4. Limit Cycles

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

FPFP

RR

CC

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 1 0x x x xµ+ − + =&& &

Relaxation oscillations( ) ( )

( )

[ ]

2 3

3

2

11 , so if we let3

1( ) , ( ) :

3

1 . So we can write vdP eq. as

( )

.

Now let :

( )

1.

dx x x x x x

dt

F x x x w x F x

w x x x x

x w F x

w x

wy

x y F x

y x

µ µ

µ

µ

µ

µµ

µ

+ − = + − = − = +

= + − = −= −= −

== −= −

&& & &

&

& && &

&

&

&

&

Relaxation oscillations

xx

yy

Notice that if the initial state is not close to the cubic Notice that if the initial state is not close to the cubic nullclinenullcline, , ||dx/dt|~O(dx/dt|~O(µµ)>>1 )>>1 and and |dy/dt|~O(|dy/dt|~O(µµ--11)<<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:

2

.

x x

y y

µ= −

= −

&

&

• This bifurcates as µ varies– FPs: stable at (x*,y*) = (√µ,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

µµ = = 11 µµ = = 00

Here’s whereHere’s wherethe ghost region the ghost region appears as appears as µ µ > 0.> 0.

2

.

x x

y y

µ= −

= −

&

&

xxxx

yy yy

Pitchfork bifurcations

µµ < 0< 0 µµ > 0> 0

µµ = 0= 0

••1D; 1D; FPsFPs ((dis)appeardis)appear in symmetrical pairsin symmetrical pairs••The prototype systems in 2D:The prototype systems in 2D:

3

3

, (supercritical; plots )

, (subcritical).

x x x y y

x x x y y

µµ

= − = − →

= + = −

& &

& &

Transcritical bifurcations

…… and just to include them all: the 2D version of and just to include them all: the 2D version of the the transcriticaltranscritical prototype system is of course prototype system is of course (in 1D (in 1D FPsFPs are there forever but change stability)are there forever but change stability)

2

.

x x

y y

µ= −

= −

&

&

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

( )( )

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

==

= =

= − = −= = + +

∂ ∂= + + +∂ ∂

∂ ∂= + +∂ ∂

&

&

& &

Hopf bifurcations

( )2 2, , . That is, the linearised

system:

, where

and is the Jacobian

g gv u v O u v uv

x y

J

f fu x y

Jv g g

x y

∂ ∂= + +∂ ∂

=∂ ∂ ∂ ∂ = = ∂ ∂ ∂ ∂

u u

u

&

&

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 bifurcationsImIm λλ

Re Re λλ

EigenvaluesEigenvalues in saddlein saddle--node, node, transcriticaltranscritical, or pitchfork , or pitchfork bifurcations.bifurcations.

ImIm λλ

Re Re λλ

EigenvaluesEigenvalues in in HopfHopfbifurcation.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 small-amplitude 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: 3

2

r r r

br

µθ ω= −

= +

&

&

Supercritical Hopf bifurcations3

2

r r r

br

µθ ω= −

= +

&

&

µ > µ > 00µ < µ < 00stable limit cycleat r = √µ

iλ µ ω= ±

Subcritical Hopf bifurcations

• By changing the sign of the cubic term it becomes destabilising

3 5

2

r r r r

br

µθ ω= + −

= +

&

&

For For µµ < 0 there are two < 0 there are two attractors: a stable limit cycleattractors: a stable limit cycleand a stable FP at the origin. and a stable FP at the origin. Between them lies an unstable Between them lies an unstable limit cycle. As limit cycle. As µ µ →→ 0 the 0 the unstable limit cycle shrinks and unstable limit cycle shrinks and engulfs the origin rendering it engulfs the origin rendering it unstable. After this the stable unstable. After this the stable limit cycle (5th order term) is the only limit cycle (5th order term) is the only attractor. Largeattractor. Large--amplitude solutions result.amplitude solutions result.

Subcritical Hopf bifurcations

µµ < 0< 0 µµ > 0> 0

The end of my scribbles for now

• Seminars from next Friday on –be there!