniqueness theorem in several dimensions reads as: The initial value problem x=f(x), x(0)=x0 in D has a unique solution on some time interval (-τ,τ) if f and all its partial derivatives ∂fi/∂xj , i, j = 1,…,n, are continuous for x in some open connected set DC Rn.

• Given that the previous theorem holds:– two trajectories never intersect– in 2D any curve starting inside a closed curve

(periodic solution) will remain there

Linearization around FP in 2D

( )( )

2 2

2 2

The system

( , )

( , )

with a FP ( *, *) : ( *, *) 0, ( *, *) 0

is linearised near FP as

*, *,

( * , * )

( *, *) , ,

, ,

x f x y

y g x y

x y f x y g x y

u x x v y y

u x f x u y v

f ff x y u v O u v uv

x y

f fu v O u v uv

x y

==

= =

= − = −= = + +

∂ ∂= + + +∂ ∂

∂ ∂= + +∂ ∂

&

&

& &


( )2 2, , . That is the linearised

system:

, where

and is the Jacobian

g gv u v O u v uv

x y

J

f fx yu

Jg gv

x y

∂ ∂= + +∂ ∂

=∂ ∂ ∂ ∂ = = ∂ ∂ ∂ ∂

u u

u

&

&


• Linearization works fine for FPs that are inside one of the regimes in the classification diagram sketched before.

• For borderline FPs one should be cautious; centres, degenerate nodes, stars, or non-isolated can be altered by small nonlinear terms.

• A phase portrait is called structurally stable if cannot be changed by an arbitrarily small perturbationto the vector field.

• So, in summary you determine the nullclines, nature of FPs and try to sketch the flows in between.

1. bifurcations - aalto · pitchfork bifurcation • pitchfork bifurcation is related to symmetries...

Documents