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Stability of Wavelengths and Spatiotemporal Intermittency in Coupled Map Lattices

In relation to spatiotemporal intermittency, as it can be observed in coupled map lattices, we study the stability of different wavelengths in competition. Introducing a two dimensional map, we compare its dynamics with the one of the whole lattice. We conclude a good agreement between the two. The reduced model also allows to introduce an order parameter which combines the diffusion parameter and the spatial wavelength under consideration.


INTRODUCTION
Coupled Map Lattices (CML) have been proposed as simplest models for space-time complexity.
Spatially extended dynamical systems are typically present in many experimental situations.
Examples of such systems can be found in hydrodynamics, plasma physics, chemical reactions or biological systems. For those non-linear systems, CML gives a useful alternative point of view to the more conventional tool of partial differential equations.
The advantage of CML is based on the use of local maps with well controlled dynamical properties, that are synchronously updated and with suitably chosen local or global coupling [1][2][3][4].
Different models of CML have been widely investigated from various point of view: existence and properties of an invariant measure for infinite lattices at low coupling [5], stability and universal properties of homogeneous states [3], [7], [8], [9], study of the dynamical regime by means of a bi-orthogonal decomposition [10], [11], [17] and, probably the most important issue in view of the physical applications as they appear, for instance in Rayleigh-Bénard convection [12], the understanding of spatiotemporal intermittency [1][2][3] and [13][14][15][16]. The emergence of spatial coherent structures from a local chaotic dynamics in CML was discussed in [6]. On the other hand, the coexistence of different domains in the lattice with coherent dynamical behaviour, separated by boundaries (kinks), was one of the first and fascinating features observed in these models.
The relation of this situation with spatiotemporal intermittency has been mentioned by several authors [1], [4], [15] and was named "natural wavelengths" by the later authors.
In a way, this mechanism can be described as follows: the size of the different coherent regions is modified by a slow displacement of the kinks, until a destabilization of the periodic structure occurs inside them. Then, after a transient regime of variable duration, a new periodic state is restored in those regions. In fig. 1-a, b and c we present three plots of the dynamics for typical situations.
There, we can see spatiotemporal representations of a one-dimensional lattice -see As already pointed out by Kaneko [1], [4] and Keller and Farmer [15], the slow motion of the kinks is due to the presence, between the two regions, of a nodal point x i , lying in the neighbourhood of the fixed point of the logistic map.
In any case, the spatial wavelength is modulated by the motion of the domain walls until the destabilization of this wavelength occurs as a threshold phenomenon.
It will be shown later that bifurcations of the temporal period of the dynamics may also occur.
It is also obvious that, inside each region, the wave number, an integer, must be constant between two consecutive unstable situations. Therefore, changing the size of the region, will involve a continuous change of the spatial wavelength λ, of the corresponding configuration.
From this point of view it seems natural to study the stability of these wavelengths.
They act as a bifurcation parameter in this transition between local order and chaos as was first observed in [15], and therefore we guess that it may help in the understanding of spatiotemporal intermittency.
The aim of the present work is to study analyticaly and numericaly some features of this mechanism. Let us mention the two main ingredients used in our paper.
First, we propose a two-dimensional mapping depending on two parameters, whose dynamics (cycles, bifurcations) may describe the main features of the whole lattice. In a sense that will become clearer later, we can say that the whole CML dynamics is enslaved by the one of this simple model.
The first parameter of the reduced map is related to the nonlinearity of the local map in the CML (here the logistic map) describing the dynamics at each site of the lattice.
The second parameter contains the diffusive constant governing the interaction between neighbouring sites together with the space wavelength and, therefore, is used as a control parameter.
The second main ingredient is used in numerical simulations. Instead of trying to describe the "real" situation as it is shown in fig.1, we build an "artificial" recipe where the whole lattice is made of only one region and the stability of the space wavelength is studied by changing the number of sites in the lattice. This is done by means of an "adiabatic compression" described in section 3 below and such that the reader can easily imagine by having a look at fig.2.
The paper is organized as follows: In section 2, after a brief presentation of the models of interest, we use the Fourier transform of spatial structures to introduce the reduced model that is described in section 4. Obviously, this step is reminescent of amplitude equations as they are used in weak turbulence [18].
In section 5 we present the simplest case for which the reduced model is exact. It will be come clear that this case corresponds to the situation described in [6]. A way of using this reduced model out of the region of stability, by means of a bi-orthogonal decomposition was proposed in [17].
In section 6 we deal with the stability in presence of a time-period two dynamics and in section 7 of period four. Section 8 deals with the case of homogeneous dynamics.
In section 9 we briefly compare the previous results with numerical computations of Lyapunov exponents and section 10 concludes.

THE MODELS AND THEIR FOURIER ANALYSIS.
In this work we consider a one-dimensional lattice of length L and a nonlinear map f of the unit interval into itself. The state of the system at time t in configuration space is The new state at time t + 1 is defined in each site by the following convex linear combination of the updated values of neighbour sites : that we often will write : where ∆ is the discrete Laplacian operator.
Here ε is the coupling between neighbour sites, 0 ≤ ε ≤ 1 and f is chosen as the logistic map : We always consider periodic boundary conditions in (2.1), namely x L+1 = x 1 , In the following we shall see that most of the results obtained for the model (2.1) are still valid for a more general one, where the coupling between sites is given by a diffusion operator. This case can be considered as more realistic if one has in mind a limiting process to obtain a continuous configuration space.
For this model, already introduced in [15], we replace formula (2.1) by the following : where N is the range of the interaction and the ρ j are real positive numbers satisfying : and ρ j = ρ −j (2.6) Clearly the model described by (2.1) is a particular case of this one.
In the numerical simulations of (2.4) we have chosen a sequence of linearly decreasing weights : where c is given by the normalization relation (2.5).
Finally the parameter µ of the logistic map will always verify µ > µ c , where µ c = 3, 5699 . . . is the critical value above which the map f has a chaotic dynamics.
In some cases we will have to consider values of µ such that µ c < µ < µ 1 = 3, 677 . . .
for which the dynamics of f is attracted inside two disjoint intervals ∆ 1 and ∆ 2 whose extreme points are determined by the four first iterated by f of the point x = 1/2.
Let's now concentrate on the model (2.1).
Since we are interested in the study of space periodic configurations, it is natural to introduce their discrete Fourier transforms.
In absence of convective terms, we can restrict ourselves to even configurations, bringing to the following representation : from where immediatly follows : Finally, using the explicit form of the logistic map, we find the equations which give the dynamics of the Fourier coefficients.
For the saving of notations we write a k = a t k and a k = a t+1 k .
Therefore we get : a k a L/2−k    and for q = 1, 2, . . . , L/2 − 1 : We denote by F this (L/2 + 1) dimensional map. Notice that, for each q, the non linear This rectangle is reduced to one of the diagonals for each of the exceptional terms a 0 and a L/2 .
It is worth to notice that the coupling parameter ε enters in F only by means of (2.11). Therefore, the relation (2.11) has the meaning of a rescaling between ε and the set of wavenumbers q.
In order to motivate the introduction later on of a reduced model we make now a remark about the relation of equations (2.10) with the corresponding dynamics in physical space.
The first equation of (2.10) describes the dynamics of the x i spatial mean value since On the other hand, writing we can rewrite the first equation in (2.10) as well as the dynamics of the variances δ i .
Thanks to the chosen boundary conditions we may have < ∆f (x) >= 0 and therefore we get where, again u, u stand for the values of u at time t and t + 1.
Coming back to (2.10) we also remark that, if at time t 0 we start with a configuration for which the only non vanishing coefficients are a 0 and a q 0 (q 0 ≤ L/4) then at time t 0 + 1 the only new created harmonics corresponds to a 2q 0 . Therefore if q 0 and ε are such that the only non vanishing amplitude for any time remains a q 0 .
In other words, in that situation, is an exact solution. This is the case, for instance for We shall see that such a selection of a spatial wavelength by means of a non linear coupling between the spatial mean value and the amplitude of the oscillation is observed in much more general situations than the ones strictly obeying (2.15).
In order to get an insight into this phenomenon it is useful to look for the behaviour of α(q, ε) as a function of q, for a given ε. As an example this function is plotted in fig.   3 for ε = 0, 667 and L = 100. The behaviour of α near to the point where α = 0 indicates we may expect the damping of all a set of harmonics.
We can consider the case λ = 2 (q 0 = L/2) to be very special, since then the system evolves according to only the first two equations in (2.10) where a L/2 = a q 0 is coupled only to the mean value a 0 . No condition as (2.15) is required for the existence of an exact solution given by (2.16). We will come back to this case in section 5.
To end up this section we briefly indicate how the previous results should be modified if, instead of the simplest model (2.1) we take the more general diffusive model (2.4).
The dynamics of Fourier coefficients obeys the same equations as (2.10) where we only need to replace α (q, ε) by : where N is the range of the diffusion operator. Let us notice that (2.18) is a good candidate for a limiting process to a continuous space.
Then, the equations for the dynamics of the mean value and variances, similar to (2.14), are the following : where the diffusion operator D is defined as and we used the normalization (2.5) and symmetry (2.6) for the ρ j as well as the periodic boundary conditions.

ADIABATIC COMPRESSION : A NUMERICAL RECIPE
As it was alredy mentioned in the introduction about the "natural" model, (i.e. the one build on a lattice of constant size L), the spatial wavelengths which are present during the dynamics are modulated by the motion of the their respective domain walls. Our aim is to study the instabilities which give rise to these spatiotemporal oscillations as the source of spatiotemporal intermittency.
In order to compare this situation in the lattice with the behaviour of the simplified model presented in Section 4 we propose to isolate that phenomenon by means of a numerical experiment. We call it adiabatic compression of the lattice that we will describe now.
When the lattice of size L reaches a periodic state (eventually after a transient regime) we carry out the following transformation : Notice that this transformation preserves the wavenumber but may change the corresponding wavelength.
After relaxation we may observe if the new wavelength of spatial oscillations is stable or not.
We repeat the operation until we notice a transition to a disordered state. Then the critical value λ cr is given by λ cr = L 1 /q where q is the wavenumber and L 1 the actual length of the lattice.
After this transition the lattice will relax (with constant length L 1 ) to a new periodic state with new wavenumber q 1 and, therefore, a wavelength λ 1 = L 1 /q 1 . The same process may be repeated now.
Coming back to fig. 2.a we can see there, two adiabatic compressions followed by the correspondent instability transitions.
In this example the model is as (2.1) with µ = 3.63 and ε = 0.667. The initial number of sites in the lattice is L = 85 and the wavenumber is q = 10 (λ = 8.5). After adiabatic compression the first destabilization occurs for L 1 = 82 and then λ cr = 8.2.
After a new adiabatic compression the system undergoes another destabilization for L 2 = 57 and therefore λ ′ cr = 57/7 = 8.1 which now relaxes to a new wavenumber q 2 = 5 which corresponds to λ ′ = 11.4. One may observe that both destabilizations are obtained for close values of λ(λ cr = 8.2 and λ ′ cr = 8.1) as well as are close the two values of the spatial wavelengths selected after relaxation (λ = 11.7 and λ ′ = 11.4).
Notice that (λ cr − λ ′ cr ) as well as (λ − λ ′ ) are as small as they are allowed to by the rational approximation determined by the maximum size of the lattice in this case and the correspondent integer wavenumbers. So we may wonder if by taking a much larger lattice at the begining a better agreement of these values may be found. This is effectively true, but the price to pay is an extremely larger relaxation time (transients) and so even for a small gain of precision, we refrain to pursue in this direction.
Instead we may notice the time period of the dynamics represented in fig. 2 also undergoes a bifurcation above λ = 11, a fact that we will adress later on. The reason why this is not observed in fig. 2 is simply because there, the stroboscopic periods are always multiples of the time periods of the dynamics.
We will use several times the adiabatic compression to test the predictions of the simplified model described in the following section, model where λ is taken as the varying control parameter.

A REDUCED MODEL
We define the following two-dimensional dynamical system : where α (λ, ε) = 1 − 2ε sin 2 (π/λ) and f is defined as the logistic map with parameter µ as before. Let's denote F the map defined by (4.1). Remark that now α is considered as a function of continuous parameters λ and ε.
The definition (4.1) comes from a matching of equations (2.10) describing the dynamics of the Fourier modes and the corresponding formulation (2.14) in terms of mean and variances.
It can also be deduced from (2.10) by means of the first terms of an expansion for small amplitudes compared with the mean value, in which case X t represents the space mean value of the configuration at time t and Y t the global amplitude of the variance at the same time.
Even if the real meaning of Y t is to be taken with caution, we got a good agreement of the values of Y t according the map (4.1) and the variance of the configuration at same time as it can be computed form direct numerical simulation with the whole lattice.
The only exception is for λ = 2 where the factor µ/2 in the first equation of (4.1) must be replaced by µ if we want to maintain this interpretation for Y . This is due to the periodic boundary conditions. Instead of changing the form of the map F for this special case λ = 2, we prefer to keep it in the same form as (4.1) since this simply corresponds to a rescaling Y ′ = Y / √ 2 which obviously do not change the properties of (4.1).
Any way, we take (4.1) as a starting point and we will show how it behaves on the parameter space. Different bifurcations and dynamical regimes of (4.1) will be interpreted and then compared to the corresponding ones for the whole lattice.
Notice the map F depends on three parameters : µ, ε and λ, but the later two enter only in (4.1) by means of the function α(λ, ε). Therefore F really depends only on two parameters : µ and α.
We shall compute different cycles of F (for different temporal periods T ) and determine their stability domains in parameter space.
When expressed as a function of ε the result will be compared to numerical simulations showing bifurcation diagrams of the whole CML as a function of ε.
On the other hand, as already written, when the results are expressed as a function of λ, they will be confirmed by applying to the CML the adiabatic compression defined in Section 3.
Before going in the more detailed study of the various dynamical regimes of the map F , we notice a special feature of it that helps to understand some of the results described below.
The second equation in (4.1) only fixes the ratio Y t+1 /Y t and terefore is invariant by scaling of the variable Y as far as the condition 0 ≤ Y ≤ 1 is fulfilled. This allows the translation described by the first equation to stabilize the logistic map governing the dynamics of X.
Then, the coupling between the two variables is closed because, in order that Y t stays in a bounded domain, this stable orbit for X should be such that αf ′ (X t ) oscillates around 1 (the case where it is always, in modulus, strictly less than 1 is addressed in Section 8).
Therefore, at the end we see that the dynamics is governed by a balance of the size of the oscillations of the mean variable X and the one of the variance amplitude that may stabilize each other.

THE SIMPLEST CASE.
We begin our analysis with the simplest dynamics of map (4.1), namely fixed points (T = 1).
The coordinates of such a fixed point obey the following system of equations : From the second equation of (5.1) we can see that an obvious fixed point holds with Y = 0. But then we discover from the first equation that it corresponds to the (unstable) fixed point of the logistic map, X = f (X). According to the previous section, this should be an homogeneous state (with vanishing variance) and it is easy to show it is not stable.
In section 8, we will come back to this issue.
Non obvious solutions (Y = 0) are easily computed as : Here α = α (λ, ε) = 1 − 2ε sin 2 π λ . Notice we are only interested in fixed points for which 0 ≤ X ≤ 1 and Y 2 ≤ 1. These conditions fix the boundaries, in parameter space, for the existence of solutions with T = 1.
The conditions for the stability of such fixed points are found by computing the eigenvalues of the tangent map of (4.1) evaluated on the solutions given by (5.2) : As it can be easily shown, for the values of µ that we are interested in and unless λ = 2 or λ = 3, the stability of these fixed points will imply ε > 1.
We have numerically confirmed the predictions also in this case, but we refrain to present them here since they go beyond the frame of the present work, see [19].
We then focus on the simplest case, λ = 2, in order to show we can recover the results shown in [6].
First we observe that : and by changing variables : the equations of the fixed points of (4.1) read : so that (5.1) reduces to a well known form. A and B are the values of the configuration for even and odd sites on the lattice and, therefore, the existence of a solution for (5.6) is equivalent to the one of the given type for the whole lattice.
In other words, as already mentioned in section 3, the reduced map (4.1) is exact when λ = 2.
From (5.6) we easily find that A and B are the roots of the equation : and which obviously agree with (5.1) where α is taken as in (5.4).
On the other hand, the stability of such fixed points is given by the condition where ν ± and the two roots of the usual equation deduced from the jacobian of (5.6) namely : where : In fig. 5 we show the diagram of bifurcations in parameter space (µ, ε), where the region corresponding to stable fixed points is filled in grey.
We also mention it is possible to show that the stability of the solution of (5.6) implies the stability of the corresponding solution for the whole lattice (the proof is given in the Appendix at the end of the paper).
For that purpose, we notice that the eigenvalue equations for the tangent map of the CML with a lattice of even length, calculated at the fixed point defined above may be reduced, by symmetry, to a set of L/2 eigenvalue problems for 2 x 2 matrices. Finally, inspecting the different cases, we found that, inside the stability domain of the reduced system all eigenvalues of the CML tangent map are, in modulus, strictly less than 1.
Furthermore, the two bifurcations of (5.6) are of different nature. For the lower end point of the stability interval the eigenvalues are real and they are complex for the upper end point.
Let us add that, in principle, it is possible to follow the same lines of equations (5.6) for solutions of the CML with T = 1 and λ > 2. This leads to a set of λ-equations replacing (5.6) that was studied in [19], but the system became rapidly cumbersome and it can only be studied by numerical computation.
The advantage of our reduced model (4.1) comes from the fact that we always have a two-dimensional system, no matter of the wavelength we consider.

TIME-PERIOD TWO DYNAMICS
We come back again to the reduced model F defined in (4.1) to analyse the case of solutions with time period 2.
Numerical simulation shows the existence of period 2 cycles satisfying the following symmetry relation : No other cycle of period 2 was found for F .
The coordinates X, Y of such cycles obey then the equations : where, again, the dependance of α = 1 − 2ε sin 2 π λ in λ and ε has been omitted. The solution of (6.2) is immediately found : The corresponding tangent map is, according to (4.1) given by : and therefore, for Q T = Q 2 = DF (X, −Y ) . DF (X, Y ), one has : and then the determinant and trace of Q T are : from which we can compute the eigenvalues ν ± as roots of the corresponding usual equation.
As   F and the dynamics of the CML is deeper than the only coincidence of the two bifurcation points. This relation will be treated elsewhere using the method described in [17].
As a second test we take ε = 0.667 to be fixed as well as µ = 3.8, the same value as before and taking now λ as order parameter.
The largest value of the modulus of the eigenvalues of Q 2 is now plotted as a function of λ in fig. 7.
Following the discussion of section 3 we compare now these values with the actual CML bifurcation diagram by using an adiabatic compression.
We begin with a snapshot of a lattice which length is L = 63 and a wavenumber q = 9 (λ = 7) as shown in fig. 8-a. Fig. 8-b shows the lattice after adiabatic compression until L = 55, where we can see the relaxed state with the same wavenumber is still stable. The destabilization occurs for L = 54 (λ ′ 1 = 6) as it is seen in fig. 8-c. Then the system, relaxes always with L = 54, to a new stable state as it is shown in fig. 8-d after a transient regime of about 4000 iterations. Fig. 8-e shows a superposition of two such states at time t and t + 2. Notice that in fig. 8-d and 8-e we are concerned with states of temporal period 4 and a more complex spatial behaviour which is reminescent of the Eckhaus instability in Rayleigh-Bénard convection [18]. in the plane (λ, ε) for µ = 3.8.
We have made a large number of numerical simulations for the CML and found they were also in good agreement with the predictions of F as they are shown in these two pictures.
We end up this section by adding that, as already noticed, changing ε in (1 − ε) in equations (5.4) transforms a fixed point in a cycle of period 2. We notice that this is a special case of a more general transformation that takes equation (5.1) of a fixed point of F in equation (6.2) of a cycle of period 2, by changing α in −α.

TIME-PERIOD FOUR DYNAMICS
Exactly in the same way of section 6 we can study the solutions with period 4 of F and then to compare the dynamics of F with CML one.
Again we observed that the cycles have the symmetry Y → −Y of the map F , so we only need to describe the cycle four positive numbers X 1 , X 2 , Y 1 , Y 2 , cycle that is runned in the following way : This combines the permutation X 1 → X 2 between ∆ 1 and ∆ 2 for the logistic map described in section 1 and the flip of period T = 2 studied before.
Therefore a cycle T = 4 is solution of or, in other words, solution of the system : This leads, after straightforward calculations, to the following equation, expressed in the variable Z = α. f ′ (X) : Denoting Z 1 , Z 2 the two real roots of (7.4) we finally get : We observe that, taking Z = ±1 we recover the equivalent expressions for T = 1 and T = 2, which is naturally related with the symmetry of the cycles and the definition of Z as the ratio of two successive values of Y .
The stability of such cycles is studied as before.
As in the previous section we only give two examples of application since all the other cases we have performed show the same accuracy.
In both cases we take µ = 3.63, a value for which the attractor of f lies inside two disjoint intervals ∆ 1 , ∆ 2 as mentioned before.
First we take λ = 8 fixed and ε as a parameter. The stability interval of the cycle of F , as it is shown in fig. 11-a, has limit points ε 1 = 0.42 and ε = 0.695 whereas the values obtained from the bifurcation diagram of the CML, shown in fig. 11-b, gave ε ′ 1 = 0.41 and ε ′ 2 = 0.69. Then, as in section 6, we fix ε = 0.667 and use λ as a control parameter. From the stability of F we get stability between λ 1 = 7.85 and λ 2 = 10.15 as it is shown in fig.   12, whereas the adiabatic compression shown in fig. 13 give, for the critical values of the wavelengths, λ ′ 1 = 65/8 = 8.1 and λ ′ 2 = 52/5 = 10.4. The same comments as in case T = 2, can be done for T = 4.
We also notice that, when the eigenvalues of Q T cross the unit circle with complex values, the whole lattice undergoes a period doubling bifurcation.
Finally we want to add that we performed the corresponding numerical simulations for the values of the parameter for which the reduced map F has cycles of higher order and we also find a good agreement with the corresponding dynamics of the CML.

HOMOGENEOUS DYNAMICS
A simple observation about the dynamics of the reduced map F is worth to be noticed.
we see that the condition implies, using the second equation of F that Y t → 0 when t → ∞ for all initial conditions Therefore, in this case, the asymptotic dynamics is defined as According to the given interpretation of these variables, such a dynamics may correspond to a dynamics of the CML for which there is synchronized motion (since Y = 0).
The corresponding dynamics is obviously given by the one of the logistic map, according to the first equation.
This regime is already observed in CML, see [20], but suprisingly, the reduced model again gives quite accurate predictions in this case. We will address this question elsewhere.
Before ending this section, let us add the following comment : condition (8.2) is which means that, for all ε > 0 there is always a λ (ε) such as for λ > λ (ε) (8.4) may not be true. But this effect can not be seen if the size of the lattice is not large enough and, moreover, only for an infinite lattice we should have no such cutoff. At least in the view of the reduced model, we see some care need to be taken when dealing with finite size lattices instead of the infinite limit case.

LYAPUNOV EXPONENTS
Finally, let us present some results obtained by computing maximal Lyapunov exponents along orbits initialized in the vicinity of the spatiotemporal periodic states which have been studied above.
The aim is to give a more precise meaning for the terms "chaotic" and "laminar" used for a phase of a CML, here simply defined by its positive or negative Lyapunov exponents.
Again we compare the results obtained from the dynamics of the reduced model F with the ones coming from F together with the explicit formulae of his tangent map. whereas ε is taken as control parameter. This is the situation described in section 6, where We may observe, again, a precise connection between the two stability intervals. Moreover, for a smaller interval (0.33.., 0.40..) we see that the Hopf bifurcation of the map F corresponds to the period doubling of F that we have already noticed.
Many other numerical computations we have performed, for different values of the parameters, confirm this agreement between the two dynamics.

CONCLUDING REMARKS
In this work we were concerned with the problem of spatiotemporal intermittency as it can be observed in CML. Directed by the idea that one of the mechanisms undersetting this phenomenon is a competition of local configurations with different wavelengths we have addressed the question of the stability of such wavelengths.
The essential point of our work came from the possibility of describing many features of this bifurcation diagram as well as of the corresponding dynamics of the CML by means a two-dimensional map.
The study of this map allows, in a very simple way, to predict the existence and stability of large ordered subsets of the lattice.
This approach propounds a particular coupling between the spatial mean value and the variance of the variables in phase space. It also suggests a particular form of a control parameter that combines the diffusion constant and the spatial wavelength of the configuration.
Even if this reduced two-dimensional model gives very accurate predictions for the CML dynamics we were not able, in the general case, to achieve a complete proof of the correspondance, as it is the case for λ = 2.
Further progress will lead to the problem of the study of the full system F as it was described in section 2 and would give a more serious basis to the knowledge of the conditions of applicability of the two-dimensional map F .

APPENDIX :
We show there the stability of state (5.5) of the whole lattice in the stability domain of the variable ε of the corresponding state of the reduced map (5.1).
Due to the fact that the tangent map DF of the whole lattice evaluated on the state (5.5) commutes with the operator of cyclic permutations of order two, whose eigenvalues are the roots of unity : W k = e 4πki L , k = 1, 2, . . . , L/2 ; the eigenvalues problem for DF may be easily reduced to the one of the set of L/2 matrices M k defined by : We remark that M L/2 is equivalent to the Jacobian of the reduced map whose trace T and determinant D are given by (5.12), and the traces T k and determinants D k of M k may be expressed in terms of T and D : where : As shown in fig. 5, for µ > 3.45, the stability domain of the reduced map extends from ε > 0.8. For such values the trace T is always negative and β k satisfye for any k in 1, . . . , L/2, the relations : where : γ = −β L/4 = − (1−ε) 2 1−2ε . Now, with the following notations : where ν ± k (resp. ν ± ) are the eigenvalues of matrices M k (resp. M L/2 ). and : τ = −T, ∆ = τ 2 − 4D, ∆ k = τ 2 − 4β k D We are going to consider successively the cases : 1./ D ≤ 0 and 2./ D > 0 with subcases 2.1/ ∆ ≥ 0 and 2.2/ ∆ < 0, and show in each of them that, under the hypothesis |ν| < 1, one finds : |ν ⋆ | < 1. Let us denote by K 1 (resp. K 2 ) the subset of the indices k where ∆ k ≥ 0 (resp. ∆ k < 0) and consider separately K 1 and K 2 : In the first case : In the second, where the eigenvalues are complex, one has : 2./ If D > 0 let us consider separately the cases of real and complex eigenvalues ν : 2.2/ for ∆ < 0 one has |ν| = √ D < 1, we again consider separately the subset K 3 and K 4 of indices k for which ∆ k ≥ 0 and ∆ k < 0.
and, as a function of ε : |ν| which ends the proof.