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typical elm signals are shown in fig.[fig1 ] . here
, we try to model type - i and type - iii elm signals of @xmath0 .
jet typical elm signals ( [ perez2004 , thesis]).,width=321 ]    we just focus on the signal itself and ignore all the other information behind , i.e. , we just discuss how to use ( as simple as possible ) equations or other approaches to reproduce the shape of these signals .
it is qualitative or at most semi - quantitative .
this is a wide used approach for modeling . for example , the famous prey - predator model for fishbone and drift wave - zonal flow system .
one may also extend it to pde to contain the spatial information .
indeed , this way has been used to model elm and l - h transit in literatures , e.g. , [ diamond1994 ] , [ itoh1991 , 1993 , 1999 ] .    if we assume that the dynamic of signal is not explicit depend on time @xmath1 , then one equation @xmath2 is not enough because that @xmath3 is not single valued with @xmath4 .
so , at least , we must use second order equation or use two first order equations . while , one second order equation is just a special case of two first order equations .
in this example , we use the second option , i.e. , two first order equations , which is also used in [ diamond1994 ] .
the equations are @xmath5    a result is shown in fig . [
fig : ode1 ] .
ode for modeling elm , example 1.,width=321 ]    the physical explanations of the equations and parameters can be found in [ diamond1994 ] , however , where @xmath6 , @xmath7 .
using one second order equation , e.g. , @xmath8 which is modified from stiff system equation , a result is shown in fig .
[ fig : ode2 ] .
the shape is very similar to the pde result in [ itoh1993 ] .
but , an apparent drawback of ( [ eq : ode2 ] ) is that the signal @xmath4 is not always positive here .
ode for modeling elm , example 2.,width=321 ]
ddes contain derivatives which depend on previous time .
we guess some equations here .
the equations are from or modified from [ shampine2000 ] .
there is also a type of dde has sawtooth solutions ( see e.g. , [ mallet - paret2011 ] ) , which may be also suitable for model the unsolved sawteeth phenomena in magnetic confined fusion study .
the equation is @xmath9     dde for modeling elm , example 1.,width=321 ]    results are shown in fig .
[ fig : dde1 ] .
equation is @xmath10 which is original to model four - year cycle of the population of lemmings .
dde for modeling elm , example 2.,width=321 ]    the result is shown in fig .
[ fig : dde2 ] .
this is a more intuitive way to model elm , which combines different time scales using a more acceptable approach .
the burst of elm signal is suggested related to peeling - ballooning mode in standard elm model ( [ connor1998 ] ) .
so , we give two time scale : transport time scale and ballooning mode time scale .    to simplify the transport process , the transport time scale is modeled using @xmath11    when pressure gradient exceeds marginal value ( using @xmath12 to represent it ) , the ballooning mode occurs .
the signals will grow exponentially @xmath13 with the characteristic time @xmath14 .    when @xmath15 exceeds a marginal value @xmath16 , the plasma crash .
and then , go to next cycle .
we also add some randomicities to @xmath12 and @xmath16 to make the whole signal more like the actual experimental signal . a result
is shown in fig .
[ fig : demc ] .
demc modeling for elm.,width=321 ]    this simple demc model captures some fundamental properties of integrated simulations , such as [ chang2008 ] and [ lonnroth2009 ] .
which one is more reasonable ? at present , i choose demc @xmath17 ode @xmath17 dde . while , maybe none of them can be final explanation of elm or h - mode .
there is another math tool called fractional differential equation ( fde ) , which has been used for anomalous transport / diffusion problems in some areas . for plasma community
, fde is not wide accepted yet , but it is possible in the future .
fde is hard to give periodic solutions as elm at present .
so , i havent given examples .
chang , c. s. ; klasky , s. ; cummings , j. ; samtaney , r. ; shoshani , a. ; sugiyama , l. ; keyes , d. ; ku , s. ; park , g. ; parker , s. ; podhorszki , n. ; strauss , h. ; abbasi , h. ; adams , m. ; barreto , r. ; bateman , g. ; bennett , k. ; chen , y. ; azevedo , e. d. ; docan , c. ; ethier , s. ; feibush , e. ; greengard , l. ; hahm , t. ; hinton , f. ; jin , c. ; khan , a. ; kritz , a. ; krsti , p. ; lao , t. ; lee , w. ; lin , z. ; lofstead , j. ; mouallem , p. ; nagappan , m. ; pankin , a. ; parashar , m. ; pindzola , m. ; reinhold , c. ; schultz , d. ; schwan , k. ; silver , d. ; sim , a. ; stotler , d. ; vouk , m. ; wolf , m. ; weitzner , h. ; worley , p. ; xiao , y. ; yoon , e. and zorin , d. , toward a first - principles integrated simulation of tokamak edge plasmas , journal of physics : conference series , 2008 , 125 , 012042 .
mallet - paret , j. and nussbaum , r. d. , superstability and rigorous asymptotics in singularly perturbed state - dependent delay - differential equations , journal of differential equations , 2011 , 250 , 4037 - 4084 .
there is no wide accepted theory for elm ( edge localized mode ) yet . 
 some fusion people feel that we may never get a final theory for elm and h - mode , since which are too complicated ( also related to the unsolved turbulence problem ) and with at least three time scales . 
 the only way out is using models . 
 ( this is analogous to that we believe quantum mechanics can explain chemistry and biology , but no one can calculate dna structure from schrodinger equation directly . ) 
 this manuscript gives some possible mathematical approaches to it . 
 i should declare that these are just math toys for me yet . 
 they may inspire to good understandings of elm and h - mode , may not . useful or useless , i do nt know . 
 one need not take too much care of it . 
 just for fun and enjoying different interesting ideas .