Introduction to XPP

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Introduction to XPP
Yanni Xiao Group meeting, 31st
January 2006
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What is XPP/ XPPAUT ?

• XPP is a tool for solving
• Differential equations
• Difference equations
• Delay equations
• Functional equations
• Boundary value problems
• Stochastic equations

XPP contains the code for the bifurcation program
AUTO (You can switch back and forth between
XPP and AUTO, using the values of one program
in the other and vice-versa)
Free download from http//www.math.pitt.edu/ba
rd/xpp/xpp.html
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Capabilities of XPP

• Handle up to 590 differential equations
• Over a dozen solvers including several for
  stiff systems(or integral equations)
• Up to 10 graphics windows can be visible at
  once
• PostScript output, or GIF
• Equilibria, linear stability and 1-d invariant
  sets can be computed
• Nullclines and flow fields aid in the
  qualitative understanding 2-d models
• Poincare maps and equations on cylinders and
  tori are supported
• Automatically generate movies of
  three-dimensional views of attractors or
  parametric changes in the attractor as some
  parameters vary.

The basic unit for Xpp a single ode file
including variables, parameters, equations and
etc.
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Creating and running an ODE file
The basic unit for XPP is a single ode file that
has

• Equations,
• Parameters,
• Variables,
• Boundary conditions,
• Functions

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Simple example Lorenz equation
Where s, r and b are parameters.
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ODE file for Lorenz equation
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Plotting options
AXES(2,3) determine whether a 2D or 3D plot will
be displayed. PHIvalue,THETAvalue
set the angles for the
three-dimensional plots XLOvalue,YLOvalue,
XHIvalue,YHIvalue set the limits for two
dimensional plots XMAXvalue, XMINvalue,
YMAXvalue, YMINvalue, ZMAXvalue, ZMINvalue
set the scaling for three-d plots.
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(No Transcript)
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Solutions (y versus t) to the Lorenz equation
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Solutions in 3-d to the Lorenz equation
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Bifurcation Calculations with AUTO Contents

• Preparation
• Choosing the hot parameters
• Choosing the plotting axes
• Setting up the numerical parameters
• Use defined points (EP, LP, HB)
• Running and continuing from a point
• Specially labeled points
• Traversing the diagram and selecting points
• Stropping a calculation
• Saving printing resetting diagrams

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AUTO window
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Example 1
A vaccination model with backward bifurcation
S Susceptibles I Infectives V
Vaccinated
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One-parameter bifurcation diagram
(Infecteds
versus the transmission rate )
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Example 2
SIRS model with three routes of transmission
S Susceptibles I Infectives R
Recovered W Infectious units in the
environment
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Two-parameter bifurcation diagram
LAS
US