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transition between magnetic and non - magnetic phase of matter is intensively studied for a long time  @xcite . at low temperatures ,
elementary atomic magnets tend to be parallel providing coherent ( magnetic ) phase of the system .
with increase of the temperature , thermal fluctuations break this order drastically .
if the temperature is higher than the phase transition critical point ( curie point ) , then chaotic orientation of elementary magnets takes place ( non - magnetic phase ) .
however , near the critical point , the elementary magnets still keep order on certain range of distances and over certain time intervals , which decrease with growth of the temperature .
thus , at the critical point , areas of coherence and areas of thermal fluctuations coexist .
areas of fluctuations in the magnetic phase or areas of coherence in the non - magnetic phase may be distingueshed only within a certain range of scales . for
large enough scales the system looks completely ordered ( as at zero temperature ) or chaotic ( as at infinite temperature ) .
if one considers the same system at a definite temperature but within various scales , it looks as the system at different temperatures .
the transformation of scales corresponds to the temperature renormalization .
let we have a cubic lattice of @xmath0 atoms with interatomic distance @xmath1 at temperature @xmath2 .
if one considers this system at rough enough scale , at which the elementary cell has the characteristic size @xmath3 and contains @xmath4 atoms , the system looks as a lattice of @xmath5 atoms at the renormalized temperature @xmath6 ( @xmath7 is a function of renormalization ) . at the moment of the phase transition
, one can observe the thermal fluctuations within any scales .
these fluctuations are self - similar .
the idea of self - similarity has been offered by kadanoff  @xcite and further advanced by wilson to the renormalization method  @xcite ( see also  @xcite ) .
concepts of universality and scaling near the critical point and a renormalization group ( rg ) method have been transferred to nonlinear dynamics by feigenbaum  @xcite .
the analogy between transition to chaos and phase transitions consist in increase of temporal scales with approach to a critical point , self - similarity , an opportunity of application of rg analysis , and in existence of universal critical indexes and scaling factors , which are determined by the most general requirements to the type of system  @xcite .    in 1952
lee and yang advanced an approach in the phase transition theory based on consideration of analytical properties of some thermodynamic values , such as partition function and free energy , depending on the temperature considered formally as a complex variable  @xcite .
the partition function is defined as follows : @xmath8 where @xmath9 is a total energy of a configuration of the system , @xmath10 is an external magnetic field , @xmath2 is a temperature , @xmath11 is the boltzmann constant , @xmath12 - is the spin variable , taking place at @xmath13-th cell of the lattice .
spins interact via magnetic field with the nearest neighbours ( @xmath14 corresponds to ferromagnetic interaction , @xmath15  antiferromagnetic ) .
partition function can be transformed to @xmath16 by the variable change @xmath17 .
it is known , that at the phase transition the divergence of the thermodynamic potential ( free energy ) @xmath18 takes place .
obviously , zeros of the partition function ( roots of the algebraic equation @xmath19 ) are the candidates for the phase transition points
. however , these roots should be positive and real , which is impossible as all factors @xmath20 . in yang - lee theory ,
the temperature and the magnetic field are considered as complex variables  @xcite .
complex zeros of the partition function ( called the yang - lee zeros ) , gather near the real axis .
it becomes possible only in the thermodynamic limit , the asymptotics of infinite number of atoms .
when the number of atoms is finite , there is a finite number of zeros in the complex plane . with increase of number of atoms ,
the set of zeros becomes dense and nestles to the real axis more and more . in the thermodynamic limit
the yang - lee zeroes form a fractal set , that cross the real temperature axis at the point of the phase transition . to find the yang - lee zeros it is appropriate to apply the wilson renormalization method , which corresponds to a consecutive decrease of the number of degrees of freedom of the partition function .
it is necessary to find the transformation of the @xmath21-particle partition function to the @xmath0-particle one ( @xmath22 ) .
this transformation is not reversible .
thus , to get zeros of @xmath23 it is necessary to have zeros of @xmath24 , and then to construct their backward images using the rg transformation .
having repeated this procedure many times one comes finally to the trivial two - atom partition function .
if zeros of @xmath25 belong to a basin of attraction of a fixed point of the rg transformation , then zeros of @xmath23 in the thermodynamic limit will coincide to the boundary of the basin of attraction in the complex plane . as shown by derrida
@xcite , this boundary , which is the julia set of the rg transformation , is identical to the set of yang - lee zeros .
unfortunately , the renormalization transformation may be performed analytically only for simple class of models  the hierarchical lattices ( for example , one and two - dimensional model with ising spins  @xcite , hierarchical lattices with potts spins  @xcite ) .
yang - lee theory appears to be fruitful for understanding the phase transitions .
it seems that a similar approach to the analysis of transition to chaos in dynamical systems would be useful for deeper understanding of the analogy with the phase transitions , and for development of the new criteria describing complexity of behavior of nonlinear systems . in the present paper
we develop such approach basing on the approximate rg analysis of transition to chaos through the period multiplication bifurcations .
in 1978 feigenbaum discovered universality of cascade of the period - doubling bifurcations and described it on a basis of the rg method .
the simplest example representing the feigenbaum universality class is quadratic map @xmath26 where @xmath27 is a real dynamical variable , and @xmath28 is a real parameter .
this map has fixed points , which can be found as roots of the equation @xmath29 . if @xmath30 , a cycle of period @xmath31 ( that is the fixed point ) loses its stability .
this @xmath32 is the parameter , at which the first period - doubling bifurcation occurs ( the multiplier of the fixed point is @xmath33 ) .
values of @xmath28 for sequent bifurcations can be found by means of approximate rg method  @xcite .
let s apply the map ( 4 ) two times : @xmath34 and neglect the last term , the fourth power of @xmath35 .
then , by the scale transformation @xmath36 this map can be rewritten in the form @xmath37 , which differs from ( 4 ) only by renormalization of @xmath28 @xmath38 thus , the operator of evolution for the double interval of discrete time can be reduced to the original operator by the renormalization transformation  ( 7 ) .
repeating this procedure with scale factors @xmath39 , one can obtain a sequence of the same form @xmath40    fixed points of these maps correspond to the @xmath41-cycles of the original map ( @xmath42 ) .
it is easy to see , that all these cycles , as well as the fixed point of the map  ( 4 ) , become unstable at @xmath43 .
solving a chain of the equations @xmath44 we get the corresponding sequence of bifurcation values of parameter @xmath28 ( with @xmath45 the @xmath41-cycle of  ( 4 ) arises ) . from iteration diagram of fig.1
it is evident , that this sequence converges with @xmath46 to a definite limit @xmath47 , the fixed point of the rg transformation .
it satisfies the equation @xmath48 , thus @xmath49 .
the scaling factors also converge to the limit : @xmath50 , where @xmath51 .
the multipliers ( floquet eigenvalues of the @xmath41-cycles ) converge to the universal value @xmath52 .    from transformation ( 8) it is possible to obtain the law of convergence of the bifurcation sequence : @xmath53 where @xmath54 is a constant , characterizing the convergence to the critical point .    in table  1
we summarize the values of critical indexes ( critical point , scale factor , parameter scaling constant and universal multiplier ) obtained by means of the exact and approximate rg analysis .
the correspondence between them is well enough .
.rigorous and approximate values of feigenbaum critical indexes [ cols="^,^,^",options="header " , ]     let us generalize the approximate rg analysis to the period @xmath0-tupling bifurcations cascades for an arbitrary @xmath0 . by induction , the expression for the scaling factor and the parameter renormalization transformation look as follows : @xmath55 @xmath56 obviously , the critical julia sets of the rg transformations  ( [ 24 ] ) with increasing @xmath0 approximate the fractal properties of the mandelbrot set more and more precisely .
for example , one of the fixed points of the rg transformation of period - quadrupling takes place on the real axes at @xmath57 .
this point corresponds to the feigenbaum critical point @xmath58 and is approximated in this renormalization scheme more precisely than previous estimate @xmath59 .
it is worth noting that for the better approximation of the mandelbrot set by julia sets of the rg transformation , for a large period multiplication the factor @xmath0 ( increasing to infinity ) must be a composite number , that is @xmath60 then the considered rg transformation will describe a number of different bifurcations cascades ( see figure  8) .
let us construct a map , which allows an rg transformation of order @xmath61 using rg transformations of low orders .
it yields @xmath62 dynamics of such a map with @xmath63 includes description of all period @xmath0-tuplings cascades .
figure  10 shows the plane of initial values of @xmath64 .
one can see that it precisely corresponds to the mandelbrot set .
in the present paper the complex variable version of approximate rg method for period - doubling bifurcation cascade is considered .
it is shown , that the julia set of renormalization transformation of parameter of complex logistic map is the approximate version of mandelbrot set of this map .
this similarity is explained by the fact , that julia set is a set of every possible unstable cycles of rg transformation , and elements of these cycles correspond to accumulation points of various bifurcations of period @xmath65 multiplication ( @xmath66 ) , located on the boundary of mandelbrot set dense everywhere .
approximate rg analysis is generalized to the case of different cascades of period - multiplication bifurcations ( for example period - tripling , period - quadrupling etc . ) , which is peculiar for the complex analytic maps .
it is shown that with increase of order of the bifurcation cascade the similarity between mandelbrot set and julia set of renormalization transformation becomes more clear .
the obtained outcomes are interpreted in a view of analogy with theory of phase transitions , namely the yang - lee theory , based on investigation of properties of thermodynamic values depending on complex temperature .
it is necessary to note , that the elements of considered unstable cycles of rg transformation which is equivalent to points of julia set of this transformation , in thermodynamic analogy corresponds to so - called yang - lee zeros , defining in a thermodynamic limit the borderline of phase transition .
it is shown , that at the points of transition to chaos the jump of an electrostatic field of a critical julia set is observed . within the framework of considered analogy ,
the jump of a derivative of a free energy at phase transition points takes place .
thus , it is possible to define a new criterion of transition to chaos .
the electrostatic potential can be regarded as order parameter for the transition .
we conclude that complex generalization of approximate rg method appears to be useful to advance understanding of the critical phenomena at threshold of chaos and for development of analogy with the theory of phase transition , which can give new approaches to investigation of these phenomena .
the authors acknowledge support from research educational center of nonlinear dynamics and biophysics at saratov state university ( rec-006 ) and rfbr ( grant no .
03 - 02 - 16074 ) .
analogy between an approximate version of feigenbaum renormalization group analysis in complex domain and the phase transition theory of yang - lee ( based on consideration of formally complexified thermodynamic values ) is discussed . 
 it is shown that the julia sets of the renormalization transformation correspond to the approximation of mandelbrot set of the original map . 
 new aspects of analogy between the theory of dynamical systems and the phase transition theory are uncovered .    
 _ institute of radio - engineering and electronics of ras , saratov branch , + zelenaya 38 , saratov , 410019 , russia + e - mail : isaevaob@info.sgu.ru_