Transcript Folie 1 - Tel Aviv University

Bifurcations & XPPAUT
Outline
•
•
•
Why to study the phase space?
Bifurcations / AUTO
Morris-Lecar
A Geometric Way of Thinking
x  sin( x)
Exact solution:
csc( x0 )  cot( x0 )
t  ln
csc( x)  cot( x)
Logistic Differential Equation
N



N  rN 1  
 K
N
N
K
K
K/2
N
t
Graphical/Topological Analysis
y
xyx  yy


0

y
y

kx
x
2
2


x . k  x
y  kx  const
y xy  kkx x x
d 2
1 d 2
y  2 yy  yy 
y
dt
2 dt


• When do we understand a dynamical system?
2
2
• Is an analytical solution better?
• Often no analytical solution to nonlinear systems.
d
0
y  kx
dt
Dynamics of Two Dimensional Systems
1.
2.
3.
Find the fixed points
in the phase space!
Linearize the system
about the fixed
points!
Determine the
eigenvalues of the
Jacobian.
Love Affairs
• Romeo loves Juliet. The more Juliet loves
him the more he wants her:
R  aJ
• Juliet is a fickle lover. The more Romeo loves
her, the more she wants to run away.
J  bR
J
R
Exercise 1
Study with AUTO (see later) the forcast for lovers
governed by the general linear system:
R  aR  bJ
J  cR  dJ
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. Sheep
We 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
Principle
of Competitive
Exclusion:
that these
conflicts occur
at a rate proportional to the
of each
population for
and the
reduce
the limited
growth rate
Twosize
species
competing
same
for each species (more severely for the rabbits!).
x  x(3  x  2 y )
y  y (2  x  y )
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)
Prototypical example:
x
x  b  x 2
x
x
*
b
Synchronisation of Fireflies
Synchronised Fireflies
Suppose
 (0)

 (t )
is the phase of the firefly‘s flashing.
is the instant when the flash is emitted.
is its eigen-frequency.
If the stimulus  (t ) 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:
    A sin    
Synchronised Fireflies II
The equation
    A sin    
can be simplyfied by introducing relative phases:     
Which yields:
Introducing
          A sin  
  At
and
 

A
We obtain the non-dimensionalised equation:      sin
 
Transcritical Bifurcatoin
x  bx  x
Prototypical example:
x
2
x
x
*
b
Pitchfork Bifurcation
x  bx  x
Prototypical example:
x
3
x
x
*
b
Hopf-Bifurcation
Prototypical example:
x*
x  x  y  x( x 2  y 2 )
y  x  y  y( x 2  y 2 )

AUTO
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…