Rigorous computation of global trajectories for FitzHugh-Nagumo system
Kaname Matsue 2015. 7/7
These code groups enable us to validate global trajectories of FitzHugh-Nagumo system for explicitly given multiscale parameter ranges. We validate the following type of global trajectories :
1. periodic orbits (fn_periodic)
2. heteroclinic cycles (fn_cycle)
3. heteroclinic orbits (fn_heteroclinic_ptoq, fn_heteroclinic_qtop)
4. homoclinic orbits (fn_homoclinic)
Sample validation results of these trajectories are listed in the following author's article:
"Rigorous numerics for fast-slow systems with one-dimensional slow variable: topological shadowing approach"
In order to play programs, we need the following library :
CAPD (Computer Assisted Proof of Dynamics) ver. 3.0
Note that it is NOT the latest version. See
http://capd.sourceforge.net/capdDynSys/download.php
for details.
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(Details of the procedure are added little by little. Latest ver. : 2015. 7/24)
The strategy for validating global orbits consists of the following processes:
1. validation of slow manifolds and slow shadowing properties
This process can be done by constructing isolating blocks and calculating inequalities involving the slow shadowing conditions. Run
./slowmfd
to complete this process. There is a choice of branches which we want to validate. Input 1 to validate the leftmost branch of nullcline. Similarly, input 2 to validate the rightmost branch. The central branch is out of our current interest.
2. Construction of extended cones
This process constructs extended stable and unstable m-cones to validate covering relations in the fast scale effectively. Run
./mcone_slowmfd
to complete this process.
This process also contains validation of singular isolating blocks on slow manifolds and extra m-cone conditions. It is necessary to validate unstable manifolds of equilibria "in the full system".
3. Fast trajectories
This process solves ODE to validate covering relations F -> S.
Initial data correspond to fast-exit faces of extended cones constructed in the previous process. We can check the covering relation graphically in fast variables as well as inclusion relations in slow variables. Run
./conn
to check covering relations from the leftmost branch to the rightmost one. Similarly, run
./conn2
to check covering relations from the rightmost branch to the leftmost one.