The folder contains all the files necessary to compute rigorously  periodic orbits for the Lorenz system and the Floquet normal form of the fundamental matrix solution of the associated linearized problem, as discussed in
"Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits." (R.Castelli, J.-P. Lessard, 2012).

The interval arithmetic package Intlab is required.

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The main file is MainComputation.m where the user needs to set up some of the variables:

rho=    set the value of rho in the Lorenz system. (The other classical parameters are fixed beta=8/3, sigma=10).
        Only admits rho in [13.9265, 24.736]

sstar=  Decay rate parameter for the Fourier coefficients of the periodic orbit (sstar>=2)

s=      Decay rate parameter for the Fourier coefficients of the solution Q(t) in the Floquet normal form (2<= s <=sstar)

Mgamma= Finite dimensional reduction parameter for the Lorenz system.

m=      Finite dimensional reduction parameter for the Floquet system

M=      Computational parameter (M>=Mgamma+m)

proof=  0 or 1. If 0 all the routines run without interval arithmetics, if 1 interval arithmetics is used.

Once the variables are set, just run MainComputation

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The main code addresses 4 different computations:

1- Compute a numerical periodic solution for the Lorenz system
2- Compute the bounds around the numerical solution within which a genuine solution exists ( rgamma)

If the computation of  1-2  is successful

3- Compute a numerical solution for the Floquet normal form of the fundamental matrix solution 
4- Compute the bound around the numerical solution of the Floquet system within which a genuine solution exists  ( rfloquet)

If all the computations are successful these output are produced:
rgamma, rfloquet
Period ( period of the  orbit)
Periodic_solution ( Fourier coefficient gamma_k of the periodic orbit, k=0…Mgamma-1)
matrixR ( numerical solution for the matrix R in the Floquet normal form)
matrixC ( Fourier coefficient Q_k of the periodic orbit function Q(t) in the Floquet normal form, k=0…m-1)

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HINT: Since the computation in interval arithmetics could be time-consuming, it is advised to perform first no rigorous  computations ( proof=0), then, if successful, to switch proof=1. 

  Example
 Set:  rho= 18.1    sstar=s=2  Mgamma=40    m=50    M=90   proof=0

 Successful computation of points 1 and 2 : rgamma=9.208046e-14
 The computation of point 4 fails. ----> Try to increase m or M
  
 Set:  rho= 18.1    sstar=s=2  Mgamma=40    m=60    M=120   proof=0

 It works: rgamma =9.208046e-14, rfloquet =4.49854771e-11
 
 Set:  rho= 18.1    sstar=s=2  Mgamma=40    m=60    M=120   proof=1
 
 It still works: rgamma=1.789786e-12, rfloquet=2.8823205e-09

