Renormalization group (RG) for the break-up of invariant tori in Hamiltonian flows

• RG_dict.py: to be edited to change the parameters of the RG computation (see below for a dictionary of parameters)

• RG.py: contains the RG classes and main functions defining the RG map

• RG_modules.py: contains the methods to execute the RG map

Once RG_dict.py has been edited with the relevant parameters, run the file as

python3 RG.py

or

nohup python3 -u RG.py &>RG.out < /dev/null &

The list of Python packages and their version are specified in requirements.txt

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Parameter dictionary

• Method: string; 'iterates', 'surface', 'region', 'line'; choice of method
  • 'iterates': starting from two Hamiltonians H₁ and H₂ defined with the modes K and amplitudes AmpInf and AmpSup respectively, the method first refines H₁ and H₂ close to the critical surface using a dichotomy procedure. Second, it iterates these two Hamiltonians by iterating and refining with the renormalization map
  • 'line': for the family of Hamiltonians defined by the modes K in the direction DirLine with ModesLine=1, determines the critical threshold
  • 'surface': computes the critical surface in the plane of Fourier modes defined by K and ModesLine (the two modes K with ModesLine=1)
  • 'region': in the plane of Fourier modes defined by K and ModesLine (the two modes K with ModesLine=1) with amplitudes in the range defined by AmpInf and AmpSup, determines the number of iterations for the Hamiltonian to converge (negative integers) or diverge (positive integers)
• Iterates: integer; number of iterates to compute for Method='iterates'
• Nxy: integer; number of points along each direction for Method='surface' or 'region'
• RelDist: float; relative distance of approach for the computation of critical values

• N: nxn integer matrix with determinant ±1
• omega0: array of n floats; frequency vector ω of the invariant torus; should be an eigenvector of ^†N (transposed matrix of N)
• Omega: array of n floats; vector Ω of the perturation in action
• K: 2-dimensional tuple of integers; wavevectors (j,ν)=(j,k₁,...,k_n) of the perturbation
• AmpInf: array of len(K) floats; minimal amplitudes of the perturbation
• AmpSup: array of len(K) floats; maximum amplitudes of the perturbation
• CoordLine: 1d array of floats; min and max values of the amplitudes of the potential used in Method='line'
• ModesLine: tuple of 0 and 1 of length len(K); specify which modes are being varied (1 for a varied mode)
• DirLine: 1d array of floats; direction of the one-parameter family used in Method='line'

• L: integer; truncation in Fourier series (angles)
• J: integer; truncation in Taylor series (actions)

• ChoiceIm: string; 'AK2000', 'K1999', 'AKW1998'; definition of I^-
• Sigma: float; definition of I^-
• Kappa: float; definition of I^-

• CanonicalTransformation: string; 'expm_onestep', 'expm_adapt', 'expm_multiply'; method to compute the canonical Lie transforms
  • 'expm_onestep': compute the exponential of the Liouville operator in one single step
  • 'expm_adapt': use an adaptative step-size method to compute the exponential of the Liouville operator with AbsTol and RelTol as tolerance parameters, and MinStep as the minimum value of the step to be used
  • 'expm_multiply': use the algorithm developed in A.H. Al-Mohy, N.J. Higham, SIAM Journal on Scientific Computing 33, 488 (2011) to compute the exponential of the Liouville operator
• MinStep: float; minimum value of the steps in the adaptive procedure to compute exponentials (for 'expm_adapt')
• AbsTol: float; absolute tolerance for the adaptive procedure to compute exponentials (for 'expm_adapt')
• RelTol: float; relative tolerance for the adaptive procedure to compute exponentials (for 'expm_adapt')
• MaxLie: integer; maximum number of Lie transforms to be performed to eliminate the non-resonant part of the perturbation

• TolMax: float; value of the norm of the Hamiltonian for divergence
• TolMin: float; value of the norm of the Hamiltonian for convergence

• Precision: integer; 32, 64 or 128; precision of calculations (default=64)
• NormChoice: string; 'sum', 'max', 'Euclidean', 'Analytic'; choice of Hamiltonian norm
• NormAnalytic: float; parameter of norm 'Analytic'

• SaveData: boolean; if True, the results are saved in a .mat file
• PlotResults: boolean; if True, the results are plotted right after the computation
• Parallelization: tuple (boolean, int); True for parallelization, int is the number of cores to be used (all of them: int='all')

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Error codes

• 0: all transformations have been properly computed (no error)
• 1: one of the Lie transforms is not accurately computed
• 2: the series of canonical transformations to eliminate the non-resonant part of the Hamiltonian is diverging
• -2: the series of canonical transformations to eliminate the non-resonant part of the Hamiltonian does not converge or diverge (MaxLie iterations reached)
• 3: the iterates of the RG map on the initially generated Hamiltonian H₁ diverge (H₁ is above the critical surface)
• -3: the iterates of the RG map on the initially generated Hamiltonian H₂ converge (H₂ is below the critical surface)
• 4: the step in the adaptive step-size computation of the Lie transform ('expm_adapt') is below the minimum defined step size (MinStep)

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References:

• C. Chandre, H.R. Jauslin, Renormalization-group analysis for the transition to chaos in Hamiltonian systems, Physics Reports 365, 1 (2002)

@article{chandre2002,
         title = {Renormalization-group analysis for the transition to chaos in Hamiltonian systems},
         author = {C. Chandre and H.R. Jauslin},
         journal = {Physics Reports},
         volume = {365},
         number = {1},
         pages = {1-64},
         year = {2002},
         issn = {0370-1573},
         doi = {https://doi.org/10.1016/S0370-1573(01)00094-1},
         url = {https://www.sciencedirect.com/science/article/pii/S0370157301000941}, 
}

• A.P Bustamante, C. Chandre, Numerical computation of critical surfaces for the breakup of invariant tori in Hamiltonian systems, SIAM Journal on Applied Dynamical Systems 22, 483 (2023)

@article{bustamante2023,
      title = {Numerical computation of critical surfaces for the breakup of invariant tori in Hamiltonian systems}, 
      author = {Adrian P. Bustamante and Cristel Chandre},
      journal = {SIAM Journal on Applied Dynamical Systems},
      volume = {22},
      number = {1},
      pages = {483-500},
      year = {2023},
      issn = {1536-0040},
      doi = {https://doi.org/10.1137/21M1448501},
}

For more information: cristel.chandre@cnrs.fr