Bifurcations & XPPAUT

Outline

• Why to study the phase space?• Bifurcations / AUTO• Morris-Lecar

A Geometric Way of Thinking

)sin(xx

)cot()csc(

)cot()csc(ln 00

xx

xxt

Exact solution:

Logistic Differential Equation

K

NrNN 1

N

N

K

K/2

N

t

K

Graphical/Topological Analysis

• When do we understand a dynamical system?• Is an analytical solution better?• Often no analytical solution to nonlinear

systems.

xkx xky

yx

xkxxy

yyyx

xkxyy 0

22

2

12 y

dt

dyyyyy

dt

d

220 kxydt

d

.22 constkxy

y

x

Dynamics of Two Dimensional Systems

1. Find the fixed points in the phase space!

2. Linearize the system about the fixed points!

3. Determine the eigenvalues of the Jacobian.

Love Affairs

• Romeo loves Juliet. The more Juliet loves him the more he wants her:

• Juliet is a fickle lover. The more Romeo loves her, the more she wants to run away.

aJR

bRJ J

R

Exercise 1 Study with AUTO (see later) the forcast for

lovers governed by the general linear system:

dJcRJ

bJaRR

Consider combinations of different types of lovers, e.g.

• The “eager beaver” (a>0,b>0), who gets excited by Juliet’s love and is spurred by his own affectionate feelings.

• The “cautious lover” (a<0,b>0). Can he find true love with an eager beaver?

• What about two identical cautious lovers?

Rabbit vs. SheepWe begin with the classic Lotka-Volterra model of

competion between two species competing for the same (limited) food supply.

1. Each species would grow to its carrying capacity in the absence of the other. (Assume logistic growth!)

2. Rabbits have a legendary ability to reproduce, so we should assign them a higher intrinsic growth rate.

3. When rabbits and sheep encounter each other, trouble starts. Sometimes the rabbit gets to eat but more usually the sheep nudges the rabbit aside. We assume that these conflicts occur at a rate proportional to the size of each population and reduce the growth rate for each species (more severely for the rabbits!).

)2(

)23(

yxyy

yxxx

Principle of Competitive Exclusion:

Two species competing for the same limited resource typically cannot coexist.

Exercise 2

Study the phase space of the Rabbit vs. Sheep problem for different parameter. Try to compute the bifurcation diagram (see later in this lecture!) with respect to some parameter.

What is a bifurcation?

Saddle Node Bifurcation (1-dim)2xbx Prototypical

example:

x

x

b

*x

Synchronisation of Fireflies

Synchronised Fireflies

Suppose is the phase of the firefly‘s flashing.

is the instant when the flash is emitted.

is its eigen-frequency.

If the stimulus with frequency is ahead in the cycle, then we assume that the firefly speeds up. Conversely, the firefly slows down if it‘s flashing is too early. A simple model is:

)(t

)0(

)(t

sinA

Synchronised Fireflies II sinA

sinA

AtA

sin

The equation

can be simplyfied by introducing relative phases:

Which yields:

Introducing and

We obtain the non-dimensionalised equation:

Transcritical Bifurcatoin

Prototypical example: 2xbxx

x

x

b

*x

Pitchfork Bifurcation

Prototypical example: 3xbxx

x

x

b

*x

Hopf-Bifurcation

)(

)(22

22

yxyyxy

yxxyxx

Prototypical example:

AUTO

*x

Exercise 3

Repeat the Bifurcation analysis for all prototypical cases mentioned above!

The Morris Lecar System

Further Exercises

• Analyse the QIF model with Auto.• Perform the bifurcation analysis for the

Morris-Lecar system. • Perform a phase space/bifurcation

analysis for the Fitzhugh-Nagumo system.

• Perform a phase space/bifurcation analysis for the Hodgkin-Huxley system.

• Use the manual for XPPaut 5.41 and try out some of the examples given in there.

Bibliography

• Nonlinear Dynamics and Chaos, Strogatz

• Understanding Nonlinear Dynamics, Kaplan & Glass

• Simulating, Analysing, and Animating Dynamical Systems, Ermentrout

• Dynamical Systems in Neuroscience, Izhikevich

• Mathematik der Selbstorganisation, Jetschke

End of this lecture…