4. Limit Cycles
• A limit cycle is an isolated closed trajectory; this means
that its neighbouring trajectories are not closed – they
spiral either towards or away from the limit cycle. Thus,
limit cycles can only occur in nonlinear systems. (In a
linear system exhibiting oscillations closed trajectories
are neighboured by other closed trajectories. (Eqs. of
type dθ/dt = f(θ).)
• A stable limit cycle is one which attracts all
neighbouring trajectories. A system with a stable limit
cycle can exhibit self-sustained oscillations – most of the
biological processes of interest are of this kind.
Limit Cycles
• Neighbouring trajectories are repelled from
unstable limit cycles.
• … and half-stable limit cycles are, of
course, ones which attract trajectories
from one side and repel those on the
other.
• As closed orbits are of such interest there
should be a method to establish that there
exists such an orbit, and there is:
Limit Cycles
•
Poincaré-Bendixson Theorem: Suppose
that:
1. R is a closed bounded subset of the plane;
2. dx/dt = f(x) is a continuously differentiable vector
field on an open set containing R;
3. R does not contain any FPs;
4. There exists a trajectory C that is confined in R,
in the sense that it starts in R and stays in R for
all future time.
Then either C is a closed orbit, or it spirals
towards closed orbit as t →∞. So, R contains a
closed orbit.
Limit Cycles
R
C
FP
Relaxation oscillations
• The ark type of oscillations occurring e.g. in firing nerve
cells is the van der Pol oscillation. Oscillations of this
type are called relaxation oscillations because the
charge that builds up slowly is relaxed during a sudden
discharge in the strongly nonlinear limit (µ>>1 in the
following).
• The van der Pol equation reads as
2
&&
&
x + µ x ( x − 1) + x = 0
Relaxation oscillations
&&
x + µ x& ( x 2 − 1) =
F ( x) =
(
)
d
x& + µ  1 x3 − x  , so if we let
 3

dt
1 3
x − x, w = x& + µ F ( x) :
3
w& = &&
x + µ x& ( x 2 − 1) = − x. So we can write vdP eq. as
x& = w − µ F ( x)
w& = − x.
Now let y = w :
µ
x& = µ [ y − F ( x)]
y& = −
1
x.
µ
Relaxation oscillations
y
x
Notice that if the initial state is not close to the cubic nullcline,
|dx/dt|~O(µ)>>1 and |dy/dt|~O(µ-1)<<1.
Relaxation oscillations
1. Bifurcations in 2D
• In the following I’ll briefly recap the bifurcations
presented earlier in 1D and then talk about the
Hopf bifurcation that’s present only in
dimensions > 1.
• In 2D not only FPs can be created or destroyed
or destabilised as parameters are varied, the
same goes for closed orbits.
• Thus, in 2D bifurcation is defined as a change in
a system’s topological structure. For this the
concept of topological equivalence is needed.
Bifurcations in 2D
• Two phase portraits are topologically equivalent if
there is a homeomorphism, that is, a continuous
deformation with a continuous inverse, mapping one
phase portrait onto the other, such that trajectories map
onto trajectories and the sense of time is preserved
(shrink, stretch but don’t tear).
• Regarding bifurcations, things are nicely generalised
when moving up in dimensionality. As everything that’s
of any importance with respect to bifurcation occurs
close to the FPs, one can just plot the variables as
functions of each other instead of the single variable’s
phase plots (x,dx/dt)
Saddle-node bifurcations
• 1D: FPs are created and destroyed
• The prototypical system in 2D:
&x = µ − x 2
y& = − y.
• This bifurcates as µ varies
РFPs: stable at (x*,y*) = (õ,0) and a saddle at
(-√µ,0) when µ > 0.
– these annihilate when µ = 0 and disappear
when µ < 0. Even after that they influence the
flow by slowing it down: they suck the
trajectories through a bottleneck called a ghost.
Saddle-node bifurcations
x& = µ − x
y& = − y.
Here’s where
the ghost region
appears as µ > 0.
2
y
y
µ=1
x
µ=0
x
Pitchfork bifurcations
•1D; FPs (dis)appear in symmetrical pairs
•The prototype systems in 2D:
x& = µ x − x3 , y& = − y (supercritical; plots → )
µ=0
x& = µ x + x 3 , y& = − y (subcritical).
µ<0
µ>0
Transcritical bifurcations
… and just to include them all: the 2D version of
the transcritical prototype system is of course
(in 1D FPs are there forever but change stability)
x& = µ − x
y& = − y.
2
Hopf bifurcations
• This type of bifurcation takes place only in
dimensions higher than one.
• Now FPs can change their stabilities with
the changing of complex eigenvalues. The
straightforward way to see this is from the
Jacobian. Recall that the perturbation
expansion around a FP of a nonlinear
system leads to a linear system written
with the aid of Jacobian:
Hopf bifurcations
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
Hopf bifurcations
∂g
∂g
2
2
&v = u
+v
+ O ( u , v , 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
Hopf bifurcations
• Recall that close to FPs flows are of exponential
form ~exp(λt), where λ is an eigenvalue of the
Jacobian. So naturally, for stable FPs Re λ < 0.
As the characteristic equation for a 2D system is
quadratic the stable eigenvalues are either both
real and negative or they are complex
conjugates in the left half plane. The bifurcations
taking place when real eigenvalues cross into
the right half plane are the ones just mentioned.
• Hopf bifurcations occur when the complex
conjugate eigenvalues simultaneously cross the
imaginary axis into the right half plane.
Hopf bifurcations
Im λ
Im λ
Re λ
Eigenvalues in saddle-node,
transcritical, or pitchfork
bifurcations.
Re λ
Eigenvalues in Hopf
bifurcation.
Supercritical Hopf bifurcations
• Supercritical Hopf bifurcation occurs when
the initial decay µ = Re λ becomes slower
and finally changes to growth at a critical
value µc. The oscillation becomes a smallamplitude sinusoidal, limit cycle oscillation
about the former steady state when µ > µc. For
example for the following system µ controls the
stability of the FP at the origin, ω gives the frequency
(and the direction) of infinitesimal oscillations, and b
determines the dependence of frequency on
amplitude for larger amplitude oscillations:
r& = µ r − r
2
&
θ = ω + br
3
Supercritical Hopf bifurcations
r& = µ r − r 3
2
&
θ = ω + br
λ = µ ± iω
µ<0
µ>0
stable limit cycle
at r = õ
Subcritical Hopf bifurcations
• By changing the sign of the cubic term it
becomes destabilising
r& = µ r + r 3 − r 5
θ& = ω + br 2
For µ < 0 there are two
attractors: a stable limit cycle
and a stable FP at the origin.
Between them lies an unstable
limit cycle. As µ → 0 the
unstable limit cycle shrinks and
engulfs the origin rendering it
unstable. After this the stable
limit cycle (5th order term) is the only
attractor. Large-amplitude solutions result.
Subcritical Hopf bifurcations
µ<0
µ>0
The end of my scribbles for now
• Seminars from next Friday on –
be there!