1. Bifurcations
• The qualitative structure of the flow on the
vector field under investigation can change
as parameters are varied: in particular, FPs
can 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
x& = r + x
2
• 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.
Saddle-node bifurcation
r<0
r=0
All diagrams ( x, x& ).
The half-stable FP above ought
to be designated by a half-filled
circle.
r>0
Saddle-node bifurcation
• 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.
x
r
Saddle-node bifurcation
•
The prototypical examples x& = r ± x 2
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:
x& = f ( x, r )
= f ( x* , rc ) + ( x − x* )
∂f
∂x
+ (r − rc )
( x* , rc )
∂f
∂r
∂f
1
+ ( x − x* ) 2
2
∂x
( x* , rc )
+ ...
( x* , rc )
, 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) ∂f / ∂x = 0.
Saddle-node bifurcation
Thus, x& = a (r − rc ) + b( x − x*) 2 + K
where a = ∂f / ∂r ( x*,r ) and b = ∂f 2 / ∂x 2
c
normal form after a scaling of x and r.
( x *, rc )
. This is of the
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
is of this general form.
Transcritical bifurcation
dx/dt
As the initially negative parameter
r increases the two FPs (at 0 and r)
coalesce and form a half-stable FP
when r = 0. When further increasing
r to positive values the two FPs split
x&
again, but now they have switched
stability.
dx/dt
r=0
x
x&
r<0
x
r>0
x
Transcritical bifurcation
• The bifurcation diagram corresponding to
the dynamics depicted in the three
diagrams above looks like this:
x
r
Again, the diagram
depicts the dynamics
when dx/dt = 0, i.e. the
behaviour of FPs.
Transcritical bifurcation
• (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.
Transcritical bifurcation
• Solution: x = 1 is a FP for all r. Introduce a new
variable u = x – 1, where u is small. Then
u& = x&
= r ln(1 + u ) + u
1 2

3 
= r u − u + O(u )  + u
2


1 2
= (r + 1)u − ru + O(u 3 ).
2
Transcritical bifurcation
By substituting u = aν the equation becomes
1  2

ν& = (r + 1)ν −  ra ν + O(ν 3 ). Now we choose a = 2 / r
2 
and let R = r + 1 and X = ν to achieve the approximate
normal form
X& = RX − X 2 + O ( X 3 ) .
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).
3
• The normal form is
x& = rx m x .
• The minus sign gives supercritical pitchfork bifurcation.
The name implies that there can exist a FP above
bifurcation.
Supercritical pitchfork bifurcation
1/2
x* = ± r 1/2
dx/dt
dx/dt
r<0
x
x
r=0
r>0
As r = 0 the solutions no longer decay exponentially fast to FP,
but rather algebraically. This is called critical slowing down.
Supercritical pitchfork bifurcation
• The corresponding
bifurcation diagram
looks like pitchfork.
x
r
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
x
Now both FP are below
the bifurcation (r =0); hence
the name subcritical.
r
2. Flows on the Circle
• The vector field on the circle is of the form
θ& = f (θ ).
Here θ is a point on the circle and dθ/dt the velocity vector
at that point. Like the line, the circle is one-dimensional, but
now a phase point unlike on the 1D line can return to its
starting place. Thus, vector fields on the circle provide the
most basic model of systems that can oscillate.
θ&
For example, the sketch of the vector
field dθ/dt = sin θ on the circle would like
this. FPs are θ*=0 and θ*=π.
3. Two-Dimensional Flows
3.1. Linear Systems
• A two-dimensional linear system is of the form
x& = ax + by
y& = cx + dy, or
a b
 x
x& = Ax, where A = 
 and x =   . For instance,
c d
 y
a 0 
for the uncoupled system with A = 

0
−
1


x& = ax and so x(t ) = x0 e at
y& = − y
y(t ) = y0 e− at .
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 < -1
Linear systems
a=-1
-1<a<0
a=0
a>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 x0
such that x(t) → x* as t → ∞. Likewise, the xaxis 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.
Classification of linear systems
• Notation: Recall that the characteristic
equation for the linear system dx/dt = Ax is
det(A-λI)=0. For a 2x2 matrix:
a b
A=
 the characteristic equation is
c d 
b 
a−λ
det 
=0
d −λ
 c
⇔ λ 2 − τλ + ∆ = 0, where
τ = trace( A) = a + d , ∆ = det( A) = ad − bc. Then
τ + τ 2 − 4∆
τ − τ 2 − 4∆
, λ2 =
.
λ1 =
2
2
Classification of linear systems
• 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 systems
x(t ) = e 2t + e −3t ,
y (t ) = e 2t − 4e −3t .
• In order to plot the phase portrait the system does
not necessarily have to be solved completely. The
knowledge of eigenvalues and –vectors is sufficient.
Because the first eigensolution (λ1=2) grows and the
second (λ1=-3) decays exponentially the origin is a
saddle point. The stable manifold is the line spanned
by the eigenvector v2=(1,-4), corresponding to the
decaying eigensolution, and the unstable manifold is
the line spanned by v1=(1,1).
Classification of linear systems
y
x
(The rightmost curve is the one corresponding to the initial
conditions – so with c1=c2=1.)
Classification of linear systems
• 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.
centre
Classification of linear systems
spiral of a (repelling) FP
Classification of linear systems
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 D
C 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
The system
x& = f ( x, y )
y& = g ( x, y )
with a FP ( x*, y*) : f ( x*, y*) = 0, g ( x*, y*) = 0
is linearised near FP as
u = x − x*, v = y − y*,
u& = x& = f ( x * +u, y * +v)
∂f
∂f
= f ( x*, y*) + u + v + O ( u 2 , v 2 , uv )
∂x
∂y
∂f
∂f
= u + v + O ( u 2 , v 2 , uv )
∂x
∂y
Linearization around FP in 2D
∂g
∂g
v& = u
+v
+ O ( u 2 , v 2 , uv ) . That is the linearised
∂x
∂y
system:
u& = Ju, where
u
u= 
v
∂f 
 ∂f
∂y 
 ∂x
and J = 
 is the Jacobian
∂g 
 ∂g
∂y 
 ∂x
Linearization around FP in 2D
• 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 nonisolated 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.