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non - linear iterated maps are now known as an universal tool in numerous scientific domains , including for instance mechanics , hydrodynamics and economy @xcite @xcite @xcite . they often appear because the differential equations describing the dynamics of a system can be reduced to non - linear iterations , with the help of poincar recurrence maps for instance . the resulting iterations combine a great mathematical simplicity , which makes them convenient for numerical simulations , with a large variety of interesting behaviors , providing generic information on the properties of the system .  in particular , they are essential to characterize one of the routes to chaos , the cascade of period doublings @xcite .    in musical acoustics , mc intyre _ et al . _ have given , in a celebrated article @xcite , a general frame for calculating the oscillations of musical instruments , based upon the coupling of a linear resonator and a non - linear excitator ( for reed instruments , the flow generated by a supply pressure in the mouth and modulated by a reed ) . in an appendix of their article they show that , within simplified models of self - sustained instruments , the equations of evolution can also be reduced to an iterated map with appropriate non - linear functions . for resonators with a simple shape such as a uniform string or a cylindrical tube , the basic idea is to choose variables that are amplitudes of the incoming and outgoing waves ( travelling waves ) , instead of usual acoustic pressure and volume velocity in the case of reed instruments . if the inertia of the reed is ignored ( a good approximation in many cases ) , and if the losses in the resonator are independent of frequency , the model leads to simple iterations ; the normal oscillations correspond to the so called helmholtz motion , a regime in which the various physical quantities vary in time by steps , as in square signals . square signals obviously are a poor approximation of actual musical signals , but this approach is sufficient when the main purpose is to study regimes of oscillation , not tone - color .    in the case of clarinet - like systems , the idea was then expanded @xcite , giving rise to experimental observations of period doubling scenarios and to considerations on the relations between stability of the regimes and the properties of the second iterate of the non - linear function ; see also @xcite and especially @xcite for a review of the properties of iterations in clarinet - like systems and a discussion of the various regimes ( see also @xcite ) . more recent work includes the study of oscillation regimes obtained in experiments @xcite , computer simulation @xcite as well as theory @xcite . the general form of the iteration function that is relevant for reed musical instruments is presented in section [ iteration ] . it it is significantly different from the usual iteration parabola ( i.e. the so - called logistic map ) . moreover , it will be discussed in more detail that the control parameters act in a rather specific way , translating the curve along an axis at @xmath0 rather than acting as an adjustable gain . the purpose of the present article is to study the iterative properties of functions having this type of behavior , and their effect on the oscillation regimes of reed musical instruments . we will study the specificities and the role of the higher order iterates of this class of functions , in particular in the regions of the so called `` periodicity windows '' , which take place beyond the threshold of chaos . these windows are known to contain interesting phenomena @xcite , for instance period tripling or a route to intermittence , which to our knowledge have not yet been studied in the context of reed musical instruments . moreover , the iterates give a direct representation of the zones of stability of the different regimes ( period doublings for instance ) , directly visible on the slope of the corresponding iterate .    for numerical calculations , it is necessary to select a particular representation of the non - linear function , which in turn requires to choose a mathematical expression of the function giving the volume flow rate as a function of the pressure difference across the reed . a simple and realistic model of the quasi - static flow rate entering a clarinet mouthpiece was proposed in 1974 by wilson and beavers @xcite , and discussed in more detail in 1990 by hirschberg _ this model provides a good agreement with experiments @xcite and leads to realistic predictions concerning the oscillations of a clarinet @xcite . using this mathematical representation of the flow rate , we will see that iterations lead to a variety of interesting phenomena . our purpose here is not to propose the most elaborate possible model of the clarinet , including all physical effects that may occur in real instruments . it is rather to present general ideas and mathematical solutions as illustration of the various class of phenomena that can take place , within the simplest possible formalism ; in a second step , one can always take this simple model as a starting point , to which perturbative corrections are subsequently added in order to include more specific details . we first introduce the model in   [ model ] , and then discuss the properties of the iteration function in   [ properties].the bifurcations curves are obtained in   [ bifurcations ] and , in [ iterated ] , we discuss the iterated functions and their applications in terms of period tripling and intermittence . in particular we see how the graph of high order iterates give visible information on the regime of oscillation ( number of period doublings for instance ) or the appearance of a chaotic regime , while nothing special appears directly in the graph of the first iterate . two appendices are added at the end . we briefly recall the basic elements of the model , the non - linear characteristics of the excitator , and the origin of the iterations within a simplified treatment of the resonator .      in a quasi static regime , the flow @xmath1 entering the resonant cavity is modelled with the help of an approximation of the bernoulli equation , as discussed e.g. in @xcite . we note @xmath2 the acoustic pressure inside the mouthpiece , assumed to be equal to the one at the output of the reed channel , @xmath3 the pressure inside the mouth of the player ; for small values of the difference : @xmath4 the reed remains close to its equilibrium position , and the conservation of energy implies that @xmath1 is proportional to @xmath5 , where @xmath6 is the sign of @xmath7 ( we ignore dissipative effects at the scale of the flow across the reed channel ) ; for larger values of this difference , the reed moves and , when the difference reaches the closure pressure @xmath8 , it completely blocks the flow . these two effects are included by assuming that if @xmath9 the flow @xmath1 is proportional to @xmath10 $ ] , and if @xmath11 , the flow vanishes .  introducing the dimensionless quantities : @xmath12 where @xmath13 is the acoustic impedance of an infinitely long cylindrical resonator having the same cross section @xmath14 than the clarinet bore ( @xmath15 is the density of air , @xmath16 the velocity of sound ) , we obtain : @xmath17 the parameter @xmath18 characterizes the intensity of the flow and is defined as : @xmath19 where @xmath20 is the opening cross section of the reed channel at rest . one can show that @xmath18 is inversely proportional to square root of the reed stiffness ) then provides @xmath21 , or @xmath22 ; but @xmath8 is roughly proportional to the reed stiffness , so that the independence of the flow with respect to the stiffness requires that @xmath18  is inversely proportional to the square root of this stiffness . ] , contained in @xmath8 . in real instruments , typical values of the parameters are @xmath23 $ ] ; @xmath24 $ ] ; values @xmath25 will not be considered here , since they correspond to multi - valued functions @xmath26 , a case that does not seem very realistic in practice . fig.[p - u - relation ] shows an example of function defined in eqs.([1 ] to [ 1ter ] ) . it is obviously non - analytic ; it is made of three separate analytic pieces , with a singular point at @xmath27 the derivative of the function @xmath26 is discontinuous at @xmath28 ( point @xmath29 in fig . [ p - u - relation ] , the index @xmath30 being used for the limit of possible beating ) ; a smoothing of the resulting angle of the function could easily be introduced at the price of a moderate mathematical complication , but this is not necessary for the present discussion . entering the resonator of a clarinet as a function of the internal pressure @xmath31 ( pressure in the mouthpiece ) . all physical quantities are expressed in dimensionless units , as explained in the text . m@xmath32 corresponds to the contact point where the internal pressure in the mouthpiece bends the reed sufficiently to close the channel , so that the flow vanishes ; it remains zero in all region @xmath33 . m@xmath34 is the inversion point where @xmath35 and where the acoustic flow changes sign . the full line corresponds to @xmath36 , the broken line to @xmath37 . here @xmath38.,width=377 ]      waves are assumed to be planar in the quasi one dimensional cylindrical resonator .  any wave can be expanded into an outgoing wave @xmath39 and an incoming wave @xmath40 , where @xmath41 is the time and @xmath42 the abscissa coordinate along the axis of the resonator ; at point @xmath43 ( at the tip of the reed ) , the acoustic pressure and flow ] are given by : @xmath44 or : @xmath45 \text { \ \ ; \ \ } p^{-}(t)=\frac{% 1}{2}\left [ p(t)-u(t)\right ] . \label{4b-2}\ ] ] we will use variables @xmath46 instead of @xmath47 and @xmath48 . if we assume that the impedance at the output of the resonator is zero ( no external radiation , the output pressure remains the atmospheric pressure ) , we obtain the reflection condition : @xmath49 where @xmath50 is the resonator length and @xmath16 the sound velocity . this equation expresses that the reflected wave has the same amplitude than the incoming wave . losses are not included in this relation , but one can also introduce them very easily by replacing ( [ 4 ] ) by : @xmath51 which amounts to introducing frequency independent losses ; a typical value is @xmath52 . for a cylindrical , open , tube with no radiation at the open end so that losses only occur inside the tube , @xmath53 , where @xmath54 is the absorption coefficient . of course this is an approximation : real losses are frequency dependent depends on both frequency @xmath55 and radius @xmath56 for normal ambient conditions ( @xmath57{0xb0}}% % beginexpansion { { } ^\circ}% % endexpansion c$ ] ) , @xmath58 ( see e.g. @xcite ) . ] and radiation occurs but , since losses remain a relatively small correction in musical instruments , using eq . ( [ 4c ] ) is sufficient for our purposes . we now assume that all acoustical variables vanish until time @xmath59 , and then that the excitation pressure in the mouth suddenly takes a new constant value @xmath60 ; this corresponds to a heaviside step function for the control parameter .  between time @xmath61 and time @xmath62 , according to ( [ 4c ] ) , the incoming amplitude @xmath63 remains zero , but the outgoing amplitude @xmath64 has to jump to value @xmath65 in order to fulfil eqs.([1 ] to [ 1ter ] ) .  at time @xmath66 , the variable @xmath63 jumps to value @xmath67 , which immediately makes @xmath64 jump to a new value @xmath68 , in order to still fulfil eqs.([1 ] to [ 1ter]).this remains true until time @xmath69 , when @xmath63 jumps to value @xmath70 and @xmath71 to a value @xmath72 , etc . by recurrence , one obtains a regime where all physical quantities remain constant in time intervals @xmath73 , in particular @xmath74 for the pressure and @xmath75 for the flow , with the recurrence relation : @xmath76 in what follows , it will be convenient to use @xmath62 as a natural time unit . we will then simply call time @xmath77  the time interval @xmath78 . notice that in order to get higher regimes ( with e.g. triple frequency ) , the previous choice of transient for @xmath79 needs to be modified ( see e.g.@xcite ) . now , by combining eqs.([1 ] to [ 1ter ] ) and [ 4b-1 ] ) , one can obtain a non - linear relation @xmath80 between @xmath81 and @xmath82 : @xmath83 which , combined with ( [ recurrence ] ) , provides the relation : @xmath84 with , by definition :  @xmath85 the equation of evolution of the system are then equivalent to a simple mapping problem with an iteration function @xmath86 . the graph of this function is obtained by rotating the non - linear characteristics of fig . [ p - u - relation ] by @xmath0 ( in order to obtain @xmath80 ) , then applying a symmetry ( to include the change of sign of the variable ) and finally a horizontal rescaling by a factor @xmath87 ; the result is shown in fig .  [ fonction - iteree].this provides a direct and convenient graphical construction of the evolution of the system @xcite ; fig . [ iteration ] shows how a characteristic point @xmath88 is transformed into its next iterate @xmath89 etc ... by the usual construction , at the intersection of a straight line with the iteration curve , i.e. by transferring the value of @xmath86 to the @xmath90 axis and reading the value of the function at this abscissa in order to obtain @xmath91 $ ] . in what follows , we consider @xmath60 as the main control parameter of the iteration ; it corresponds to a change of pressure in the mouth of the instrumentalist . a  second control parameter is @xmath18 , which the player can also change in real time by controlling the lip pressure on the reed.for a given note of the instrument , parameter @xmath92 remains fixed , but of course depends on which lateral holes of the clarinet are closed , in other words on the pitch of the note . the oscillations where the functions remain constant and jump to a different value at regular interval of times are reminiscent of the raman regime for the oscillation of bowed strings @xcite . mc intyre et al . have indeed noticed that , if one replaces the non - linear function by that corresponding to a bowed string , one obtains the raman oscillation regime of a string bowed at its center @xcite . the analytical expression of the iteration function is given in appendix [ app1 ] . figure [ fonction - iteree ] shows the function for given values of the parameters @xmath60 and @xmath18 , and three different values of the loss parameter @xmath93     for @xmath38 and @xmath94  solid line @xmath95 ( no loss in the resonator ) ; mixed line @xmath96 ; dotted line @xmath97 . the circles on the right indicate the contact point m@xmath98 , those on the left the flow inversion point m@xmath99.,width=377 ]    in the literature , the most commonly studied functions have the following properties ( see e.g. collet and eckman @xcite or berg et al @xcite ) :    * they are defined on a finite interval and map this interval into itself ; * they are continuous ; * they have a unique maximum ; * their schwarzian derivative is negative . a function verifying these properties will be called a standard  function ; the function @xmath86 of interest in our case does not fulfil all these requirements .    [ [ domain ] ] domain of iteration    usually , the iteration function defines an application of the interval @xmath100 over itself .  here , @xmath86 is defined on an infinite interval @xmath101 $ ] even if , obviously , very large values of the variables are not physically plausible . nevertheless , analyzing the different cases corresponding to eqs.([1 ] to [ 1ter ] ) , one can show that the function  @xmath86 has a maximum @xmath102 obtained for : @xmath103   \label{xmax}\ ] ] with value : @xmath104 where @xmath105 is defined by : @xmath106 \text { . } \label{5b}\ ] ]    it can be shown that this maximum is unique for large value of @xmath18 ( @xmath107 ; for smaller values , a second maximum exists at a very large negative values of @xmath90 , i.e. for very large negative flow , but we will see below that such values of the flow can not be obtained after a few iterations .  therefore we focus our attention only on the maximum @xmath102 , which varies slowly as a function of @xmath18 because @xmath108 increases monotonically from 0 for @xmath109 to a small value ( @xmath110 for @xmath111 ) . the geometrical construction of fig . [ iteration ] shows that , after a single iteration , the characteristic point m necessarily falls at an abscissa @xmath112 . let us call @xmath113 the ordinate of the point on the iteration function with abscissa @xmath102 . the two vertical lines @xmath114 and @xmath115 , together with the two horizontal lines @xmath116 and @xmath117 , define a square in the @xmath118 plane , from which an iteration can not escape as soon as the iteration point has fallen inside it . , which means that the iteration curve crosses the left side of the square , as is the case in fig [ iteration ] . ] conversely , since every characteristic point has at least two antecedents , the iteration can bring a point that was outside the square to inside . in other words , the square determines a part of the curve which is invariant by action of the function .  for usual initial conditions , such as @xmath119 , the starting point already lies within the square , so that all points of the iteration keep this property . we have checked that , even if one starts with very large and unphysical pressure differences ( positive or negative ) , the iterations rapidly converge to the inside the square . in what follows , we call it the iteration square . the net result is that , if we do not consider transients , we can consider that the function defines an application of the interval @xmath120 $ ] over itself . we are then very close to the usual mapping situation , except that here the interval depends on the control parameters ( since the value of @xmath102 depends on @xmath60 and @xmath18 ) , but with a relatively slow variation .    , after a few steps the iteration remains inside an  iteration square " shown in broken lines . this square has its upper side tangent to the maximum of the function , at point @xmath121 , which after one iteration becomes point @xmath122 defining the lowest side of the square ( ordinate @xmath123 ) . depending on the parameters , the iteration square contains or does not contain the contact point @xmath29 and the flow inversion point @xmath124 . here @xmath125 , @xmath126 , @xmath127.,width=377 ]    [ [ singularities ] ] singularities    an interesting feature of the iteration function is the discontinuity of its first derivative occurring at the beating limit point @xmath29 at @xmath128 , given by : @xmath129 when the reed closes the channel ( @xmath130 , @xmath131 ) , @xmath132 , @xmath133 , the iteration function is linear . another singularity , i.e. a discontinuity of the second derivative , is obtained at the crossover between positive and negative flow , the inversion point @xmath124 where sign of the flow changes . its abscissa @xmath134 is given by : @xmath135 for @xmath136 , @xmath137 is negative and @xmath138 positive : therefore the initial point of the iteration ( @xmath139 ) lies in the interval @xmath140 $ ] , with neither contact with the mouthpiece nor negative flow , as one could expect physically .    [ [ schwarzian - derivative ] ] schwarzian derivative    the schwarzian derivative @xcite of @xmath86 is equal to : @xmath141 ^{2}\text{,}\ ] ] where @xmath142 , @xmath143 and @xmath144 indicate the first , second and third derivatives of @xmath86 , respectively . if @xmath145 , it is zero ; if @xmath146 , using the change of variables given in appendix [ app1 ] , @xmath147 can be shown to be equal to : @xmath148 , \ ] ] where @xmath149 is a function of @xmath150 - see eqs . ( [ a1 ] ) to ( [ a3 ] ) . therefore its sign does not depend on the loss parameter @xmath92 . after some calculations , the schwarzian derivative is found to be negative for all @xmath151 $ ] when @xmath152 otherwise , it is negative up to a certain value , then positive up to @xmath153 the calculation of @xmath147 for the case @xmath154 shows that it is positive , except for a small interval . the iteration function therefore differs from a standard function because of the sign of the schwartzian derivative ; this is related to the nature of the bifurcation at the threshold of oscillation @xcite , which can be either direct or inverse . [ [ beating - and - negative - flow - limits - beating - limits ] ] beating and negative flow limits [ beating - limits ]    in fig . [ iteration ] we see that , depending whether the contact point @xmath155  and flow inversion point @xmath124 of the iteration curve fall inside or outside the iteration square , a beating behavior of the reed and a sign inversion of the air flow are possible or not . point @xmath29 falls inside the iteration square if its abscissa @xmath138 given by ( [ 5b ] ) is smaller than @xmath156 , which leads to : @xmath157     and @xmath158 , the line representing @xmath159 gives a border between the upper region , where reed beating may occur , and the lower region where it can not - see eq . ( [ 5d ] ) .  as a point of comparison , the line labelled @xmath160 correspond to the limit obtained in @xcite for the particular case of a 2-state regime , and given by eq.([5f ] ) . the figure also shows the line @xmath161 and @xmath162 associated with the possibility of negative flow ( the first one turns out to be very close to that associated with reed beating ) . small values for the losses have been assumed ( @xmath163).,width=377 ]    the limiting value @xmath164 is less than unity ( it tends to @xmath165 when @xmath166 tends to 0 and @xmath92 tends to unity ) . this necessary condition is completely independent of the nature of the limit cycle , and less stringent than the limit  @xmath167 obtained in @xcite , for a 2-state cycle : @xmath168 ,   \label{5f}\ ] ] where @xmath169 and @xmath170 fig . [ limits ] gives a comparison between the two limits . similarly , a necessary condition for possible inversions of the sign of the air flow is that point @xmath124 falls inside the iteration square of fig [ iteration ] , in other words that @xmath137 is larger than @xmath171 . we show in appendix [ app2 ] that this happens if : @xmath172 the expression of the two limits @xmath173 and @xmath174 are given in the appendix and can be seen on fig . [ limits ] . they are solutions of @xmath175 , and exist only if the following condition holds : @xmath176  therefore , for a given @xmath177 negative flow is possible only above a certain value of @xmath92 ; this value is @xmath178 for @xmath179 , and tends to unity when @xmath18 tends to zero . using a more realistic shape for the function @xmath86 with a rounding of the kink at @xmath138 ( no discontinuity of the derivative ) should lead to a shorter range of negative flow , making the phenomenon even less likely , as illustrated in fig . [ limits ] .    of course , the two above conditions ( [ 5d ] ) and ( [ 10j ] ) are necessary , but not sufficient ; they do not ensure that either beating or flow inversion will indeed take place , since this will be true only if the corresponding regions of the non - linear curve are reached during the iterations . generally speaking , this will have more chance to occur in chaotic regimes , where many points are explored in the iterations , than in periodic regimes . since no observation of negative flow has been reported in the literature , it is not clear whether this actually happens in real instruments .    in conclusion of this section , the iteration function is similar to those usually considered in the context of iterated maps , without really belonging to the category of standard  functions.the major difference is actually the effect of the control parameters on the function , since usually the control parameters acts as a gain , expanding the vertical axis of the graph ; here the parameter @xmath60 ( pressure in the mouth of the instrumentalist ) translates the iteration function along an axis at @xmath0 of the coordinate axis , while the other control parameter @xmath18 ( the pressure of the lip on the reed ) expands the function along the perpendicular axis . it is therefore not surprising that we should find a parameter dependence of the dynamical behaviors that is significantly different from the standard results . figure [ fig - bif ] shows an example of bifurcation curves , for @xmath180 and @xmath181 , and illustrates the relative complexity of the possible regimes . the upper curve corresponds to the outgoing amplitude @xmath182 ( or @xmath90 ) , the middle curve to acoustic pressure @xmath31 , and the lower curve to the acoustic flow @xmath183 . the three curves show the last 20 values obtained after computing 400 iterations for each value of the mouth pressure @xmath60 . by calculating @xmath184 iterations for a given value of the parameter @xmath60 , we have checked that the limit cycle is then reached . obviously this method leads to stable regimes only .     and @xmath185 . for increasing values of the abscissa @xmath186 ( blowing pressure ) , corresponding to a crescendo , the curve show the values after 400 iterations of the outcoming wave @xmath187 ( top ) the pressure @xmath31 in the mouthpiece ( middle ) , and the volume flow @xmath183 ( bottom ) . above @xmath188 , the flow can be negative and the reed can beat . the top figure also shows @xmath189 and @xmath190 ( mixed lines ) associated to the iteration square.,width=377 ]    when the control parameter @xmath60 increases , the beginning of these curves follows a classical scenario of successive period doublings , leading eventually to chaos ; as expected , high values of the parameters @xmath18 and @xmath92 favour the existence of chaotic regimes , as well as beating reed or negative flow . when @xmath60 continues to increase , another phenomenon takes place : chaos disappears and is replaced by a reverse scenario containing a series of frequency ( instead of period ) doublings . we call this phenomenon a backwards cascade  ( in order to distinguish it from the usual inverse cascade , which takes place within periodicity windows inside chaos @xcite ) ; this backwards cascade is a consequence of the specificities of the effect of the control parameter on the iteration function in our model , and of the particular shape of the iteration function ( for instance a straight line beyond the beating limit point ) . as a matter of fact , different kinds of cascades have been studied in the literature ( see e.g. @xcite and @xcite , in particular fig.5 ) . [ fig - bif ] , the variations of @xmath60 correspond to a crescendo : for a given value of @xmath60 , the initial value for the iteration , @xmath191 , is chosen to be equal to the last value @xmath192 obtained with the previous value of @xmath60 . but we have also studied the decrescendo  regime and observed that , in the chaotic regimes , the plotted points differ from the crescendo points ; on the other hand , they remain the same in the periodic regimes , indicating a direct character of the bifurcations ( no hysteresis ) . we have found an exception to this rule : between values @xmath193 and @xmath194 , 2-state and 4-state regimes coexist , indicating an inverse bifurcation . another inverse bifurcation , between a 2-state regime and a static regime , occurs beyond the limit of the figure , the two regimes coexisting between @xmath195 and @xmath196 ; this is not shown here ( the shape of the curve can be found in ref . @xcite , see upper fig . the two limits of the function @xmath86 , @xmath102 and @xmath197 are also plotted in the upper figure ( @xmath187 ) showing that , as expected , the corresponding values remain inside the iteration square (   [ domain ] ) . in the figure at bottom , the results for the flow @xmath183 exhibit lower limits for negative flow and for beating , which are very close to the theoretical limits , respectively @xmath198 and @xmath199 and are located within the chaotic regime . negative flow disappears at the bifurcation between the 4-state and the 2-state regime , @xmath200 , a much lower value than the higher limit for negative flow @xmath201 . @xmath202=@xmath203=@xmath2040@xmath2050.4540@xmath2060.3545@xmath2070.4542@xmath2080.4272@xmath2090.4544@xmath2100.4384@xmath2110.466216@xmath2120.4403@xmath2130.4664@xmath2140.4408@xmath2150.46945@xmath2160.4409@xmath217_i1=0.4454@xmath2180.4695@xmath2190.4467@xmath2200.4696@xmath2140.4474@xmath2210.46985@xmath2220.4476@xmath2230.5000@xmath2240.4479@xmath225_b=0.4503@xmath2180.53@xmath2260.4538@xmath2271.@xmath228_i2=1.189@xmath2290.4539@xmath2306.3544@xmath231    table 1 shows the critical values of @xmath60 corresponding to changes of regime . up to the first chaotic regime ( @xmath232 ) , the behavior follows the usual period doubling cascade scenario . between @xmath233 and @xmath234 a periodicity windows  @xcite is obtained , with 6-state , then 12-state and 24-state regimes ( but no 3-state regime ) . above the value @xmath232 for which chaos starts , an inverse cascade  type scenario is observed , then intermittences occur , chaos again , and finally the backwards cascade  to the static regime . we did not try to obtain the same accuracy for the values of all different thresholds , because the ranges for @xmath60 have very different widths ; for some values of @xmath235 , it has been necessary to make up to 2000 iterations , and sometimes it is not obvious to distinguish between a chaotic regime , a long transient , or an intermittency regime . we now discuss how the iterated functions can be used to study the different regimes and their stability . we write @xmath236 the second iterate of @xmath55 , and more generally @xmath237 its iterate of order @xmath77 ; the derivative of @xmath55 with respect to @xmath90 is @xmath238 . around the fixed point @xmath239 of the first iterate @xmath86 , a taylor expansion gives : @xmath240 which provides the well known stability condition for a fixed point @xmath241 of @xmath86 : @xmath242 since the derivative of the iterate of order @xmath77 is given by : @xmath243 one can show by recurrence that , when @xmath244 , it is equal to the @xmath77-th power of the derivative of @xmath55 , so that : @xmath245 ^{n}+ .. \label{ddd}\ ] ] if the fixed point is stable ( resp . unstable ) with respect to @xmath86 , it is also stable ( resp . unstable ) with respect to any iterate . if @xmath90 is a vector , instead of a scalar , this linearized approach leads to the floquet matrix , and @xmath238 should be replaced by the eigenvalues of the matrix . @xmath246 and @xmath38 . the 1st iterate @xmath86 is shown with a mixed line , the 2nd iterate @xmath236 with a ( blue ) solid line , and @xmath247 with a ( red ) , thin mixed line . the dotted lines are the first diagonal and the straight line perpendicular at the fixed point @xmath248 of @xmath86 , solution of @xmath249 the tangent lines to iterate 1 at the point @xmath250 is shown with a solid line.,width=377 ]     @xmath251 and @xmath38 , with the same plots that in figure [ iteree_gamma=031 ] . the tangent lines at the new fixed points of @xmath236 , @xmath252 and @xmath253 , are also shown.,width=377 ]    examples of iterated functions of order 1 , 2 , and 4 are shown in figures [ iteree_gamma=031 ] and [ iteree_gamma=042 ] , with the same values of @xmath166 and @xmath92 as in figure [ fig - bif ] ; in the former , the blowing pressure @xmath60 is @xmath254 , in the latter , @xmath60 is @xmath255 . the first iterate has a unique fixed point , @xmath256 , @xmath257 , located by definition on the first diagonal . the fixed point is stable if the absolute value of the derivative at @xmath248 is smaller than unity , in other words if the tangent line lies between the first diagonal ( with slope @xmath258 ) and its perpendicular ( with slope @xmath259 ) .  when @xmath260 @xmath261 , we see in fig . [ iteree_gamma=031 ] that the fixed point @xmath262 is stable , so that no oscillation takes place . when @xmath79 increases , m * becomes instable and , at the same time , gives rise to three fixed points of @xmath263 .  for @xmath264 , fig . [ iteree_gamma=042 ] shows that the tangent is outside the angle between the diagonal and its perpendicular , so that the fixed point is now unstable ; on the other hand , the second iterate @xmath263 now has two more fixed points @xmath252 and @xmath253 with slopes less than @xmath88 ( in absolute value ) : we therefore have a stable 2-state regime . the same scenario then repeats itself when @xmath60 continues to increase : at some value , points @xmath252 and @xmath253 become instable in turn ( the corresponding slope exceeds @xmath88 in absolute value ) , and both points @xmath252 and @xmath253 divide themselves into three fixed points of @xmath265 ; the two extreme new points have small slopes for this iterate , which leads to a 4-state stable regime .  by the same process of successive division of fixed points of higher and higher iterates , one obtains an infinite number of period doublings , until eventually chaos is reached . this is the classical feigenbaum route to chaos . some general remarks are useful to understand the shape of the iterates in the figures :    * if the value of @xmath86 for the abscissa @xmath90 verifies @xmath266 , i.e. if the point @xmath267(@xmath90 , @xmath86 ) is on a horizontal line @xmath268 , all iterates go through the same point ; * the extrema of @xmath236 verify either @xmath269 ( i.e. @xmath270 ) or @xmath271 , because @xmath272 f^{\prime } ( x)$ ] ; therefore the extrema of @xmath236 are at either the same abscissa or the same ordinate as those of @xmath86 ; * more generally , for @xmath273 , if @xmath274 , then @xmath275 , and it is at a maximum ( its first derivative vanishes and the second one is negative ) , and if @xmath276 , then @xmath277 ,  and it is at a minimum ( its first derivative vanishes and the second one is positive ) ; * the kink of the first iterate ( beating limit point ) is also visible on the iterates ; * a well known property of the schwarzian derivative is as follows : if the schwarzian derivative of @xmath86 is negative , the schwarzian derivatives of all iterates are negative as well . @xmath251 and @xmath38 , of order 1 , 2 , 4 , 8 and 16 . the convergence to the 2-state regime is visible.,width=377 ]    figure [ iteree_16 ] shows the higher order iterates ( of order 4 , 8 and 16 ) in the same conditions as figure [ iteree_gamma=042 ] . we observe that the iterates become increasingly close together when their order increases , with smaller and smaller slopes at the fixed points corresponding to the 2-state regime . moreover , they resemble more and more a square function , constant in various domains of the variable . this was expected : in the limit of very large orders , whatever the variable is ( i.e. whatever the initial conditions of the iteration are ) one reaches a regime where only two values of the outgoing wave amplitude are possible ; these values then remain stable , meaning that the action of more iterations will not change them anymore . so , one can read directly that the limit cycle is a 2-state on the shape of @xmath278 , which has two values ; it would for instance have 4 in the limit cycle was a 4-state regime for these values of the parameters . for the clarity of the figure , we have shown only iterates with orders that are powers of @xmath279 , but it is of course easy to plot all iterates . for a 2-state regime , even orders are sufficient to understand the essence of the phenomenon , since odd order iterates merely exchange the two fixed points @xmath280 and @xmath253 .    in table 1 , the existence of two different stable regimes for the same value of the parameters signals an inverse bifurcation ; figure [ iteree_16b ] shows an example of such a situation .  for @xmath281 , both the static and 2-state regimes are then stable , depending on the initial conditions . for the static regime , the curve @xmath282 coincides with the second diagonal @xmath283 , a case in which the fixed point is presumably stable ( the stability becomes intuitive when one notices that the tangents of the higher order curves lie within the angle of the two diagonals ) . for the 2-state regime , the state of positive pressure value corresponds to a beating reed . @xmath284 and @xmath38 , of order 1 , 2 , 3 , 4 , and 8 . the curves of @xmath285 and @xmath286 are almost perfectly superimposed . around @xmath139 , the convergence to the static regime appears to be very slow . on the contrary the convergence to the 2-state regime is rapid . , width=377 ]     @xmath287 and @xmath38 , of order 1 , 2 , 4 , 8 and 16 . , width=377 ]    finally fig.[iteree_16c ] shows another case of existence of two different regimes for the same value of the parameters . a 2-state regime can occur , as well as a 4-state regimes can occur . it appears that the second one is more probable than the first one , when initial conditions are varied . we now investigate some regimes occurring in a narrow range of excitation parameter @xmath60 . \(i ) we first examine a chaotic regime occurring just before a 6-state regime ( period tripling ) and the transition between the two regimes . figure [ iteree_12a ] shows the iterated functions of order 1 , 2 , 6 , and 12 . the 6th iterated function crosses the first diagonal at the same points than the first and the second iterates only , which means that no 6-state regime is expected . by contrast , the 12th iterate cuts the diagonal at more points , but with a very high slope , indicating that the corresponding fixed points can not be stable . this , combined with the fact that no convergence to a square function ( constant by domains ) , such as @xmath278 in figure [ iteree_16 ] , suggests an aperiodic behavior ; the time dependent signal shown in fig.[figtransit_04445 ] looks indeed chaotic ( nevertheless the flow always remains positive ) . the periodic / chaotic character of the signal can be distinguished by examining the time series , but a complementary method is the computation of an fft . for the signal of fig.[figtransit_04445 ] , the spectrum is more regular than the spectrum of a 6-state periodic regime . nevertheless the frequencies of the latter ( the `` normal '' frequency @xmath288 of the 2-state regime with the frequencies @xmath289 and @xmath290 ) remains visible in the spectrum of the first one , as it is often the case for signals corresponding to very close values of the parameter . a consequence is that these frequencies clearly appear when listening the sound . @xmath291 and @xmath38 , of order 1 , 2 , 6 and 12 . a convergence to an aperiodic regime is visible . the arrow indicates a region where @xmath292 is very close to the first diagonal , but does not yet cross it.,width=377 ]     for @xmath293 @xmath38 , @xmath294 , @xmath295 ; the upper part shows the the pressure @xmath31 , the lowest part the values of the flow @xmath183 . the regime looks chaotic.,width=377 ]    figure [ iteree_12b ] is similar to figure [ iteree_12a ] , but with a slightly larger value of @xmath60 ( @xmath296 instead of @xmath297 ) . in the region indicated by the arrow , one notices that the 6th iterated function now cuts the first diagonal . they are 12 points of intersection ( plus 1 common point with the first iterate as well as two common points with the second iterate , all unstable ) ; the slope of the tangent shows that 6 of them are stable , so that one obtains a 6-state , periodic , regime . the variations of higher order iterates , e.g. @xmath298 , remain very fast ; the convergence to the limit cycle is then much slower than for fig . [ iteree_16 ] , except if the initial point is close to a limit point ( e.g. that shown by an arrow : it turns out that the 12th iterated function is very close to the 6th one ) . as a consequence , the initial transient to the 6-state regime can be rather chaotic , as shown in fig . [ figtransit_04469 ] , but convergence to a periodic regime does occur later . this existence of periodic regimes above the threshold for chaos is called periodicity windows , which appears as a narrow whiter region in fig . [ fig - bif ] . a difference with the usual @xmath299-state regimes ( when @xmath60 is below the chaotic range ) , for instance corresponding to fig . [ iteree_gamma=042 ] , is that one obtains @xmath299 intersections with the diagonal , stable or unstable ; by contrast , for the 6-state regime , they are 6 stable and 6 unstable points . @xmath300 and @xmath38 , of order 1 , 2 , 6 and 12 . a convergence to a 6-state regime is observed . the arrow indicates a region where @xmath292 cuts the first diagonal.,width=377 ]     for @xmath293 @xmath38 , @xmath301 , @xmath302 the regime is periodic ( 6-state).,width=377 ]    \(ii ) we now examine the transition between a 6-state regime and a 4-state regime through chaotic regimes or intermittency regimes . for @xmath303 , a 6-state regime is obtained . [ iteree_6a ] shows the iterates of order 1 , 2 , 4 and 6 . the 4th and 6th iterates have common intersections with the first and second iterates , since both 4 and 6 are multiples of 2 . the 6th iterate intersects the first diagonal at 12 other points , while the 4th cuts the diagonal at 4 points only . these 4 points are unstable , thus no 4-state regime can exist . on the contrary , for the 6th iterate , half of the 12 points are stable ( i.e. with a small slope of the tangent line ) , so that one obtains a 6-state stable regime . @xmath304 and @xmath38 , of order 1 , 2 , 4 and 6 . a convergence to a 6-state regime is observed . , width=377 ]    what happens for a higher value of @xmath305 namely @xmath306 corresponding to a 4-state regime is shown in fig . [ iteree_6b ] , with again the iterates of order 1 , 2 , 4 , 6 . the 4th iterate curve crosses the diagonal for the same number of points than previously , but the 4 points are now stable . the 6th order iterate does not intersect the diagonal , except at the common points with the two first iterates . @xmath307 and @xmath38 , of order 1 , 2 , 4 and 6 . a convergence to a 4-state regime is observed . , width=377 ]    between the two preceding values of the parameter @xmath60 , both chaotic and intermittent regimes can exist . for @xmath308 , figure [ figtransit_inter2 ] shows intermittencies between a chaotic and a 6-state behaviors ( upper curve ) , and figure [ iteree_046623 ] shows that the 6th iterate is tangent to the first diagonal in 6 points , so that the resulting permanent regime can be interpreted as a kind of hesitation  between two behaviors . the 4 intersections of the 4th iterate remain unstable . + the lower curve in figure [ figtransit_inter2 ] shows another , more visible , example of intermittencies , obtained with slightly different values of the parameters , between a chaotic regime and a 4-state one ( actually it is a 8-state one , very close to a 4-state regime ) .     for @xmath293 @xmath38 , @xmath309 @xmath295 ( upper curve ) : intermittencies between chaos and a 6-state regime are observed . however the lower curve ( for @xmath310 @xmath38 , @xmath311 @xmath295 ) shows a more clear situation of intermittencies between chaos and a 4-state regime . , title="fig:",width=377 ]   for @xmath293 @xmath38 , @xmath309 @xmath295 ( upper curve ) : intermittencies between chaos and a 6-state regime are observed . however the lower curve ( for @xmath310 @xmath38 , @xmath311 @xmath295 ) shows a more clear situation of intermittencies between chaos and a 4-state regime . , title="fig:",width=377 ]     @xmath312 and @xmath38 , of order 1 , 2 , 4 and 6 , corresponding to intermittencies . the sixth iterate is tangent to the diagonal . , width=377 ] the study of the iteration model of the clarinet should not be limited to the first iterate : higher order iterates give interesting information on possible regimes of oscillation . in the limit of very high orders , their shape gives a direct indication of the number of states involved in the limit regime , or of chaotic behavior . one can also predict an intermittent regime of the iterations , which takes place when an iterate is almost tangent to the first diagonal , so that the iterations are trapped  for some time in a narrow channel . the phenomenon might be related to some kinds of multiphonic sounds produced by the instrument . it is true that this phenomenon takes place only in a rather narrow domain of parameters , but this is also the case of the period doubling cascade , which has been observed experimentally . one can therefore reasonably hope that the present calculations will be followed by experimental observations . this work was supported by the french national agency anr within the consonnes project . we thank also the conservatoire neuchtelois and the high school arc - engineering in neuchtel . finally we wish to thank sami karkar and christophe vergez for fruitful discussions . our purpose is to obtain an analytical expression of the iteration function @xmath313  from the basic model ( eqs . ( [ 1 ] to [ 1ter ] , [ 4b-1 ] , [ recurrence ] ) ) , the following quantities can be defined : @xmath314 @xmath315 @xmath316 can be obtained from the knowledge of the function @xmath317 given by the solving of : @xmath318 for the non - beating reed case , the study of function @xmath319 leads to a direct analytical solution , as explained below , at least if @xmath320 ( otherwise it is a multi - valued function ) .    finally , with the notation @xmath321 and @xmath322 , if @xmath323 is the heaviside function , the iteration function is obtained , as : @xmath324      for this case , both @xmath150 and@xmath326  are positive and smaller than unity , because @xmath327 writing @xmath328 , eq . ( [ a2 ] ) is written as : @xmath329 .\ ] ] the study of function @xmath330 shows that it is monotonously increasing from @xmath61 to @xmath88 when @xmath331 increases from @xmath61 to @xmath332 therefore the equation @xmath333 has a unique solution when @xmath334 with this condition , it appears that the equation has three real solutions , and that the interesting solution ( located between @xmath61 and @xmath88 ) is the intermediate one . as a conclusion , it is possible to use the classical formula for the solution of the cubic equation : @xmath335 + \frac{1}{3\zeta } \text { \ ; } \\ \psi & = & \frac{1}{\zeta ^{2}}\text { ; } \eta = \sqrt{3+\psi } \text { ; } \mu = % \frac{9}{2}(3y-1).\end{aligned}\ ] ]      for this case , both @xmath150 and@xmath326  are negative@xmath337 writing @xmath338 , eq . ( [ a3 ] ) is written as follows : @xmath339 .\ ] ] the study of the function @xmath340 shows that it is monotonously decreasing from @xmath61 when @xmath331 increases from @xmath341 therefore the equation @xmath342 has a unique real , positive solution when @xmath343 the two other solutions are either real and negative or complex conjugate , with a negative real part , because the sum of the three solutions is negative ( @xmath344 ) . as a conclusion , the solution can be written by using the following formulae :    [ [ if - the - discriminant - is - positive ] ] if the discriminant is positive    @xmath345    @xmath346 \text { \ ; } r=-\frac{\psi + \mu } { % 27\zeta } \text { } . \\ \sqrt{-x } & = & z = s_{1}-\frac{q}{s_{1}}-\frac{1}{3\zeta } \text { ; } s_{1}=\left [ r+\sqrt{discr}\right ] ^{1/3}.\end{aligned}\ ] ]    [ [ if - the - discriminant - is - negative ] ] if the discriminant is negative    @xmath347    @xmath348 -\frac{1}{% 3\zeta } \text { ; } \\ \eta ^{\prime } & = & \sqrt{-3+\psi } .\end{aligned}\ ] ] the condition of existence of negative flow is given by @xmath349.this is equivalent to the condition on the antecedents , @xmath350 , where @xmath351 is the larger antecedent of @xmath137 , such as @xmath352 because @xmath86 is decreasing for all @xmath353 ( see fig . [ fonction - iteree])@xmath337 therefore the volume flow is negative at time @xmath354 .    in order to determine the limit value @xmath355 , the following equations are to be used : @xmath356 @xmath60 being positive ( a reasonable hypothesis for the normal playing ) , the unknown @xmath150 needs to be larger than the quantity @xmath357 eliminating @xmath60 in the above equations implies the following equation , with @xmath358 : @xmath359 or @xmath360 with :        an example of function @xmath366 is shown in fig . [ figfonctionh(x ) ] . it appears that no solutions exist if @xmath367 and two solutions exist if @xmath368 , i.e. if inequation ( [ 10h ] ) holds . the two solutions can be obtained analytically . however , for sake of simplicity , we give the exact solution for the larger one , @xmath174 , and an approximation for the smaller one , @xmath173 , obtained at the first order in  @xmath369 : @xmath370 \text { , }   \label{10a } \\ & & \text { with } \varepsilon = \frac{\lambda -1 + 2\lambda a_{\zeta } } { ( \lambda + 1)\zeta + \lambda -1}.\end{aligned}\ ] ]    this error is found to be less than @xmath371 in comparison with the exact value . condition ( [ 10h ] )  can be shown to be necessary and sufficient . we do not give the entire proof , but it can be shown that another necessary condition for having two solutions is @xmath372 , or  @xmath373 , but it is implied by condition ( [ 10h ] ) . [ limits ] shows that the first negative flow threshold @xmath173 is very close to the threshold @xmath164 , and slightly smaller . for a given @xmath92 , the limit value of @xmath18 such as @xmath374 corresponds to the equality between the beating reed threshold and the negative flow one . for a given @xmath177 negative flow is possible above a certain value of @xmath92 . for rather strong losses , if @xmath375 , no negative flow can occur . for a cylindrical resonator , this implies that @xmath376 . collet , p. and eckmann , j.p . , `` properties of continuous maps of the interval to itself '' , _ mathematical problems in theoretical physics _ , k. osterwalder ( ed . ) , springer - verlag , heidelberg , 1979 ; _ iterated maps on the interval as dynamical systems _ , birkhuser , basel , 1980 . feigenbaum , j. ,  `` the universal metric properties of nonlinear transformations '' . _ journal of statistical physics , _ 21 , 1979 , 669 - 706 ; `` the metric universal properties of period doubling bifurcations and the spectrum for a route to turbulence '' , _ annals of the new york academy of science _ , 357 , 1980 , * *  * * 330 - 336 . kergomard , j. , `` elementary considerations on reed - instrument oscillations '' . in _ mechanics of musical instruments _ , vol . * 335 * ( a.  hirschberg/ j. kergomard/  g. weinreich , eds),of _ cism courses and lectures _ , pages 229290 . springer - verlag , wien , 1995 . lize , a. , doublement de priode dans les instruments  anche simple de type clarinette , master degree thesis , paris 2004 , http://www.atiam.ircam.fr/archives/stages0304/lizee.pdf [ http://www.atiam.ircam.fr/archives/stages0304/lizee.pdf ] idogawa , t.  , kobata , t.  , komuro , k.   and masakazu , i. , nonlinear vibrations in the air column of a clarinet artificially blown , _ journal of the acoustical society of america , _ 93 , 1993 , 540551 . kergomard , j. ,  dalmont , j.p . , gilbert , j.  and guillemain , ph . . `` period doubling on cylindrical reed instruments '' . _ in proceedings ot the joint congress cfa / daga04 , _ pages 113114 , strasbourg , 22th 25th march 2004 . dalmont , j .- p . , gilbert , j. ,  kergomard , j.  and  ollivier , s. `` an analytical prediction of the oscillation and extinction thresholds of a clarinet '' . _ journal of the acoustical society of america _ , 118 , 2005 , 32943305 . hirschberg , a.  , van  de laar , r.  w.  a. , marrou - maurires , j.  p. , wijnands , a.  p.  j. , dane , h.  j. , kruijswijk , s.  g. and houtsma , a.  j.  m.  a quasi - stationary model of air flow in the reed channel of single - reed woodwind instruments  . _ acustica _ , 70 , 1990 , 146154 . dalmont , j .- ,  gilbert , j.  and ollivier , s.  . `` nonlinear characteristics of single - reed instruments : quasistatic volume flow and reed opening measurements '' , _ journal of the acoustical society of america , _ 114 , 2003 , 22532262 . dalmont , j .- p .   and frapp , c. , ``  oscillation and extinction thresholds of the clarinet : comparison of analytical results and experiments '' . _ journal of the acoustical society of america _ , 122 , 2007 , 11731179 . causs , r. ,  kergomard , j. , lurton , x. , `` input impedance of brass musical instruments - comparison between experiment and numerical model '' , _ journal of the acoustical society of america , _ 75 , 1984 , 241 - 254 . mayer - kress , g. and haken , h. , `` attractors of convex maps with positive schwarzian derivative in the presence of noise '' , _ physica 10d _ , 1984 , 329 - 339 . parlitz , u. , englisch , v. , scheffczyk , c. and lauterborn , w. , `` bifurcation structure of bubble oscillators '' , _ journal of the acoustical society of america , _ 88 , 1990 , 1061 - 1077 . scheffczyk , c. , parlitz , u. , kurz , t. , knop , w. , lauterborn , w. , `` comparison of bifurcation structures of driven dissipative nonlinear oscillators'',_physical review a _ , 43 , 1991 , 6495 - 6502 .
the dynamical equations of clarinet - like systems are known to be reducible to a non - linear iterated map within reasonable approximations . 
 this leads to time oscillations that are represented by square signals , analogous to the raman regime for string instruments . in this article 
 , we study in more detail the properties of the corresponding non - linear iterations , with emphasis on the geometrical constructions that can be used to classify the various solutions ( for instance with or without reed beating ) as well as on the periodicity windows that occur within the chaotic region . in particular 
 , we find a regime where period tripling occurs and examine the conditions for intermittency . 
 we also show that , while the direct observation of the iteration function does not reveal much on the oscillation regime of the instrument , the graph of the high order iterates directly gives visible information on the oscillation regime ( characterization of the number of period doubligs , chaotic behaviour , etc . ) . 
 keywords : bifurcations , iterated maps , reed musical instruments , clarinet , acoustics .