Bifurcations

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Bifurcations XPPAUT
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Outline

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

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A Geometric Way of Thinking
Exact solution
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Logistic Differential Equation
K
K
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Graphical/Topological Analysis

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

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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.

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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.

J
R
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Exercise 1

• Study with AUTO (see later) the forcast for
  lovers governed by the general linear system

• Consider combinations of different types of
  lovers, e.g.
• The eager beaver (agt0,bgt0), who gets excited by
  Juliets love and is spurred by his own
  affectionate feelings.
• The cautious lover (alt0,bgt0). Can he find true
  love with an eager beaver?
• What about two identical cautious lovers?

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Rabbit vs. Sheep

• We begin with the classic Lotka-Volterra model of
  competion between two species competing for the
  same (limited) food supply.

• Each species would grow to its carrying
  capacity in the absence of the other. (Assume
  logistic growth!)
• Rabbits have a legendary ability to reproduce,
  so we should assign them a higher intrinsic
  growth rate.
• 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!).

Principle of Competitive Exclusion Two species
competing for the same limited resource typically
cannot coexist.
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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.

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What is a bifurcation?
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Saddle Node Bifurcation (1-dim)
Prototypical example
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Synchronisation of Fireflies
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Synchronised Fireflies
Suppose is the phase of the fireflys
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 its
flashing is too early. A simple model is
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Synchronised Fireflies II
The equation can be simplyfied by introducing
relative phases Which yields
Introducing and
We obtain the non-dimensionalised equation
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Transcritical Bifurcatoin

• Prototypical example

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Pitchfork Bifurcation
Prototypical example
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Hopf-Bifurcation
Prototypical example
AUTO
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Exercise 3

• Repeat the Bifurcation analysis for all
  prototypical cases mentioned above!

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The Morris Lecar System
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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.

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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

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End of this lecture