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# Kolmogorov–Arnold–Moser theorem

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 Title: Kolmogorov–Arnold–Moser theorem Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Kolmogorov–Arnold–Moser theorem

The Kolmogorov–Arnold–Moser theorem (KAM theorem) is a result in dynamical systems about the persistence of quasi-periodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory of classical mechanics.

The problem is whether or not a small perturbation of a conservative dynamical system results in a lasting quasiperiodic orbit. The original breakthrough to this problem was given by Andrey Kolmogorov in 1954. This was rigorously proved and extended by Vladimir Arnold (in 1963 for analytic Hamiltonian systems) and Jürgen Moser (in 1962 for smooth twist maps), and the general result is known as the KAM theorem. The KAM theorem, as it was originally stated, could not be applied directly as a whole to the motions of the solar system because of the presence of degeneracy in the unperturbed Kepler problem. However, it is useful in generating corrections of astronomical models, and to prove long-term stability and the avoidance of orbital resonance in solar system. Arnold used the methods of KAM to prove the stability of elliptical orbits in the planar three-body problem.

## Statement

The KAM theorem is usually stated in terms of trajectories in phase space of an integrable Hamiltonian system. The motion of an integrable system is confined to a doughnut-shaped surface, an invariant torus. Different initial conditions of the integrable Hamiltonian system will trace different invariant tori in phase space. Plotting the coordinates of an integrable system would show that they are quasi-periodic.

The KAM theorem states that if the system is subjected to a weak nonlinear perturbation, some of the invariant tori are deformed and survive, while others are destroyed. The ones that survive are those that meet the non-resonance condition, i.e., they have “sufficiently irrational” frequencies. This implies that the motion continues to be quasiperiodic, with the independent periods changed (as a consequence of the non-degeneracy condition). The KAM theorem specifies quantitatively what level of perturbation can be applied for this to be true. An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.

The methods introduced by Kolmogorov, Arnold, and Moser have developed into a large body of results related to quasi-periodic motions, now known as KAM theory. Notably, it has been extended to non-Hamiltonian systems (starting with Moser), to non-perturbative situations (as in the work of Michael Herman) and to systems with fast and slow frequencies (as in the work of Mikhail B. Sevryuk).

The non-resonance and non-degeneracy conditions of the KAM theorem become increasingly difficult to satisfy for systems with more degrees of freedom. As the number of dimensions of the system increases, the volume occupied by the tori decreases.

Those KAM tori that are not destroyed by perturbation become invariant Cantor sets, named Cantori by Ian C. Percival in 1979.

As the perturbation increases and the smooth curves disintegrate we move from KAM theory to Aubry-Mather theory which requires less stringent hypotheses and works with the Cantor-like sets.

## References

• Arnold, Weinstein, Vogtmann. Mathematical Methods of Classical Mechanics, 2nd ed., Appendix 8: Theory of perturbations of conditionally periodic motion, and Kolmogorov's theorem. Springer 1997.
• require('Module:No globals')

local function getCatForId( id )

```   local title = mw.title.getCurrentTitle()
local namespace = title.namespace
if namespace == 0 then
```

return ''

```   elseif namespace == 2 and not title.isSubpage then
```

return ''

```   else
```

return ''

```   end
```

end

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'VIAF' )
```

end

```   return '.. id .. ' id'
```

end

```   return '.. id .. '&lng=en ' .. id .. ''
```

end

local function tlsLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'WIKI') end)

```   return '.. id2 .. ' ' .. id .. ''
```

end

```   return '.. id .. '?l=en ' .. id .. ''
```

end

```   return '.. id .. ' ' .. id .. ''
```

end

```   return '.. id .. ' ' .. id .. ''
```

end

```   return '.. id .. ' ' .. id .. ''
```

end

```   return '.. id .. ' ' .. id .. ''
```

end

local function botanistLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'PATH') end)

```   return '.. id2 .. ' ' .. id .. ''
```

end

```   -- TODO Implement some sanity checking regex
return '.. id .. ' ' .. id .. ''
```

end

```   -- TODO Implement some sanity checking regex
return '.. id .. '&CON_LNG=ENG ' .. id .. ''
```

end

local function leonoreLink( id ) -- Identifiants allant de LH/1/1 à LH/2794/54 (légionnaires) -- Identifiants allant de C/0/1 à C/0/84 (84 légionnaires célèbres) -- Identifiants allant de 19800035/1/1 à 19800035/385/51670 (légionnaires décédés entre 1954 et 1977, et quelques dossiers de légionnaires décédés avant 1954)

```   if not string.match( id, '^LH/%d%d?%d?%d?/%d%d?%d?\$' ) and
not string.match( id, '^C/0/%d%d?\$' ) and
not string.match( id, '^19800035/%d%d?%d?%d?/%d%d?%d?%d?%d?\$' ) then
return false
end
return '.. id .. ' ' .. id .. ''
```

end

```   if not string.match( id, '^IT\\ICCU\\%d%d%d%d%d%d%d%d%d%d\$' ) and not string.match( id, '^IT\\ICCU\\%u%u[%d%u]%u\\%d%d%d%d%d%d\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'SBN' )
```

end

local function nkcLink( id ) return '.. id .. '&CON_LNG=ENG ' .. id .. '' end

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '&CON_LNG=ENG ' .. id .. ''
```

end

local function ndlLink( id ) return '.. id .. ' ' .. id .. '' end

```   if not string.match( id, '^%d%d%d%d%d%d%d%d[%dxX]\$' ) then
return false
end
return '.. id .. ' ' .. id .. ''
```

end

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '.php ' .. id .. ''
```

end

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '.450.0.html ' .. id .. ''
```

end

local function splitLccn( id )

```   if id:match( '^%l%l?%l?%d%d%d%d%d%d%d%d%d?%d?\$' ) then
id = id:gsub( '^(%l+)(%d+)(%d%d%d%d%d%d)\$', '%1/%2/%3' )
end
if id:match( '^%l%l?%l?/%d%d%d?%d?/%d+\$' ) then
return mw.text.split( id, '/' )
end
return false
```

end

local function append(str, c, length)

```   while str:len() < length do
str = c .. str
end
return str
```

end

```   local parts = splitLccn( id )
if not parts then
return false
end
local lccnType = parts[1] ~= 'sh' and 'names' or 'subjects'
id = parts[1] .. parts[2] .. append( parts[3], '0', 6 )
return '.. lccnType .. '/' .. id .. ' ' .. id .. '' .. getCatForId( 'LCCN' )
```

end

```   -- TODO Implement some sanity checking regex
return '.. id .. ' ' .. id .. '' .. getCatForId( 'MusicBrainz' )
```

end

--Returns the ISNI check digit isni must be a string where the 15 first elements are digits local function getIsniCheckDigit( isni )

```   local total = 0
for i = 1, 15 do
local digit = isni:byte( i ) - 48 --Get integer value
total = (total + digit) * 2
end
local remainder = total % 11
local result = (12 - remainder) % 11
if result == 10 then
return "X"
end
```

end

--Validate ISNI (and ORCID) and retuns it as a 16 characters string or returns false if it's invalid --See http://support.orcid.org/knowledgebase/articles/116780-structure-of-the-orcid-identifier local function validateIsni( id )

```   id = id:gsub( '[ %-]', ):upper()
if not id:match( '^%d%d%d%d%d%d%d%d%d%d%d%d%d%d%d[%dX]\$' ) then
return false
end
if getIsniCheckDigit( id ) ~= string.char( id:byte( 16 ) ) then
return false
end
return id
```

end

```   id = validateIsni( id )
if not id then
return false
end
return '.. id .. ' ' .. id:sub( 1, 4 ) .. ' ' .. id:sub( 5, 8 ) .. ' '  .. id:sub( 9, 12 ) .. ' '  .. id:sub( 13, 16 ) .. '' .. getCatForId( 'ISNI' )
```

end

```   id = validateIsni( id )
if not id then
return false
end
id = id:sub( 1, 4 ) .. '-' .. id:sub( 5, 8 ) .. '-'  .. id:sub( 9, 12 ) .. '-'  .. id:sub( 13, 16 )
return '.. id .. ' ' .. id .. '' .. getCatForId( 'ORCID' )
```

end

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'GND' )
```

end

local function selibrLink( id ) if not string.match( id, '^%d+\$' ) then

```       return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'SELIBR' )
```

end

```   --Add cb prefix if it has been removed
if not string.match( id, '^cb.+\$' ) then
id = 'cb' .. id
end
```
```   return '.. id .. ' ' .. id .. ' .. id .. ' (data)' .. getCatForId( 'BNF' )
```

end

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'BPN' )
```

end

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'RID' )
```

end

```   return '.. id .. '&feltselect=bs.autid ' .. id .. '' .. getCatForId( 'BIBSYS' )
```

end

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'ULAN' )
```

end

local function nlaLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'NLA' ) end

local function rkdartistsLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'RKDartists' ) end

local function getIdsFromWikidata( item, property )

```   local ids = {}
if not item.claims[property] then
return ids
end
for _, statement in pairs( item.claims[property] ) do
if statement.mainsnak.datavalue then
table.insert( ids, statement.mainsnak.datavalue.value )
end
end
return ids
```

end

local function matchesWikidataRequirements( item, reqs )

```   for _, group in pairs( reqs ) do
local property = 'p' .. group[1]
local qid = group[2]
if item.claims[property] ~= nil then
for _, statement in pairs ( item.claims[property] ) do
if statement.mainsnak.datavalue ~= nil then
if statement.mainsnak.datavalue.value['numeric-id'] == qid then
return true
end
end
end
end
end
return false
```

end

local function createRow( id, label, rawValue, link, withUid )

```   if link then
if withUid then
return '* ' .. label .. ' ' .. link .. '\n'
else
return '* ' .. label .. ' ' .. link .. '\n'
end
else
```

return '* \n'

```   end
```

end

--In this order: name of the parameter, label, propertyId in Wikidata, formatting function local conf = {

```   { 'VIAF', 'VIAF', 214, viafLink },
{ 'LCCN', 'LCCN', 244, lccnLink },
{ 'ISNI', 'ISNI', 213, isniLink },
{ 'ORCID', 'ORCID', 496, orcidLink },
{ 'GND', 'GND', 227, gndLink },
{ 'SELIBR', 'SELIBR', 906, selibrLink },
{ 'SUDOC', 'SUDOC', 269, sudocLink },
{ 'BNF', 'BNF', 268, bnfLink },
{ 'BPN', 'BPN', 651, bpnLink },
{ 'RID', 'ResearcherID', 1053, ridLink },
{ 'BIBSYS', 'BIBSYS', 1015, bibsysLink },
{ 'ULAN', 'ULAN', 245, ulanLink },
{ 'HDS', 'HDS', 902, hlsLink },
{ 'LIR', 'LIR', 886, lirLink },
{ 'MBA', 'MusicBrainz', 434, mbLink },
{ 'MGP', 'MGP', 549, mgpLink },
{ 'NLA', 'NLA', 409, nlaLink },
{ 'NDL', 'NDL', 349, ndlLink },
{ 'NCL', 'NCL', 1048, nclLink },
{ 'NKC', 'NKC', 691, nkcLink },
{ 'Léonore', 'Léonore', 640, leonoreLink },
{ 'SBN', 'ICCU', 396, sbnLink },
{ 'RLS', 'RLS', 947, rslLink },
{ 'Botanist', '[[Author ci
```
• require('Module:No globals')

local function getCatForId( id )

```   local title = mw.title.getCurrentTitle()
local namespace = title.namespace
if namespace == 0 then
```

return ''

```   elseif namespace == 2 and not title.isSubpage then
```

return ''

```   else
```

return ''

```   end
```

end

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'VIAF' )
```

end

```   return '.. id .. ' id'
```

end

```   return '.. id .. '&lng=en ' .. id .. ''
```

end

local function tlsLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'WIKI') end)

```   return '.. id2 .. ' ' .. id .. ''
```

end

```   return '.. id .. '?l=en ' .. id .. ''
```

end

```   return '.. id .. ' ' .. id .. ''
```

end

```   return '.. id .. ' ' .. id .. ''
```

end

```   return '.. id .. ' ' .. id .. ''
```

end

```   return '.. id .. ' ' .. id .. ''
```

end

local function botanistLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'PATH') end)

```   return '.. id2 .. ' ' .. id .. ''
```

end

```   -- TODO Implement some sanity checking regex
return '.. id .. ' ' .. id .. ''
```

end

```   -- TODO Implement some sanity checking regex
return '.. id .. '&CON_LNG=ENG ' .. id .. ''
```

end

local function leonoreLink( id ) -- Identifiants allant de LH/1/1 à LH/2794/54 (légionnaires) -- Identifiants allant de C/0/1 à C/0/84 (84 légionnaires célèbres) -- Identifiants allant de 19800035/1/1 à 19800035/385/51670 (légionnaires décédés entre 1954 et 1977, et quelques dossiers de légionnaires décédés avant 1954)

```   if not string.match( id, '^LH/%d%d?%d?%d?/%d%d?%d?\$' ) and
not string.match( id, '^C/0/%d%d?\$' ) and
not string.match( id, '^19800035/%d%d?%d?%d?/%d%d?%d?%d?%d?\$' ) then
return false
end
return '.. id .. ' ' .. id .. ''
```

end

```   if not string.match( id, '^IT\\ICCU\\%d%d%d%d%d%d%d%d%d%d\$' ) and not string.match( id, '^IT\\ICCU\\%u%u[%d%u]%u\\%d%d%d%d%d%d\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'SBN' )
```

end

local function nkcLink( id ) return '.. id .. '&CON_LNG=ENG ' .. id .. '' end

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '&CON_LNG=ENG ' .. id .. ''
```

end

local function ndlLink( id ) return '.. id .. ' ' .. id .. '' end

```   if not string.match( id, '^%d%d%d%d%d%d%d%d[%dxX]\$' ) then
return false
end
return '.. id .. ' ' .. id .. ''
```

end

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '.php ' .. id .. ''
```

end

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '.450.0.html ' .. id .. ''
```

end

local function splitLccn( id )

```   if id:match( '^%l%l?%l?%d%d%d%d%d%d%d%d%d?%d?\$' ) then
id = id:gsub( '^(%l+)(%d+)(%d%d%d%d%d%d)\$', '%1/%2/%3' )
end
if id:match( '^%l%l?%l?/%d%d%d?%d?/%d+\$' ) then
return mw.text.split( id, '/' )
end
return false
```

end

local function append(str, c, length)

```   while str:len() < length do
str = c .. str
end
return str
```

end

```   local parts = splitLccn( id )
if not parts then
return false
end
local lccnType = parts[1] ~= 'sh' and 'names' or 'subjects'
id = parts[1] .. parts[2] .. append( parts[3], '0', 6 )
return '.. lccnType .. '/' .. id .. ' ' .. id .. '' .. getCatForId( 'LCCN' )
```

end

```   -- TODO Implement some sanity checking regex
return '.. id .. ' ' .. id .. '' .. getCatForId( 'MusicBrainz' )
```

end

--Returns the ISNI check digit isni must be a string where the 15 first elements are digits local function getIsniCheckDigit( isni )

```   local total = 0
for i = 1, 15 do
local digit = isni:byte( i ) - 48 --Get integer value
total = (total + digit) * 2
end
local remainder = total % 11
local result = (12 - remainder) % 11
if result == 10 then
return "X"
end
```

end

--Validate ISNI (and ORCID) and retuns it as a 16 characters string or returns false if it's invalid --See http://support.orcid.org/knowledgebase/articles/116780-structure-of-the-orcid-identifier local function validateIsni( id )

```   id = id:gsub( '[ %-]', ):upper()
if not id:match( '^%d%d%d%d%d%d%d%d%d%d%d%d%d%d%d[%dX]\$' ) then
return false
end
if getIsniCheckDigit( id ) ~= string.char( id:byte( 16 ) ) then
return false
end
return id
```

end

```   id = validateIsni( id )
if not id then
return false
end
return '.. id .. ' ' .. id:sub( 1, 4 ) .. ' ' .. id:sub( 5, 8 ) .. ' '  .. id:sub( 9, 12 ) .. ' '  .. id:sub( 13, 16 ) .. '' .. getCatForId( 'ISNI' )
```

end

```   id = validateIsni( id )
if not id then
return false
end
id = id:sub( 1, 4 ) .. '-' .. id:sub( 5, 8 ) .. '-'  .. id:sub( 9, 12 ) .. '-'  .. id:sub( 13, 16 )
return '.. id .. ' ' .. id .. '' .. getCatForId( 'ORCID' )
```

end

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'GND' )
```

end

local function selibrLink( id ) if not string.match( id, '^%d+\$' ) then

```       return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'SELIBR' )
```

end

```   --Add cb prefix if it has been removed
if not string.match( id, '^cb.+\$' ) then
id = 'cb' .. id
end
```
```   return '.. id .. ' ' .. id .. ' .. id .. ' (data)' .. getCatForId( 'BNF' )
```

end

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'BPN' )
```

end

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'RID' )
```

end

```   return '.. id .. '&feltselect=bs.autid ' .. id .. '' .. getCatForId( 'BIBSYS' )
```

end

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'ULAN' )
```

end

local function nlaLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'NLA' ) end

local function rkdartistsLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'RKDartists' ) end

local function getIdsFromWikidata( item, property )

```   local ids = {}
if not item.claims[property] then
return ids
end
for _, statement in pairs( item.claims[property] ) do
if statement.mainsnak.datavalue then
table.insert( ids, statement.mainsnak.datavalue.value )
end
end
return ids
```

end

local function matchesWikidataRequirements( item, reqs )

```   for _, group in pairs( reqs ) do
local property = 'p' .. group[1]
local qid = group[2]
if item.claims[property] ~= nil then
for _, statement in pairs ( item.claims[property] ) do
if statement.mainsnak.datavalue ~= nil then
if statement.mainsnak.datavalue.value['numeric-id'] == qid then
return true
end
end
end
end
end
return false
```

end

local function createRow( id, label, rawValue, link, withUid )

```   if link then
if withUid then
return '* ' .. label .. ' ' .. link .. '\n'
else
return '* ' .. label .. ' ' .. link .. '\n'
end
else
```

return '* \n'

```   end
```

end

--In this order: name of the parameter, label, propertyId in Wikidata, formatting function local conf = {

```   { 'VIAF', 'VIAF', 214, viafLink },
{ 'LCCN', 'LCCN', 244, lccnLink },
{ 'ISNI', 'ISNI', 213, isniLink },
{ 'ORCID', 'ORCID', 496, orcidLink },
{ 'GND', 'GND', 227, gndLink },
{ 'SELIBR', 'SELIBR', 906, selibrLink },
{ 'SUDOC', 'SUDOC', 269, sudocLink },
{ 'BNF', 'BNF', 268, bnfLink },
{ 'BPN', 'BPN', 651, bpnLink },
{ 'RID', 'ResearcherID', 1053, ridLink },
{ 'BIBSYS', 'BIBSYS', 1015, bibsysLink },
{ 'ULAN', 'ULAN', 245, ulanLink },
{ 'HDS', 'HDS', 902, hlsLink },
{ 'LIR', 'LIR', 886, lirLink },
{ 'MBA', 'MusicBrainz', 434, mbLink },
{ 'MGP', 'MGP', 549, mgpLink },
{ 'NLA', 'NLA', 409, nlaLink },
{ 'NDL', 'NDL', 349, ndlLink },
{ 'NCL', 'NCL', 1048, nclLink },
{ 'NKC', 'NKC', 691, nkcLink },
{ 'Léonore', 'Léonore', 640, leonoreLink },
{ 'SBN', 'ICCU', 396, sbnLink },
{ 'RLS', 'RLS', 947, rslLink },
{ 'Botanist', '[[Author ci
```
• Rafael de la Llave (2001) A tutorial on KAM theory.
• KAM theory: the legacy of Kolmogorov’s 1954 paper
• Kolmogorov-Arnold-Moser theory from Scholarpedia
• H Scott Dumas. The KAM Story - A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov–Arnold–Moser Theory, 2014, World Scientific Publishing, ISBN 978-981-4556-58-3. Chapter 1: Introduction
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