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a multistable system is one that possesses a large number of coexisting attractors for a fixed set of parameters . there is ample evidence for such phenomena in the natural sciences , with examples coming from neurosciences and neural dynamics @xcite - @xcite , optics @xcite @xcite , chemistry @xcite @xcite @xcite , condensed matter @xcite and geophysics @xcite . multistability also seems to be an essential complexity - generating mechanism in a large class of agent - based models @xcite .    in view of this , it is important to identify the dynamical mechanisms leading to multistability and , in particular , to construct simple models where this phenomenon might be under control . the first mathematical result in this direction was obtained by newhouse @xcite @xcite @xcite who proved that , near a homoclinic tangency , a class of diffeomorphisms in a two - dimensional manifold has infinitely many attracting periodic orbits ( sinks ) , a result that was later extended to higher dimensions @xcite . it has also been proved @xcite that , in addition to infinitely many sinks , infinitely many strange attractors exist near the homoclinic tangencies . the stability of the phenomena under small random perturbations has been studied @xcite @xcite . a second dynamical mechanism leading to multistability is the addition of small dissipative perturbations to conservative systems . conservative systems have a large number of coexisting invariant sets , namely periodic orbits , invariant tori and cantori . by adding a small amount of dissipation to a conservative system one finds that some of the invariant sets become attractors . not all invariant sets of the conservative system will survive when the dissipation is added . however , for sufficiently small dissipation , many attractors ( mainly periodic orbits ) have been observed in typical systems @xcite @xcite @xcite . the problem of migration between attractors and their stability in multiple - attractor systems has also been studied by other authors @xcite @xcite . most of results are based on numerical evidence . however , using the techniques of deformation stability @xcite @xcite @xcite @xcite some rigorous mathematical results @xcite may be obtained . finally , it has been found recently @xcite that , for parameter values near the feigenbaum period - doubling accumulation point , quadratic maps coupled by convex coupling may have a large number of stable periodic orbits . this is one of the phenomena we study in detail in this paper . the emphasis on quadratic maps near the feigenbaum accumulation point has a motivation close to the idea of control of chaos @xcite @xcite . the typical situation in control of chaos , is that of a strange attractor with an infinite number of embedded periodic orbits , all of them unstable . these orbits are then stabilized by several methods . if , instead of a large number of unstable periodic orbits , one has , for example , a large number of sinks , the controlling situation would seem more promising and robust , because the control need not be so accurate . it would suffice to keep the system inside the desired basin of attraction . at the period - doubling accumulation point the feigenbaum attractor , because of the properties of the flip bifurcations , coexists with an infinite set of unstable periodic orbits . by coupling , as we will show , an arbitrarily large number of orbits may become stable . the existence of a large number of stable periodic orbits for just two coupled quadratic maps , provides a simple model where multistability is well under control , in the sense that not only the nature of the phenomenon is completely understood as one may also compute the range of parameters that provides any desired number of stable orbits . this should be contrasted , for example , with concrete models for the newhouse phenomenon @xcite .    rather than merely focusing on multistability , we also study the phenomenology of two coupled quadratic maps , in particular the bifurcations of periodic orbits and the regime of synchronization .. the stabilization of orbits in the coupled system is similar to that obtained in higher dimensional coupled map lattices @xcite with the exception that , due to the restricted dimension of the phase space , the types of bifurcations are different in our system . the results concerning the multistability phenomenon at @xmath0 also considerably extend , and also correct , some imprecise statements in @xcite . coupled map lattices ( cml ) are discrete dynamical systems generated by the composition of a local nonlinearity and a coupling . the phase space of the cml considered in this letter is the square @xmath1^{2}$ ] and the dynamics is generated by the map @xmath2 defined as follows . given a point @xmath3 , its image by @xmath2 , denoted @xmath4 is given by @xmath5 where @xmath6 , @xmath7 and @xmath8 . the map @xmath9 maps @xmath1 $ ] into itself . therefore , the convex combination in ( [ def ] ) ensures that @xmath10^{2})\subset [ -1,1]^{2}$ ] and the dynamics is well - defined . we denote the orbit issued from the initial condition @xmath11 by the sequence @xmath12 , that is to say , @xmath13 and @xmath14 for all @xmath15 .    for the sake of simplicity , we will often employ the variables @xmath16 and @xmath17 . the previous notation of orbits also applies to these variables for which relation ( [ def ] ) becomes @xmath18 where @xmath19 . finally , note that the dynamics commutes with the symmetry @xmath20 or @xmath21 in the original variables . if @xmath22 , then @xmath23 and @xmath24 . in this case , the orbit is said to be synchronized ( from @xmath25 on ) . more generally , an orbit is said to synchronize if @xmath26 and if all orbits synchronize , then we say to have synchronization of the map . synchronization is the simplest dynamical regime exhibited by two - dimensional cml .    to determine a sufficient condition for synchronization in our system , we note that for any orbit , one has @xmath27 for all @xmath15 . it follows from ( [ def2 ] ) that the condition @xmath28 ensures an exponential decay of @xmath29 , and hence synchronization .. since @xmath30 , the condition @xmath28 is equivalent to the following ones ( see figure 1 ) . @xmath31    from now on , we assume that @xmath32 . the condition @xmath33 is not necessary . indeed , if for instance @xmath34 , then @xmath9 has an attracting fixed point in @xmath1 $ ] , and one can prove that synchronization occurs . when @xmath35 is sufficiently small , this happens even though @xmath36 . starting from synchronization and modifying the parameters , non - synchronized ( periodic ) orbits appear from bifurcations of synchronized ( periodic ) ones . to understand this phenomenon , as well as the bifurcations of subsequent orbits , we now study analytically the periodic orbits of period 1 and 2 . let @xmath37 be the synchronized fixed point denoted in @xmath38-variables . it exists for any values of the parameters and @xmath2 has no other fixed point in @xmath1^{2}$ ] . in @xmath38-variables , the jacobian of @xmath39 at this fixed point is diagonal . one eigenvalue is @xmath40 and the corresponding eigendirection is the diagonal @xmath41 . the other eigenvalue is @xmath42 and the corresponding direction , orthogonal to the diagonal is referred as the anti - diagonal . the condition @xmath43 then determines a period - doubling bifurcation , which is of co - dimension 1 if @xmath44 . this is the well - known period - doubling bifurcation of @xmath9 which creates a synchronized period-2 orbit of @xmath2 . moreover , one checks that the derivative @xmath45 is negative for any @xmath46 . hence the conditions @xmath44 and @xmath47 , i.e.   @xmath48 determine another co - dimension 1 period - doubling bifurcation of the synchronized fixed point . indeed the conditions of the corresponding bifurcation theorem ( see e.g.  @xcite ) are satisfied when the curve @xmath49 is crossed upward .    the period-2 orbit created at this bifurcation is non - synchronized and symmetric . to show this , denote by @xmath50 and @xmath51 its components . since the multiplier @xmath42 is negative and the bifurcating direction is the anti - diagonal , we have @xmath52 ( sufficiently close to the bifurcation ) . because of the @xmath53 symmetry , the map @xmath2 also has a period-2 orbit with components @xmath54 and @xmath55 . consequently if @xmath56 , the system would have two periodic orbits created by a co - dimension 1 bifurcation . this is impossible by the unicity in the bifurcation theorem . therefore , sufficiently close to the bifurcation , we have @xmath57 and @xmath58 which is the desired conclusion .    by continuity in the parameters of @xmath2 , sufficiently close to the bifurcation , this symmetric orbit is stable with respect to perturbations in one direction ( the anti - diagonal direction at the bifurcation ) and since @xmath59 , it is unstable in the direction orthogonal to the latter . the bifurcations will now be computed . the orbit with @xmath60 and @xmath58 exists for any @xmath61 and is the unique ( up to time translations ) period-2 non - synchronized symmetric orbit of @xmath2 in @xmath1^{2}$ ] . computing the corresponding jacobian , one obtains the equation for the multipliers @xmath62 \lambda + \left ( 1 - 2\varepsilon \right ) ^{2}(s^{2}-d^{2})^{2}=0\ ] ] where @xmath63 direct calculations show that , if @xmath64 , the multipliers , say @xmath65 and @xmath66 , have zero imaginary part iff @xmath67 . under this condition , we have @xmath68 if @xmath69 and @xmath70 iff @xmath71 . ( the inequality @xmath72 indeed holds if @xmath73 and @xmath74 , see figure 1 . ) consequently , by increasing @xmath46 , the symmetric orbit suffers an inverse pitchfork bifurcation at @xmath75 . this bifurcation is generic for a symmetric orbit in a system with symmetry @xcite and the conditions of the bifurcation theorem hold when the curve @xmath76 is crossed upward . this bifurcation creates two non - symmetric period-2 orbits ( one orbit and its symmetric ) . we have checked that these orbits exist for any @xmath77 and @xmath78 . for @xmath79 , their components are combinations of a fixed point of @xmath9 and the components of a period-2 orbit .    when the imaginary part of @xmath80 and @xmath81 is not zero , we have @xmath82 . ( once again , if @xmath73 and @xmath83 , the inequality @xmath84 is satisfied , see figure 1 . ) the symmetric orbit is thus stable in the interval @xmath85 . if @xmath44 and the curve @xmath86 is crossed upward , this orbit suffers a hopf bifurcation creating a locally stable invariant circle . a numerical calculation shows that the latter is destroyed when @xmath46 is sufficiently large or when @xmath87 is sufficiently small . obviously , if @xmath79 , it does not exist and the bifurcation at @xmath88 which is a period - doubling bifurcation of @xmath9 creates a period-4 orbit .    note that invariant circles in two - dimensional cml resulting from the destabilization of a symmetric orbit and their normal form had already been reported in @xcite . in that work , the system is also defined by ( [ def ] ) , but the local map is @xmath89 and @xmath35 may be larger than @xmath90 . figure 2 shows an example of the phenomenology described above . numerically , it is more convenient to follow the orbits from @xmath79 to increasing values of the coupling . in this picture , as well as in the following ones the map parameter is @xmath0 ( the accumulation point of the period - doubling cascade ) . in figure 2 , from @xmath79 ( the circle ) to @xmath91 ( the point labelled 1 ) , the symmetric orbit is unstable . the figure also shows the invariant circle for @xmath92 . between the points 1 and 2 , the symmetric orbit is stable . at point 2 , the pitchfork occurs , the symmetric orbit becomes unstable and the non - symmetric orbits are created . finally , the point 3 ( @xmath93 ) corresponds to the collapse on the synchronized fixed point . the previous phenomenology is not restricted to small periods but extend to any power of 2 . in particular , the synchronized period-@xmath94 orbit may destabilize to create a symmetric ( non - synchronized ) orbit of twice the period . given @xmath95 , let @xmath96 be the components of the period-@xmath94 orbit of @xmath9 . the points @xmath97 are the components of the synchronized period-@xmath94 orbit of @xmath39 . by the chain rule and since each jacobian at @xmath98 is diagonal , the corresponding multiplier along the anti - diagonal direction is @xmath99 the condition that this multiplier equals @xmath100 determines , if @xmath44 , a co - dimension 1 period - doubling bifurcation . applying the reasoning of the previous section to each component @xmath98 , we conclude that this bifurcation creates an orbit with the property @xmath101 and @xmath102 for all @xmath15 , which is called a phase - opposition period-@xmath103 orbit . since @xmath104 , this bifurcation occurs only if the bifurcation along the diagonal direction has occurred ( the local period - doubling bifurcation of @xmath96 ) . in other words , the phase opposition period-@xmath94 orbit exists only if the synchronized period-@xmath94 orbit does . moreover it follows from figure 3 that , at least for @xmath0 , the phase opposition period-@xmath103 orbit exists only if the phase opposition period-@xmath94 orbit does . this is confirmed analytically for the period@xmath105 orbit whose existence condition is the instability of the synchronized period@xmath106 orbit in the anti - diagonal direction . one obtains @xmath107 and @xmath108 if @xmath109 and @xmath110 and @xmath111 if @xmath112 ( see figure 1 ) . furthermore , a numerical calculation at @xmath0 , reported in figure 3 , shows that the succession of bifurcations of a phase opposition orbit does not depend on the period . on this picture , we have plotted the values of @xmath35 for the hopf bifurcation , the pitchfork bifurcation and the period - doubling bifurcation creating the orbit , versus the power of the period . for each period , the phenomenology is identical to that described in the previous section , with an adequate change of scale in @xmath87 . in addition , the picture shows that several phase opposition orbits may be stable for @xmath44 fixed . this stabilization is an effect of the coupling that will be discussed below . finally , since the phase opposition orbits are the first orbits to appear when the parameters are varied from synchronization and since the first such orbit that is created is of period 2 , it follows that a necessary and sufficient condition for synchronization is @xmath113 , the condition for the existence of the latter . we now analyze the existence and the stability of other period-@xmath94 orbits for @xmath0 . our interest for this value of @xmath46 is that the scaling properties of @xmath9 are reflected on scaling laws for the periods and values of @xmath35 at which the bifurcations occur ( see figure 3 and 8) . we only consider the orbits which for @xmath79 have the same period on projection to both axis @xmath114 and @xmath115 . these orbits are followed numerically when @xmath35 increases and are referred using the phase shift of their components at @xmath79 .    for @xmath0 , the map @xmath9 has a period-@xmath94 orbit for each @xmath116 , whose components for @xmath117 up to 5 are shown in figure 5 . in this picture , the numbers reflect the order in which the components are visited and the tree structure represents the origin of each component in the bifurcation cascade . an important notion is the dyadic distance @xmath118 between the components of an orbit . @xmath118 is the number of steps one has to go back in the bifurcation tree to meet a common component . the dyadic distance is used to characterize the families of periodic orbits that we are considering . for instance , the coordinates of each component of a synchronized orbit are at distance 0 , those of a phase - opposition orbit are at distance 1 . accordingly , when we speak of distance@xmath119 orbit we refer to the dyadic distance of the coordinates of its components .    for any @xmath120 , there are @xmath121 different orbits with distance @xmath118 which have coordinates out of phase by @xmath122 steps , with @xmath123 . the distance of a period-@xmath124 orbit is at most @xmath117 ( @xmath125 ) .    for @xmath79 , the only symmetric orbits are those at distance 0 and 1 . this property is preserved for @xmath44 as shows figure 6 for @xmath126 . the succession of bifurcations of orbits with distance @xmath127 should then differ from those with distance 1 . the differences are seen in figure 6 which shows the evolution of the eigenvalues . for @xmath79 , the orbit is unstable . when @xmath35 increases , it suffers two collisions with orbits of twice the period when the eigenvalues cross @xmath100 and then becomes stable . ( when decreasing @xmath35 , these collisions would be period - doubling bifurcations . ) if @xmath35 increases further , the orbit collides with an unstable one of the same period in a saddle - node bifurcation when the larger eigenvalue reaches 1 . for larger values of @xmath87 , the orbit does not exist . the unstable orbit with which it collides is the one that at @xmath79 has period @xmath94 in one projection and @xmath128 in the other . for higher dyadic distances , the overall variation of the eigenvalues is similar to the @xmath129 case . figure 7 shows a typical example of these phenomena for the case @xmath129 . between @xmath79 and the point labelled 1 in the figure , the orbit is unstable . the point 1 corresponds to the smaller eigenvalue crossing -1 ( see figure 6 ) . therefore , between the point 1 and 2 , the orbit is stable . it disappear at the point 2 when it collides with an unstable orbit of the same period . we have seen that the coupling stabilizes the orbits with distance larger than 0 at @xmath0 . there are indeed two mechanisms responsible for this stabilization .. the determinant of the jacobian of a period-@xmath94 orbit is @xmath130 the term @xmath131 coming from the coupling decreases when @xmath117 increases . however , there is yet a second stabilizing mechanism . denote by @xmath132 the remaining factor in the determinant @xmath133 without coupling , @xmath134 is simply the square of the multiplier of @xmath9 for the periodic orbit . from the properties of the feigenbaum - cvitanovic functional equation it follows @xcite that this factor converges to a fixed value around @xmath135 when @xmath117 increases . the coupling however , changes the position of the orbit components in such a way that this factor also decreases . it is the combined action of this decrease with the contraction of the coupling that brings the eigenvalues into the interior of the unit circle and stabilizes the orbits .    for small @xmath35 there is a simple geometrical interpretation for the variation of @xmath136 . the reason why in the one dimensional map the product @xmath137 remains constant , when @xmath117 grows , is because each time the period doubles , the doubling in the number of factors greater than one is compensated by the fact that the component of the orbit closest to zero approaches zero a little more .    for the unstable orbits along the period - doubling chain , the orbit components closest to zero alternate on each side of the origin . the contracting effect of the convex coupling tends to bring the orbits back in the period - doubling hierarchy . therefore , because the component closest to zero has to move across the origin for the orbit to approach the one with half the period , this implies that the product of the coordinates is going to decrease . the greater the dyadic distance between the orbit projections on the axis , the greater will be the perturbation that the original ( one - dimensional ) orbits suffer . therefore one expects the contracting effect in @xmath136 to increase with the dyadic distance . this effect is quite apparent on figure 8 which shows the stabilizing and destabilizing lines for orbits with distance from 1 to 4 . the shift downwards of the stable regions for successively larger dyadic distances implies that the smaller @xmath35 is , the larger the number of distinct stable orbits that are obtained . an accurate numerical estimate of the number of distinct orbits is obtained by computing the derivative @xmath138 at @xmath79 for each @xmath117 and dyadic distance @xmath139 . actually this derivative provides an accurate estimate of @xmath140 itself , because this one varies almost linearly with @xmath35 for most of the stable range of the orbits . on figure 9 , the scaling properties , when @xmath117 grows , of this derivative are shown . from these results one computes @xmath141 with @xmath142 notice that in figure 9 there is more than one data point for each pair @xmath143 which correspond to non - equivalent orbits with the same dyadic distance .      * the value of the smallest @xmath35 parameter that stabilizes an orbit of dyadic distance @xmath118 equal to the power @xmath117 @xmath144 * the value of the largest @xmath35 parameter for which a @xmath145 orbit is stable @xmath146    from this , one obtains the result that at least @xmath147 distinct stable orbits are obtained if @xmath148 @xmath147 is only a lower bound on the number of distinct stable periodic orbits , because here we have studied only orbits with the same period under projection in the two axis .    in conclusion : _ for sufficiently small _ @xmath35 _ _  an arbitrarily large number of distinct stable periodic orbits is obtained__. however , for any fixed @xmath35 , it is an arbitrarily large number that is obtained , not an infinite number . most orbits either synchronize ( and are then unstable ) or disappear as @xmath35 grows . as a result , a reasoning based on the implicit function theorem , as used in @xcite is misleading . given a sequence of orbits of different periods , even if they remain as orbits for a small perturbation , that does not mean that their ( smallest ) periods remain distinct .
the phenomenology of a system of two coupled quadratic maps is studied both analytically and numerically . 
 conditions for synchronization are given and the bifurcations of periodic orbits from this regime are identified . 
 in addition , we show that an arbitrarily large number of distinct stable periodic orbits may be obtained when the maps parameter is at the feigenbaum period - doubling accumulation point . 
 an estimate is given for the coupling strength needed to obtain any given number of stable orbits .