Antonis D Pinotsis

Infinite Feigenbaum Sequences and Spirals in the Vicinity of the Lagrangian Periodic Solutions

Keywords: celestial mechanics, restricted three-body problem, chaos, Lagrange points L 4, L 5

We studied systematically cases of the families of non-symmetric periodic orbits in the planar restricted three-body problem. We took interesting information about the evolution, stability and termination of bifurcating families of various multiplicities. We found that the main families of simple non-symmetric periodic orbits present a similar dynamical structure and bifurcation pattern. As the Jacobi constant changes each branch of the characteristic of a main family spirals around a focal point-terminating point in x-at which the Jacobi constant is C ∞ =3 and their periodic orbits terminate at the corotation (at the Lagrangian point L 4 or L 5 ). As the family approaches asymptotically its termination point infinite changes of stability to instability and vice versa occur along its characteristic. Thus, infinite bifurcation points appear and each one of them produces infinite inverse Feigenbaum sequences. That is, every bifurcating family of a Feigenbaum sequence produces the same phenomenon and so on. Therefore, infinite spiral characteristics appear and each one of them generates infinite new inner spirals and so on. Each member of these infinite sets of the spirals reproduces a basic bifurcation pattern. Therefore, we have in general large unstable regions that generate large chaotic regions near the corotation points L 4 , L 5 , which are unstable. As C varies along the spiral characteristic of every bifurcating family, which approaches its focal point, infinite loops, one inside the other, surrounding the unstable triangular points L 4 or L 5 are formed on their orbits. So, each terminating point corresponds to an asymptotic non-symmetric periodic orbit that spirals into the corotation points L 4 , L 5 with infinite period. This is a new mechanism that produces very large degree of stochasticity. These conclusions help us to comprehend better the motions around the points L 4 and L 5 of Lagrange.

Introduction

In general, the stable periodic orbits are followed by many quasi periodic orbits, while the unstable periodic orbits are followed by large numbers of chaotic orbits. There are some routes to chaos in dissipative and conservative dynamical systems. The well-known Feigenbaum sequence of infinite period doubling bifurcations is a particular route to chaos in dissipative and conservative dynamical systems [START_REF] Feigenbaum | Quantitative Universality for a Class of Nonlinear Transformations[END_REF][START_REF] Coullet | Iterations d' Endomorphismes et Groupe de Renormalisation[END_REF]. Some other routes for the transition from order to chaos in conservative dynamical systems of 2-degrees of freedom have been found by [START_REF] Contopoulos | Periodic Orbits and Ergodic Components of a Resonant Dynamical System[END_REF], Contopoulos (1983aContopoulos ( ,b, 1991)), [START_REF] Heggie | On the Bifurcations of a Certain Family of Periodic Orbits[END_REF], [START_REF] Bier | Remerging Feigenbaum Trees in Dynamical Systems[END_REF] and [START_REF] Pinotsis | Bifurcations and instabilities in the restricted three-body problem[END_REF]. These scenarios of routes to chaos are mentioned by [START_REF] Contopoulos | Order and Chaos[END_REF][START_REF] Contopoulos | Order and Chaos in Dynamical Astronomy[END_REF]. Also, some of the routes to chaos in dissipative and conservative systems we find in an article by [START_REF] Eckmann | Roads to Turbulence in Dissipative Dynamical Systems[END_REF].

It is well known, that in the planar restricted 3-body problem Feigenbaum sequences of families of simple and double symmetric periodic orbits appear, as the Jacobi constant C varies, with a universal ratio consistent with the value δ 8.72. Bifurcated families have in general large unstable regions and generate large chaotic regions [START_REF] Contopoulos | Infinite Bifurcations in the Restricted Three-Body Problem[END_REF][START_REF] Pinotsis | Bifurcations, stability and universality of families of periodic orbits in the restricted three-body problem[END_REF][START_REF] Pinotsis | Successive bifurcations and evolution of double and quadruple periodic orbits in the restricted three-body problem[END_REF].

Although there exist many studies of the types of bifurcations for the families of symmetric periodic orbits in various Hamiltonian systems of two degrees of freedom, there are limited explorations of the stability and bifurcations of families of non-symmetric periodic orbits. Thus, our study centers on the changes of stability and bifurcations of the families of non-symmetric periodic orbits, which bifurcate from the original families b, c, l and n.

The stability of motions in the neighborhood of the Lagrangian points L 4 and L5 has been investigated for some values of the mass ratio μ by a number of investigators [START_REF] Pinotsis | Bifurcations and instabilities in the restricted three-body problem[END_REF][START_REF] Contopoulos | The generation of spiral characteristics[END_REF][START_REF] Sandor | The phase space structure around L 4 in the restricted three-body problem[END_REF][START_REF] Bardin | On motions near the Lagrange equilibrium point L 4 in the case of Routh's critical mass ratio[END_REF][START_REF] Henrard | Families of periodic orbits emanating from homoclinic in the restricted problem of three bodies[END_REF][START_REF] Efthymiopoulos | Formal Integrals and Nekhoroshev stability in a mapping model for the Trojan asteroids[END_REF][START_REF] Erdi | A parametric study of stability and resonances around L 4 in the elliptic restricted three-body problem[END_REF]. Also, [START_REF] Sicardy | Stability of the triangular Lagrange points beyond Gascheaus's value[END_REF] examined the stability of these points for some values of μ beyond the Gascheau's value in the planar restricted three-body problem and found that the periodic orbits follow a Feigenbaum cascade leading to chaos at a value of μ .

We made a preliminary study for the non-symmetric short period orbits (SPO) of the family b [START_REF] Pinotsis | Bifurcations and instabilities in the restricted three-body problem[END_REF]) and we found infinite spiral characteristics around infinite focal points. Also, spiral characteristics of families of periodic solutions have been found in the case of a satellite oscillations [START_REF] Bruno | The limit problems for the equation of oscillations of a satellite[END_REF][START_REF] Bruno | Families of periodic solutions to the Beletsky equation[END_REF]. In this paper we make a systematic study of the families of non-symmetric periodic orbits that bifurcate at the points b1, c1, l2, n 2 4 [START_REF] Hénon | Exploration Numerique du probleme restreint, (II) Masses egales, stabilite des orbites periodiques[END_REF]) and z 1 [START_REF] Pinotsis | Bifurcations, stability and universality of families of periodic orbits in the restricted three-body problem[END_REF], in the planar circular restricted 3-body problem and in the case of equal masses of the primaries. We found interesting information about the evolution and termination of the bifurcating families of various multiplicities of non-symmetric periodic orbits. The relevant topological characteristics change not only quantitatively but also qualitatively. We found a new mechanism that introduces a large degree of stochasticity, which could not be predicted a priori. Thus, we found a new route for the transition from order to chaos. This work required very large computing time because as the Jacobi constant C varies infinite successive stable and unstable segments along the characteristic of each bifurcating family appear, while each bifurcation point generates infinite inverse Feigenbaum sequences. So, each bifurcating family forms infinite spirals and each one of them generates infinite new inner spirals and so on. Therefore, we have in general large unstable regions that generate large chaotic regions near the corotation points L 4 , L 5 and so these points are unstable.

The method

For the planar restricted 3-body problem the equations of motion are [START_REF] Szebehely | Theory of orbits[END_REF]

y x 2 y , x y 2 x       (1)
The potential function Ω is given by 2 2 1 2 1 2 1 1 ((1 )r r ) r r 2

(2) and r 1 2 =(x-μ) 2 +y 2 , r 2 2 =(x-μ+1) 2 +y 2

The Jacobian integral of equations ( 1) is 2 2

C 2 (x y ) 

(3)

where C is the so-called Jacobian constant. We study non-symmetric periodic solutions with respect to the x-axis (the axis joining the two primaries), in the physical plane (x,y), for the Copenhagen case (μ=0.5). If we vary the Jacobi constant C, the values of x, x  and y  vary continuously and the characteristic of the family is a curve in the 3-dimensional space (C, x, x  ), since the value of y  can be calculated from the Jacobi integral. For the numerical calculations, in double precision, we used the recurrent power series method with variable step in time.

In order to find non-symmetric periodic orbits we applied essentially a Newton method. A similar method has been applied already in 3-dimensional Hamiltonian systems [START_REF] Magnenat | Periodic orbits in triaxial galactic models[END_REF]. In order to calculate a periodic orbit approximate, initial conditions (x 0 , y 0 =0, 0

x  , 0 y  ) are given. Each orbit intersects again the y=0 axis with y  in the same direction as 0 y  and at the point x, y  , x  . If we consider x, x  as functions f i (i=1, 2) of initial values of x 0 , 0 x  (we can find 0 y  from the Jacobi constant) then,

x=f 1 (x 0 , 0 x  ), x  =f 2 (x 0 , 0 x  ) (4)
In order to find a periodic orbit we give a small change of the initial conditions, Δx 0 , Δ 0 x  which implies a change of the final point Δx, Δ x  . In order for the orbit to be periodic we must have,

x 0 +Δx 0 = f 1 (x 0 +Δx 0 , 0 x  +Δ 0 x  ) 0 x  +Δ 0 x  = f 2 (x 0 +Δx 0 , 0 x  +Δ 0 x  ) (5)
Neglecting terms of order higher than one in Taylor's expansion, in matrix form equations ( 5) can be written

0 0 x x  + 0 0 x x  = x x  + 4 3 2 1 a a a a 0 0 x x  (6) or 1 a a a 1 a 4 3 2 1 0 0 x x  = x x x x 0 0   (7) where 0 1 x f =a 1 , 0 1 x f  =a 2 , 0 2 x f =a 3 , 0 2 x f  =a 4
The quantities a 1 , a 2 , a 3 , a 4 can be found by calculating two orbits near the original one (i.e. we need to compute 3 orbits) with deviations (Δx 0 , 0) and (0, Δ 0 x  ). In order to find approximate initial conditions for different bifurcating families at the bifurcation points we used a method of trial and error. Since we neglected higher order terms in Taylor's expansion the solution of the system (7) does not give exactly the periodic orbit. Adding the increments Δx 0 , Δ 0

x  , given by equation ( 7), to x 0 , 0 x  we obtain better initial conditions for the next iteration. This iterative process has to be repeated a number of times in order to give a periodic orbit with the required accuracy.

The stability of the families of non-symmetric periodic orbits can be studied in general by evaluating the [START_REF] Hénon | Exploration Numerique du probleme restreint, (II) Masses egales, stabilite des orbites periodiques[END_REF] stability parameters a, b, c, d or by evaluating the monodromy matrix of the variational equations [START_REF] Hadjidemetriou | Periodic Orbits and Stability[END_REF]. In this work the stability of the families is studied by evaluating the stability parameters which are functions of the initial values 0 , 0 x x  and the Jacobian C (Hénon 1965). For a non-symmetric periodic orbit the stability condition is

A a d 2 1 (8)

Infinite sets of spirals and self-similar orbits

We consider the original families of simple symmetric periodic orbits b, l and c. At the previously mentioned bifurcation points -critical points-with initial conditions: b1 (C in , x up , x  , for the upper branch and x l for the lower one) (Table I), l2 (C in , x up , x  , x l ) (Table IV) and c1 (C in , x up , x  , x l ) (Table VI) bifurcate new families of simple non-symmetric periodic orbits of the same period with their corresponding symmetric periodic orbits (we call these families main families fb1, fl2 and fc1 respectively) (Figures 1,4,7).

Each main family of the above three cases and each bifurcating families of various multiplicities presents the Feigenbaum scenario of infinite period doubling pitchfork bifurcations and each bifurcation ratio tends to approach the universal number δ 8.72 at a finite value of the Jacobian constant, which we have already found for the families of symmetric periodic orbits [START_REF] Contopoulos | Infinite Bifurcations in the Restricted Three-Body Problem[END_REF][START_REF] Pinotsis | Bifurcations, stability and universality of families of periodic orbits in the restricted three-body problem[END_REF][START_REF] Pinotsis | Successive bifurcations and evolution of double and quadruple periodic orbits in the restricted three-body problem[END_REF].

Along the characteristic curve of each main family, as the serial number of its bifurcation points increases the distances between two neighbouring characteristics of the bifurcating families become smaller and smaller. So, it is more and more difficult to separate the bifurcating families of different multiplicity. Also, they are strongly unstable and the instability of the families becomes larger and larger from one bifurcating family to the next one. Therefore more accuracy is necessary to study them, that is, a new magnification of the space (x-C diagram) is needed and so on. Thus, it is difficult to follow further some of these families in order to find their termination point because of the very large computing time needed.

As we mention below in these cases we took many interesting information about the evolution and termination of the bifurcating families of various multiplicities of periodic orbits.

Also, we studied the main family of double non-symmetric periodic orbits, fz 1 , which bifurcates at the critical point z 1 (C in , x, x  ) (Table VIII). We found here the Feigenbaum phenomenon. We followed the evolution of this main family of double non-symmetric periodic orbits beyond its first 2 4 transition point. The family becomes more and more unstable, as C increases, while the period of the orbits continuously increases. We did not follow further this family because of the very large computing time needed. We stopped the numerical calculations at the point (C st , x, x  ).

As regards the main family of double non-symmetric periodic orbits, fn 2 4, which bifurcates at the critical point n 2 4, it is unstable and becomes more and more unstable as the Jacobi constant C varies. So, it does not appear that it has any bifurcation of other families of periodic orbits. In Tables (I-VIII) we give some of the bifurcation points of all these families. The tables "are published as attachments to the online version of this article".

Evolution and stability of the main family fb1 I. Simple periodic orbits

The orbits of the main family fb1 are the short period orbits (SPO). The second branch of its characteristic is outside the Figure 1. As the Jacobi constant increases the stability parameter decreases and at the 1 2 point (C 1 , x 1 , x  ) (Tables I, II, III) (first bifurcation point) bifurcates a new stable family of double non-symmetric periodic orbits (we call this family fbd1) (Table II). This new family, as C increases, it becomes critically unstable at the 2 4 transition point (C 2 4 , x up , x  , x l ) (Table III). Here we have the Feigenbaum scenario. The characteristic of each bifurcating family has two branches which extend towards larger values of C (Figure 1).

Looking for possible inverse Feigenbaum sequences, we followed the evolution of the main family fb1. At first, as C increases the characteristic of the family extends to the right and downwards (the absolute value of x decreases) in the x-C diagram. At the 1 2 bifurcation point (C 2 , x 2 , x  ) (Tables I, III) -second bifurcation point-a new stable family of non-symmetric double periodic orbits-we call this family fbd2-with two branches towards smaller values of the Jacobi constant appears. As C decreases the new stable family fbd2 becomes unstable and so on. Here the Feigenbaum scenario appears.

As C increases beyond the second bifurcation point, the characteristic of the main family continues to the right and downwards while the family is stable until the first turning point (C t1 , x t1 , x  ) and the family has a maximum C. Following the evolution of the main family beyond the first turning point the characteristic of the family has an initial part with C decreasing for decreasing absolute value of x. As C decreases further the characteristic turns upwards in x. At the point (C t2 , x t2 , x  ), second turning point, the family becomes critically stable again and the characteristic has a minimum C. After the second turning point, for C increasing the absolute value of x increases and the family has a small stable segment until the third 1 2 critical point (C 3 , x 3 , x  ) at which it bifurcates a new stable family of non-symmetric double periodic orbits with two branches towards larger values of C. This family loses its stability at the 2 4 bifurcation point (C 2 4 , x up , x  , x l ) (Table III) and so on. A cascade of Feigenbaum bifurcations is generated here.

After the third bifurcation point, for C increasing, the main family becomes more and more unstable but after a while the family becomes stable again at the fourth 1 2 bifurcation point (C 4 , x 4 , x  ). At this point it bifurcates a new stable family of non-symmetric double periodic orbits with two branches towards smaller values of C. At this point the Feigenbaum scenario appears.

After the fourth bifurcation point the main family is stable until the third turning point (C t3 , x t3 , x  ) at which it has a maximum C and so on (Figure 1). The same phenomenon occurs in the other (lower) branch of the main family. We stopped the numerical calculations at the point C st , x st , x  ) (Table I) after the fifth bifurcation point (C 5 ,, x 5 , x  ) at which we found the Feigenbaum scenario.

As C changes along the spiral characteristic, which approaches asymptotically its focal point-terminating point, infinite changes from stability to instability and vice versa appear, while each bifurcation point generates infinite inverse Feigenbaum sequences. That is, each branch of a Feigenbaum sequence presents the same phenomenon and so on. Therefore there exist infinite spirals and each one of them produces infinite new inner spirals and so on. After infinite successive small stable and large unstable segments the characteristic spirals inwards around its terminating point. Each branch of the main family spirals into its focal point C ∞ =3, x up ≈-1.624989 (or x l ≈ -0.684635 for the lower branch). The values of x (for each branch of a family) were calculated as the average of the extreme -closest to the focal point-values of the corresponding final spiral, which was found numerically. [START_REF] Contopoulos | The generation of spiral characteristics[END_REF] investigated periodic orbits of a galactic type potential and explained how the spirals are generated as soon as the Langrangian points L 4 , L 5 become unstable.

Figure 2 shows a set of some simple periodic orbits. We found that near the first turning point a cusp appears close to the Lagrangian point L 4 , which turns to loop in the second turning point. As C varies, the size of the loop of the orbit grows and encloses the triangular libration point L 4 . As C varies further, a new inner cusp appears which evolves to a new inner loop, near the fourth turning point, surrounding the triangular point L 4 and so on. As we approach the terminating point infinite loops, one inside the other, emanate, surrounding the Lagrangian point L 4 . Thus, an infinite set of self-similar orbits is constructed, each member of this set reproducing a basic pattern. The orbits spiral around the triangular point L 4 . So, the terminating point corresponds to an asymptotic non-symmetric periodic orbit spiraling into the point L 4 , which has the same Jacobi constant as the Langrangian point L 4 (C =3). In this limiting point we have an infinite self-similar orbit with infinite period and so the termination principle T is satisfied [START_REF] Szebehely | Theory of orbits[END_REF].

II. Families of double periodic orbits fbd1 and fbd2

We followed the evolution of the family fbd1 beyond the 2 4 transition point (first bifurcation point of the family fbd1) (Tables II, III and Figure 3a). As C increases, the two branches of the characteristic of the family fbd1 extend to the right in the x-C diagram. The upper branch initially goes upwards, with decreasing period and then turns downwards while the period continuously increases. The lower branch goes downwards. These branches are outside and far from the characteristic of the main family (Figure 1). As C increases further the family becomes critically stable again at the 2 4 bifurcation point, second bifurcation point, (C 2 , x l , x  , x up ) (Table II) a new stable family of quadruple non-symmetric periodic orbits with two branches towards smaller C is generated and so on. Here the Feigenbaum scenario appears.

After its second bifurcation point, the family fbd1 has a stable segment and at its first turning point (C t1 , x l , x  , x up ) the family has a maximum C. After this point C decreases with the absolute value of x decreasing for the upper branch and increasing for the lower one (Figure 1). At the second turning point (C t2 , x l , x  , x up ) the family has a minimum C. After this point the family has a stable segment until the third 2 4 bifurcation point (C 3 , x l , x  , x up ) and so on. At this point a new stable family with two branches towards larger values of C emanates, giving birth to a Feigenbaum sequence. We stopped at the point (C st , x l , x  , x up ) (Table II, Figure 3c) which is after the fourth 2 4 bifurcation point.

In this case a similar bifurcation pattern as in the case of the main family of simple periodic orbits fb1 appears. The two branches of the family fbd1, which pass outside the spirals of the main family, spiral around their focal points with C ∞ =3, x up ≈-1.6558 and x l ≈-1.5038. As the spiral characteristic approaches asymptotically its focal point infinite changes from stability to instability and vice versa appear while from each bifurcation point infinite inverse Feigenbaum sequences are generated. Each branch has infinite successive small stable and large unstable segments. Every bifurcating family of a Feigenbaum sequence produces the same phenomenon and so on. Therefore, infinite spiral characteristics appear and each one of them generates infinite new inner spirals and so on. Each member of these infinite sets of the spirals reproduces a basic bifurcation pattern. Therefore, we have in general large unstable regions that generate large chaotic regions near the corotation point L 4 . We have here self-similar orbits, which grow loops, as C varies. These loops surround the Langrangian point L 4 . The terminating point corresponds to an infinite self-similar asymptotic orbit with C ∞ =3, while its period is infinite.

We also followed the evolution of the family fbd2 (Table III, Figure 1). As C decreases the two branches of the characteristic curve of the family extend to the left in the x-C diagram, while its period continuously increases. This family becomes critically unstable at the 2 4 bifurcation point (C 2 4 , x l , x  , x up ) (Figure 3d). We have here the Feigenbaum sequences (Figure 1).

As C decreases, the family fbd2 becomes more and more unstable and its two branches continue further to the left and upwards in the x-C diagram. The further study of this family is very difficult because of the very large degree of stochasticity and so the very large computing time needed to extend it to its terminating point. We stopped the numerical calculations at the point (C st , x l , x  , x up ) (Table III, Figure 3e).

The two branches of this family are close to the characteristic of the main family fb1 and inside of the two branches of the previous family fbd1 (Figure 1). Also, the shapes and the initial values of x of the orbits (b) and (e) of Figure 3, that belong to the two families fbd1 and fbd2 respectively and they have the same value of Jacobi constant C, are different. So, the family fbd2 can not join the family fbd1 in order to form a bubble (Contopoulos 1983a). Thus, it is topologically necessary that the evolution of the family fbd2 should present a similar bifurcation pattern and dynamical structure of spirals as the previous mentioned cases of the families of simple and double periodic orbits fb1 and fbd1. So, each branch of the characteristic of the family fbd2 spirals into its focal point with C ∞ =3. The terminating point corresponds to the initial conditions of a limiting asymptotic non-symmetric orbit spiraling into the Langrangian point L 4 . These results help us to comprehend the motions around the point L 4 of Lagrange.

Evolution and stability of the main family fl2 I. Simple periodic orbits

The second branch of the characteristic curve does not appear in Figure 4. As C decreases, at the 1 2 critical point -first bifurcation point-(C 1 , x 1 , x  ) (Tables IV,V) bifurcates a new stable family of non-symmetric double periodic orbits-family fld1. After that, for smaller values of C, the family fld1 loses its stability at the 2 4 bifurcation point (C 2 4 , x up , x  , x l ) (Table V) and so on (Figure 4). The Feigenbaum scenario appears.

We followed the main family after its first bifurcation point. As C decreases, the characteristic of the family continues to the left and upwards (the absolute value of x increases) in the x-C diagram while the period continuously increases (Figure 4). The family becomes critically stable at the second bifurcation point (C 2 , x 2 , x  ) (Tables IV). At this 1 2 transition point a new stable family of double non-symmetric periodic orbits-family fld2-with two branches towards larger values of C is generated. For somewhat larger C the new family fld2 also becomes unstable at the 2 4 bifurcation point (C 2 4 , x up , x  , x l ) (Table V, Figure 6d) and so on. Thus, we have here the Feigenbaum scenario (Figure 4).

As C decreases the main family is stable until the first turning point (C t1 , x t1 , x  ) at which it has minimum C. After this point, the characteristic of the family turns to the right and upwards. As C increases further the absolute value of x decreases and the family becomes critically stable again at the second turning point (C t2 , x t2 , x  ) and it has maximum C. With C decreasing the family has a stable segment until the third 1 2 bifurcation point (C 3 , x 3 , x  ) (Tables IV,V). At this point a new stable family of double non-symmetric periodic orbits emanates, while its two branches are towards smaller value of C. The new family becomes critically unstable at the 2 4 transition point (C 2 4 , x up , x  , x l ) (Table V) and so a Feigenbaum scenario appears.

With C decreasing beyond the third 1 2 bifurcation point the family becomes stable again at the fourth 1 2 bifurcation point (C 4 , x 4 , x  ), giving birth to a Feigenbaum sequence towards larger values of C. Αs C decreases further, the family fl2 has a stable segment until the third turning point (C t3 , x t3 , x  ) at which the family has minimum C and so on. We stopped the numerical calculations at the point (C st , x st , x  ) (Table IV) which is after the fifth turning point (C t5 , x t5 , x  ).

The same phenomenon occurs in the other branch of the main family. Thus the evolution of the family fl2 has a similar bifurcation pattern and dynamical structure as the families fb1 and fbd1. Namely, as the Jacobi constant changes each branch of the main family spirals around its focal-terminating-point at C ∞ =3, x up =-2.279677, x l =2.28174. As we approach the focal point of the family infinite changes of stability to instability and vice versa appear while each bifurcation point produces infinite inverse Feigenbaum sequences and so on. Therefore there exist infinite spirals and each one of them produces infinite new inner spirals and so on. That is, a basic pattern is continuously reproduced. The stable parts along the characteristic of the family are smaller and smaller as we approach to the terminating point.

In Figure 5 we give a set of simple periodic orbits. We found that with C decreasing, a cusp near the Lagrangian point L 4 appears which turns to a loop surrounding this point (Figures 5b,c). As C varies further, a new inner cusp appears which evolves to a new inner loop surrounding the point L 4 (Figure 5d) and so on. Thus, as we approach the terminating point of the family we pass through an infinite set of self-similar orbits, every member of this set reproducing a basic pattern. The terminating point of the family corresponds to an infinite nonsymmetric self-similar orbit, with infinite period and C ∞ =3, approaching asymptotically the triangular point L 4 .

II. Families of double periodic orbits fld1 and fld2

We follow now the evolution of the family fld1 beyond the 2 4 its bifurcation point (Table V, Figure 6a). As C decreases, the two branches of its characteristic continue to the left in the x-C diagram (Figure 4). The upper branch turns upwards while the lower one extends initially downwards and then turns upwards, that is, they have a similar behavior as the previous case of the family fbd1. The two branches seem to pass respectively above and below the spirals of the main family. We stopped the evolution of the family at the point (C st , x up , x  , x l ) because of the very large degree of stochasticity and so of the very large computing time needed. The instability parameter reaches an absolutely large value (Table V, Figure 6c).

Figure 6 shows a set of non-symmetric double periodic orbits. We see that the orbits (a), (b), (c) of the family fld1 present a similar evolution as the orbits of the main family fl2 (Figure 5) as well as the orbits of the families fb1 and fbd1 (Figures 2,3). That is, with decreasing C three cusps on the orbits appear near the triangular libration points L 4 , L 5 , which evolve to three small loops. As C decreases further the loops grow and enclose these points. It seems that as C varies new inner loops continuously appear on the orbits surrounding the Lagrangian points. We conclude that the family has a similar bifurcation pattern and dynamical structure as the previous cases of the families fl2, fb1 and fbd1. Each branch of the characteristic of the family will approach asymptotically its focal point, with C ∞ =3, spiralling around it.

Also, the evolution of the family fld2 shows that its two branches extend to the right and in general downwards in the x-C diagram (Figure 4). The two branches are close to the characteristic of the main family and inside the two branches of the previous family fld1. We stopped the evolution of the family fld2 at the point (C st , x up , x  , x l ) because of the very large computing time needed (Figure 6e). The instability parameter reaches an absolutely large value (Table V).

In Figure 6 we give two non-symmetric double periodic orbits of the family fld2. We see that as C increases the two loops grow and enclose the Lagrangian point L 4 . The periodic orbits (c) and (e) (Figure 6) of the families fld1 and fld2 respectively have different shapes. Thus, the two families do not seem to join each other in order to form a bubble. Thus, it is topological necessity the two families to have a similar bifurcation pattern as in the previous case of the families fbd1, fl2 and fb1.

Evolution and stability of the main family fc1 I. Simple periodic orbits

As C increases the stability index of the main family decreases and at the point (C 1 , x 1 , x  ) (Tables VI, VII) we have the 1 2 critical point (first bifurcation point) bifurcates a new family of double non-symmetric periodic orbits-family fcd1-which is also initially stable. After this point, as C increases, the family of double periodic orbits becomes unstable at the 2 4 bifurcation point (C 2 4 , x up , x  , x l ) (Table VII). We have here the Feigenbaum scenario.

We continued the characteristic of the family fc1 beyond its first bifurcation point (Figure 7). The other branch of the characteristic is outside of the x-C diagram of Fig. 7. As C increases the characteristic curve extends to the right and downwards in the x-C diagram (the absolute value of x decreases) and at the second 1 2 bifurcation point (C 2 , x 2 , x  ) (Tables VI, VII) the family becomes critically stable again. In this point a new stable family of non-symmetric double periodic orbits bifurcates-family fcd2-from the main family with two branches towards smaller values of C. As C decreases this new family becomes critically unstable at the 2 4 transition point (C 2 4 , x up , x  , x l ) (Table VII, Figure 9c) and so the Feigenbaum scenario appears (Figure 7).

The main family is stable until the first turning point (C t1 , x t1 , x  ) and it has maximum C. As C decreases the characteristic curve turns to the left and downwards in the x-C diagram. With C decreasing further the absolute value of x increases and the family becomes critically stable again at the second turning point (C t2 , x t2 , x  ) and it has minimum C. After the second turning point, C increases with the absolute value of x increasing while the family has a small stable segment until the third 1 2 bifurcation point (C 3 , x 3 , x  ) (Tables VI,VII), at which a new stable family of non-symmetric double periodic orbits with two branches towards larger values of C bifurcates. This stable family loses its stability at the 2 4 transition point (C 2 4 , x up , x  , x l ) (Table VII) and so on. Here a Feigenbaum scenario appears.

After the third bifurcation point, as C increases, the main family becomes stable again at the point (C 4 , x 4 , x  ). At this fourth 1 2 bifurcation point a bifurcated new stable family of double non-symmetric periodic orbits extends towards smaller values of C. This stable family loses its stability and the Feigenbaum phenomenon appears.

After the 1 2 transition point the family fc1 becomes critically unstable again at the third turning point (C t3 , x t3 , x  ) at which we have a maximum C and so on. We stopped the numerical calculations at the point (C st , x st , x  ) (Table VI). This point is after the fifth 1→2 bifurcation point at which we found the usual Feigenbaum scenario.

Thus the evolution of the main family presents a similar bifurcation pattern and dynamical structure as in the cases of the families fb1 and fl2. That is, as C varies, each branch of the family spirals into its focal point at C ∞ = 3, x up =-0.214405 and x l =0.215312. Therefore, infinite spirals appear and each one of them generates infinite new inner spirals and so on. Each member of these infinite sets of the spirals reproduces a basic bifurcation pattern. So, we found a new route to Chaos

Figure 8 shows four simple periodic orbits. The last orbit (d) has two loops surrounding the Lagrangian point L 5 . As C varies new inner cusps appear which evolve into new inner loops surrounding the point L 5 and so on. Thus, we have here self-similar orbits. The terminating point of the family corresponds to an infinite non-symmetric self-similar orbit, with infinite period, approaching asymptotically the triangular point L 5 (C ∞ = 3).

II. Families of double periodic orbits fcd1 and fcd2

We continued the family fcd1 beyond the 2 4 its bifurcation point-first bifurcation point of this family (Table VII, Figure 9a). The two branches of its characteristic curve extend to the right in the x-C diagram. As C increases, the upper branch turns initially upwards in x and then downwards, as in the previous mentioned cases of the families of double periodic orbits. The lower branch moves continuously downwards while the period of the orbits initially decreases and then increases. The two branches are outside and far from the characteristic of the main family (Figure 7). So, it seems that the two branches will pass outside of the spirals of the main family as in the previous case of the family fb1. Because of the very large computing time needed, we stopped at the point (C st , x l , x  , x up ) (Table VII, Figure 9b).

On the contrary, the evolution of the family fcd2 (Tables VI, VII) shows that its two branches extend to the left and upwards for the upper branch and downwards initially and then upwards for the lower one, while the period continuously increases. The two branches of the family are close to the characteristic of the main family and inside of the two branches of the previous family fcd1 (Figure 7). This family becomes more and more unstable, as C decreases. We did not follow further the family because of the very large degree of stochasticity and consequently of the very large computing time needed. We stopped at the point (C st , x up , x  , x l ) (Table VII, Figure 9d)

Figure 9 shows some double non-symmetric periodic orbits along the characteristic of the two families fcd1 and fcd2. It seems that the two families fcd1 and fcd2 present similar bifurcation structure and dynamical behavior as the previously mentioned families of double periodic orbits of the main family fb1. Comparing the variation of the periods of the orbits of the two families (Table VII) as well as the different shape of their orbits b and d of Figure 9, it seems that they do not join each other in order to form a bubble. Thus, it is topologically necessary that the families should have a similar bifurcation pattern as in the previous case of the main family fc1 as well as in the case of the double periodic orbits of the family fb1. That is, each bifurcated family seems to turn around its focal point with C ∞ =3, while infinite bifurcation points and so infinite inverse Feigenbaum sequences and infinite spirals appear. Consequently, in the case fc1 we found a new mechanism which leads to a large degree of stochasticity. The conclusions help us to comprehend the motions around the point L 5 of Lagrange.

Conclusions

We studied systematically cases of non-symmetric periodic orbits of the families b, l, c and n of the planar restricted 3-body problem and in the case of equal masses of the primaries. The study gives a new interesting dynamical structure of the phase space and the topological characteristics change not only quantitatively but also qualitatively.

For the main families of simple non-symmetric periodic orbits fb1, fl2, fc1, and fz 1 that bifurcate at the points b1, l2, c1, [START_REF] Hénon | Exploration Numerique du probleme restreint, (II) Masses egales, stabilite des orbites periodiques[END_REF] and z 1 [START_REF] Pinotsis | Bifurcations, stability and universality of families of periodic orbits in the restricted three-body problem[END_REF]) respectively, we found the evolution and their stability. In their first bifurcation point these families present a similar bifurcation structure. We did not find the termination point of the family fz 1 because of the very large computing time required.

As regards the main families fb1, fl2, fc1, as the Jacobi constant C varies each branch of the characteristic of a main family of simple non-symmetric periodic orbits spirals around its focal point-termination point in x-at which the Jacobi constant is C ∞ =3. As the characteristic of each main family approaches asymptotically its termination point infinite changes from stability to instability and vice versa and so infinite bifurcation points appear. So, infinite successive very small stable and very large unstable segments along the characteristic of each main family are formed. The stable parts along the characteristic of each family are smaller and smaller as we approach to the terminating point. Each bifurcation point generates infinite inverse Feigenbaum sequences. That is, each bifurcating family presents the same phenomenon and so on. Thus, infinite sets of inverse Feigenbaum sequences and infinite sets of spirals appear. Each member of these infinite sets of the spirals reproduces a basic bifurcating pattern. Thus, the families are strongly unstable and so the regions near the Lagrangian points L 4 , L 5 are mostly chaotic.

Every bifurcating family of double non-symmetric periodic orbits produces also the same phenomenon, i.e. infinite inverse Feigenbaum sequences and infinite spirals that have infinite different focal points. Also, each bifurcated family of higher multiplicity presents the same phenomenon. That is, each bifurcated family leads to spiral around its focal point in x with C =3. Each spiral has infinite changes of stability to instability and vice versa. Infinite bifurcation points appear and each one of them generates infinite inverse Feigenbaum sequences and so on. Therefore, we have infinite spirals with infinite different focuses and each one of them generates infinite new inner spirals and so on. Thus, infinite sets of spirals appear and each member of these infinite sets of the spirals reproduces a basic bifurcating pattern.

The orbits of each main family near the corotation -Lagrangian points L 4 , L 5 present a new dynamical structure (Figures 2,5,8). As C changes along the characteristic of each main family a cusp close to the Lagrangian point L 4 or L 5 on the orbits emanates that evolves to loop surrounding one of these points. As C changes continuously, approaching asymptotically its terminating point, a new cusp inside the loop appears that turns to new inner loop enclosing one of the triangular points L 4 or L 5 and so on. This process reproduces continuously a basic pattern. As we approach the terminating point, an infinite set of self-similar orbits is constructed. The same phenomenon presents the families of double non-symmetric periodic orbits (Figures 3b,c and 6b,c) as well as the orbits of each bifurcated family of higher multiplicity. So, the terminating point of each bifurcating family corresponds to a non-symmetric self-similar orbit with infinite period and C ∞ =3, that approaches asymptotically the point. Thus, the termination principle for a family of periodic orbits, which states that either the period of an orbit or its size or the energy goes to infinity [START_REF] Szebehely | Theory of orbits[END_REF], is satisfied.

Bubbles do not exist in this case of families of periodic orbits that terminate at L 4 or L 5 . In general, there do not exist bubbles in the case μ=0.5. We have found indications that bubbles are formed after a critical value of μ.

The maximum absolute value of the instability parameter of each family is very large of order 10 6 . This value grows further as we approach the focus of its characteristic as well as the multiplicity of the bifurcating families increases. Therefore, there must be large chaotic regions of stochastic orbits around these periodic orbits.

This mechanism of the infinite sets of inverse Feigenbaum sequences and infinite sets of spirals of each (main) family are generated near corotation points L 4 or L 5 . Each multiplicity family terminates after infinite spirals at its focal point-constant point in x-with the value of Jacobi constant C ∞ =3. The Lagrangian points L 4 , L 5 as well as the regions near them are strongly unstable.

This mechanism produces, in general, a very large degree of stochasticity that generates large chaotic regions near the corotation points (L 4 or L 5 ). This is a new route to chaos. We have thus a new mechanism that produces very large degree of stochasticity. These conclusions help us to comprehend better the motions around the corotation points L 4 and L 5 . We anticipate that this mechanism will appear also in other rotating dynamical systems of 2-degrees of freedom.

Figure captions

Figure 1 :

 1 Figure 1: Evolution of the families of non-symmetric simple (main family fb1), double, quadruple,… periodic orbits schematically (the actual values are given in the Tables). ( ___ ) stable, (….) unstable orbits. The families of double periodic orbits fbd1 and fbd2, emanated at the 1→2 first and second bifurcation points of the family fb1. The two branches of the

Figure 2 :

 2 A set of simple periodic orbits in the x-y plane: (a) A symmetric periodic orbit of the original family b in the neighborhood of the bifurcation point b1. Also, three non-symmetric periodic orbits along the characteristic of the main family fb1. We see a change of the form of the cusps or of the loops as C varies: (b) near the first turning point, (c) near the second turning point, and (d) at the point at which we stopped the numerical calculations.

Figure 3 :

 3 Some double non-symmetric periodic orbits in the x-y plane: Along the characteristic curve of the family fbd1: (a) near the 2→4 bifurcation point, (b) at the point C=3.047539, x=-1.498219454462, x  -0.03068701204, T=13.3517 and (c) at the point at which we stopped the numerical calculations: Also, along the characteristic of the family fbd2: (d) near the 2→4 bifurcation point, and (e) at the point at which we stopped the numerical calculations. The orbits (b) and (e) of the two families fbd1 and fbd2 correspond to the same C.

Figure 4 :

 4 Evolution of the families of non-symmetric simple (main family fl2), double, quadruple,… periodic orbits schematically (the actual values are given in the Tables). ( ___ ) stable, (….) unstable orbits. The families of double periodic orbits fld1 and fld2 emanated at the 1→2 first and second bifurcation points of the family fl2. The two branches of the characteristic curve of the family fld2 remain inside the two branches of the characteristic curve of the family fld1.

Figure 5 :

 5 Simple periodic orbits: (a) A symmetric periodic orbit of the original family l near the bifurcation point l2. Also three non-symmetric periodic orbits of the main family fl2: (b) a cusp appears near the point C=3.19, x=-2.079926229183, x  0.191954854072, T=10.8154, (c) near the first turning point and (d) near the fifth turning point. Figere 6: Double non-symmetric periodic orbits: Orbits belonging to the family fld1: (a) near the bifurcation point 2→4, (b) three cusps appear near the point C=3.1230853, x=-2.13360173108, x  0.19734606655 and (c) at the point at which we stopped the numerical calculations: Orbits along the characteristic of the family fld2: (d) near the 2→4 bifurcation point of the family fld2 and (e) at the point at which we stopped the numerical calculations.

Figure 7 :

 7 Evolution of the families of non-symmetric simple (main family fc1), double, quadruple,… periodic orbits schematically (the actual values are given in the Tables). ( ___ ) stable, (….) unstable orbits. The families of double periodic orbits fcd1 and fcd2 emanated at the 1→2 first and second bifurcation points of the family fc1. The two branches of the characteristic curve of the family fcd2 remain inside the two branches of the characteristic curve of the family fcd1.

Figure 8 :

 8 A set of simple periodic orbits in the x-y plane: (a) A symmetric periodic orbit of the original family c in the neighborhood of the bifurcation point c1. Also, three non-symmetric periodic orbits along the characteristic of the main family fc1. We see a change of the form of the cusps or of the loops as C varies: (b) near the first turning point, (c) near the second turning point and (d) at the point at which we stopped the numerical calculations.

Figure 9 :

 9 Some double non-symmetric periodic orbits in the x-y plane: Orbits belonging to the family fcd1: (a) near the bifurcation point 2→4 and (b) at the point at which we stopped the numerical calculations: Orbits along the characteristic of the family fcd2: (c) near the 2→4 bifurcation point and (d) at the point at which we stopped the numerical calculations.

  Figure 1

  the families of double periodic orbits fld1, fld2 and fld3 Initial conditions of the bifurcation points of the family fld1

Table IV Bifurcations of the main family fl2. C 1 ,C 2 ,.., Jacobi constant of the bifurcation points. C t1 ,C t2 ,.., Jacobi constant of the turning points. Initial conditions of the critical points Period T

 IV

	8
	C st =2.99997945696	x up =-2.27968469583	x  =0.19384087578	26.2157 4.7 10 5
		Table V
	Bifurcations of
				Stability
				index A
	C in =3.552593	x up =-1.907961968103	x  =0	10.5675 +1
	x l =1.9079619681 X 1 =-1.91796399 x 2 =-2.2764855257 x t1 =-2.27808186 x t2 =-2.27955289 x 3 =-2.2795231 x 4 =-2.2796852 x t3 =-2.2796858 x t4 =-2.27967606904 x 5 =-2.27967606 x 6 =-2.2796774342327 C t5 =2.9999679358126962 x t5 =-2.2796774343397 x  =0.1938436770498 25.2888 +1 C 1 =3.53401 x  =0.05286984 10.5765 -1 C 2 =2.855512 x  =0.2132893264 11.8966 -1 C t1 =2.85545487 x  =0.21300771 11.9193 +1 C t2 =3.01707925685 x  =0.19140455 15.3749 +1 C 3 =3.01707913 x  =0.19140926 15.3783 -1 C 4 =2.997887875 x  =0.19413788 18.6593 -1 C t3 =2.997887874567 x  =0.19413779 18.6599 +1 C t4 =3.000260188348925 x  =0.19380232 21.9771 +1 C 5 =3.000260188348 x  =0.19380233 21.9772 -1 C 6 =2.9999679358126972 x  =0.1938436770669 25.2887 -1

Table VI Bifurcations of the main family fc1. C 1 ,C 2 ,.., Jacobi constant of the bifurcation points. C t1 ,C t2 ,.., Jacobi constant of the turning points. Initial conditions of the critical points Period T

 VI

					Stability
					index A
	C in =2.6217	x up =-0.2624399239	x  =0	5.2617	+1
	C 1 = 2.6249439 C 2 =3.0433717756 C t1 =3.043371908421 C t2 =2.994695024137 C 3 =2.99469502439 C 4 =3.000654379316 C t3 =3.00065437931646	x l =0.262412784 X 2 =-0.262203 x 2 =-0.21174171 x t1 =-0.21173066 x t2 =-0.21476724 x 3 =-0.21476741 x 4 =-0.21438139001 x t3 =-0.2143813874	x  =0.03937928 x  =0.27078923 x  =0.27073376 x  =0.25165819 x  =0.25165904 x  =0.25403459 x  =0.25403457867	5.271 7.9827 7.9849 11.2868 11.2871 14.6037 14.6038	-1 -1 +1 +1 -1 -1 +1
	C t4 =2.

Table VIII Bifurcations of the family of double periodic orbits fz 1 Initial conditions of the bifurcation points Period T Stability index A C in =2.2330236 X=-0.5233143 x  =0 12.2682 +1 C 2 4 =2.234761 X=-0.52225233

 VIII

			x  =0.02279438	12.3055 -1
	C 4 8 =2.234933 C st =2.285669	x=-0.52176309 x=-0.51127768	x  =0.02353585 x  =0.07348204	24.6135 -1 12.8737 -6.25•10 4

Acknowledgements: I wish to thank Prof. G. Contopoulos for his useful comments and Manolis Zoulias for making the Figures 1, 4 and 7.


Elke S Schaeffner
email: elke.schaeffner@charite.de

Markus Giet

Jens Gaedeke

Markus Tölle

Natalie Ebert

Martin K Kuhlmann

Peter Martus

Markus Van Der Giet

Dr Elke Pd

MS Schaeffner

The Berlin initiative study: the methodology of exploring kidney function in the elderly by combining a longitudinal and cross-sectional approach

Keywords: CKD, design, epidemiological study, formula, glomerular filtration rate, incidence, older persons, prevalence

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Word count: Abstract: 256, Text: 3966

Running title: Exploring kidney function in the elderly

INTRODUCTION

There is an ongoing demographic shift in Germany that is characterized by less children and a growing number of elderly people with higher life expectancy.

Whereas in 2005 there were almost as many newborns as 60 year olds, in 2050 Germany expects double the number of 60 year olds compared to newborns (1).

Chronic kidney disease (CKD) is a large and growing problem among aging populations. A number of studies show an age-dependent loss of kidney function after 30 years of age. The process of renal aging is complex and includes anatomical, structural as well as functional changes in the kidneys [START_REF] Lindeman | Anatomic and physiologic age changes in the kidney[END_REF]. It is well known that the number of glomeruli decreases with age [START_REF] Goyal | Changes with age in the human kidney[END_REF]. The cause of age-related increases in renal fibrosis leading to glomerulosclerosis, interstitial fibrosis, tubular atrophy, vascular sclerosis, and consequently to loss of renal function are poorly understood; however, in animal models, collagen seems to accumulate with age in the glomerulus, peritubular capillary, and tubulointerstitium [START_REF] Abrass | Aging-associated changes in renal extracellular matrix[END_REF]. Telomere shortening, implicated in somatic cellular senescence pathways, is observed in aging human kidneys [START_REF] Melk | Telomere shortening in kidneys with age[END_REF] to mention just a few possible mechanisms of decline. However, not everyone seems to follow that pattern, some even do not show any loss of kidney function but rather an increase as could be shown in a large longitudinal study [START_REF] Danziger | The age-associated decline in glomerular filtration in healthy normotensive volunteers. Lack of relationship to cardiovascular performance[END_REF]. It is unclear if the loss in kidney function is the result of immunologic, infectious, toxic or ischemic influences or is associated with the observed hyperfiltration resulting in glomerulosclerosis.

This unclarity about renal aging does not only refer to causes, co-morbidities, or associated risk factors. Already the assessment of kidney function in people ≥70 years of age is problematic: In general glomerular filtration rate (GFR) is considered the best overall index of kidney function in health and disease. Since it cannot be measured easily in clinical practice it is nowadays estimated from equations on the basis of serum-creatinine, age, gender, race and body size [START_REF] Levey | A more accurate method to estimate glomerular filtration rate from serum creatinine: a new prediction equation. Modification of Diet in Renal Disease Study Group[END_REF]. One such formula, the Modification of Diet in Renal Disease (MDRD)-equation has gained worldwide acceptance. One major limitation is that in people over 70 years of age the equation is not validated and clearly underestimates kidney function. Even though more accurate than the Cockcroft-Gault formula, an older and also widely used equation [START_REF] Cockcroft | Prediction of creatinine clearance from serum creatinine[END_REF] this underestimation could also be demonstrated for healthy individuals (9), [START_REF] Rule | Measured and estimated GFR in healthy potential kidney donors[END_REF], [START_REF] Vervoort | Assessment of glomerular filtration rate in healthy subjects and normoalbuminuric diabetic patients: validity of a new (MDRD) prediction equation[END_REF]. A very recently developed formula, the CKD-Epi equation, is supposed to be more precise and accurate than the MDRD equation. Again, elderly people were underrepresented in this study [START_REF] Levey | A new equation to estimate glomerular filtration rate[END_REF].

Elderly persons are generally underrepresented in studies about estimation equations [START_REF] Rule | Using serum creatinine to estimate glomerular filtration rate: accuracy in good health and in chronic kidney disease[END_REF]. In addition, stages of CKD developed by National Kidney Foundation's "kidney disease outcomes quality initiative" (KDOQI) do not take into account age related loss of kidney function [START_REF] Kdg | GUIDELINE 1. DEFINITION AND STAGES OF CHRONIC KIDNEY DISEASE[END_REF]. For an 80 year old woman with a serum-creatinine of 1.2 mg/dL the MDRD equation calculates a GFR of only 45 ml/min/1.73m 2 . Therefore the patient will be classified as kidney disease, stage 3. In reality this serum-creatinine of 1.2 mg/dL may correspond to an age-adapted normal kidney function without any evidence of kidney disease. None of the huge previous population based studies regarding prevalence of chronic kidney disease have used a validated equation for GFR-estimation in people ≥70 years of age [START_REF] Coresh | Prevalence of chronic kidney disease and decreased kidney function in the adult US population: Third National Health and Nutrition Examination Survey[END_REF]. One of the few existing studies analysing cardiovascular risk factors and kidney function in the elderly is based on serum creatinine levels only [START_REF] Baggio | Atherosclerotic risk factors and renal function in the elderly: the role of hyperfibrinogenaemia and smoking. Results from the Italian Longitudinal Study on Ageing (ILSA)[END_REF]. The interpretation of a calculated MDRD-GFR is difficult without reference values. A current analysis compared GFR of persons with risk factors such as hypertension, diabetes, and cardiovascular disease with the GFR of a "reference-population" without such risk factors [START_REF] Wetzels | Age-and genderspecific reference values of estimated GFR in Caucasians: The Nijmegen Biomedical Study[END_REF]. The GFR was calculated with the MDRD equation categorizing patients between the age of 18 and 85. Results showed that the KDOQI-staging is not suited for women >50 and men >60 years of age.

Therefore epidemiologic data on normal as well as reduced kidney function in people ≥70 years of age are highly needed in order to estimate GFR not only on the basis of serum creatinine. Without this prerequisite neither assumptions regarding prevalence, progression and risk factors of CKD can be made nor can preventive or therapeutic strategies be developed.

MATERIALS AND METHODS

Design overview

We describe design and methods of the "Berlin Initiative Study", a population based cohort study that started its pilot in November 2009. The main study will start January 2010 and will presumably end in 2014. The study follows a two part design: First there is a longitudinal prospective cohort study following the participants over a prespecified period of time seeing them twice for a baseline-and a follow-up visit (figure 1). The longitudinal approach aims to assess incidence of CKD, risk factors for development and progression of disease as well as associated co-morbidities.

Secondly, there will be a cross-sectional design aiming to develop a new estimating equation to estimate GFR more accurately and precisely and thus assess prevalence of CKD in the population of elderly people ≥70 years of age.

The study was approved by the local ethics committee.

Aims and Hypotheses

Longitudinal design 1. To assess progression of CKD in the elderly and the natural course of kidney function in this population 2. To assess the incidence of clinically relevant decrease in GFR (>3 ml/min/year) applying the classical MDRD equation and the recently published CKD-Epi equation and/or requirement of dialysis within the follow-up of 2 years 3. To assess the incidence of cardiovascular events and mortality.

Longitudinal design

It is planned that 2000 participants will be seen for a baseline visit at one of 13 study sites across Berlin. This baseline visit includes a computer-based standardised questionnaire including questions regarding demographics, lifestyle variables, comorbidities, emphasizing cardiovascular events, and medication. There will be a short examination including blood pressure (identical electronic blood pressure meter available at every study site), heart rate, body mass index, and waist-to-hip-ratio measurement. All measurements are carried out according to pre-specified standard operating procedures. Blood and urine samples will be taken and GFR will be estimated using the MDRD-and the new CKD-Epi equation. The urine samples include testing for (micro)albumin and creatinine as this is part of the current KDOQI classification of CKD and since it was shown that proteinuria in the elderly is predicting progression in kidney function decline [START_REF] Halbesma | Macroalbuminuria is a better risk marker than low estimated GFR to identify individuals at risk for accelerated GFR loss in population screening[END_REF]. Additional to an instant laboratory analysis, storage of blood and urine samples at -80°C will be done for further analyses including genetics. For genetics we use DNA from the white bood cell fraction to perform comprehensive SNP analyses in order to look for gene polymorphysms that might determine renal function in the elderly or correlate with factors influencing renal function in the elderly.

After two years these 2000 participants are planned to be seen again (follow-up visit) for an identical procedure. Baseline and follow-up visit will be carried out by trained personnel only. Information on all participants will be under a pseudonym. Prior to the study informed consent will be obtained from all participants.

In case of death of a study participant between baseline and follow-up visit this information will be retrieved from the insurance company, his/her general practitioner or the register office.

Cross-sectional design (additional measurements at baseline)

Of the 2000 study participants in whom GFR was estimated using the conventional MDRD-equation a subsample of 600 subjects will be defined with oversampling of subjects under risk for CKD based on MDRD-GFR. The participation in the longitudinal part is therefore a prerequisite for the participation in the cross-sectional study part. In this subset of 600 participants kidney function will be assessed by measuring plasma clearance with the exogenous filtration marker Iohexol.

Iohexol is a non-ionic, non-radioactive low osmolar x-ray contrast medium which is eliminated from plasma mainly by glomerular filtration with extra-renal elimination equalling that of Inulin. It was selected for GFR measurement because it is safe and has been used in x-ray investigations for millions of patients all over the world. Its clearance from plasma has been used for measuring kidney function for more than 20 years. In Sweden, where data has been published of a 15 year experience of its use involving more than 8000 patients, no significant adverse effects have been reported [START_REF] Nilsson-Ehle | Iohexol clearance for the determination of glomerular filtration rate: 15 years' experience in clinical practice[END_REF]. By contrast urinary clearance of Inulin, the gold standard, would require extensive technical assistance, the use of a bladder catheter and a difficult chemical assay. Alternatively exogenously administered radionuclides such as 51 CR-EDTA, 99m TC-DTPA, and 125 I-Iothalamate have been used to measure GFR. Major drawbacks of using radio-isotopes are associated with complicated procedures for handling, storage and disposal of contaminated wastes. GFR-measurement with nonradioactive Iothalamate has been applied frequently but also needs urinary clearance measurement requiring a much more complicated protocol probably difficult to fulfil in elderly participants. One shortcoming about the Iohexol plasma clearance method that should be mentioned, is the fact that in patients with a GFR <25 ml/min/1.73 m 2 BSA, the final slope is so flat that it cannot be reliably measured [START_REF] Aurell | Accurate and feasible measurements of GFR--is the iohexol clearance the answer?[END_REF].

Participants will be instructed to abstain from caffeine, paracetamol, smoking and high protein foods for 6 hours prior to and during the hospital visit. Weight and height will be recorded. Baseline venous blood samples will be taken for measurement of hematocrit and serum creatinine. A bolus of 5 ml of 300 mg/ml Iohexol will be administered via an intravenous cannula. Venous blood samples will be collected from the contralateral arm by trained study nurses approximately 2, 3 and 4 hours later and the precise times will be recorded. Participants with a known allergy to iodine or a thyroid stimulating hormone <0.3 mU/l will not be allowed to participate in the cross-sectional part.

Plasma Clearance measurement

GFR will be calculated from plasma clearance following a bolus intravenous injection of Iohexol, with the clearance computed from the amount of the marker administered and the area under the curve of plasma concentration versus time. The decline in serum levels is secondary to the immediate disappearance of the marker from the plasma into its volume of distribution (fast component) and to renal excretion (slow component). This is best estimated using a two compartment model which requires early (usually 2 to 3 time points until 60 minutes) and late (1 to 3 time-points from 120 minutes onward) blood sampling. Advantages include low expense, wide availability, stability in biologic fluids, and rare adverse reactions when given as a small dose (5 ml 300 mg/ml iodine when assayed with a sensitive HPLC assay) (21), ( 22), ( 23), [START_REF] Krutzen | Plasma clearance of a new contrast agent, iohexol: a method for the assessment of glomerular filtration rate[END_REF].

The creatinine-assay will be calibrated with the creatinine-assay of the Cleveland Clinic Laboratory, which is the original MDRD-Laboratory [START_REF] Murthy | Variation in the serum creatinine assay calibration: a practical application to glomerular filtration rate estimation[END_REF]. The necessity of a correct calibration of the creatinine assay has been stated repeatedly [START_REF] Coresh | Calibration and random variation of the serum creatinine assay as critical elements of using equations to estimate glomerular filtration rate[END_REF], [START_REF] Hallan | Validation of the Modification of Diet in Renal Disease formula for estimating GFR with special emphasis on calibration of the serum creatinine assay[END_REF].

Creatinine (male / female): >1.17 -1.5 mg/dl / >0.95 -1.5 mg/dl: moderate kidney disease Creatinine >1.5 mg/dl: manifest kidney disease

Oversampling

The aim of our sampling plan is to obtain equal sizes of age/sex strata defined by 5 year intervals. In the entire population the number of subjects declines with increasing age. Thus in our random sample we give higher weights to the elderly to obtain equal numbers. Moreover, as we expect less response among the elderly, weights are increased by a second factor based on the experience with a community based study in the same city that is an ongoing project of one of our institutions (Dpt.

Biostatistics and Clinical Epidemiology) with the Robert Koch Institute of Germany which is the main institution of health reporting in Germany. These weights are fixed and thus back-calculation of frequencies and their standard errors will be possible.

Recruitment of the subset

600 out of 2000 participants will be asked to participate in an additional kidney function assessment determined by a gold standard method measuring plasma clearance with the exogenous filtration marker Iohexol. This will take place at all three Charité University Hospital study locations with the participant again being able to choose the location.

There will be a second step of oversampling with higher weights for subjects with risk for CKD. This will be based on percentiles of the preliminary determination of kidney function using the MDRD equation.

Sample size 1) Estimation of kidney function:

We expect a prevalence of CKD of 15% in our population. This assumption is based on data from the Third National Health and Nutrition Examination Survey (NHANES III) which is the most comprehensive source of epidemiologic data regarding CKD [START_REF] Jones | Serum creatinine levels in the US population: third National Health and Nutrition Examination Survey[END_REF]. NHANES estimated that 25.22% of men >70 years of age and 8.25% of women >70 years have a serum creatinine ≥1.5 mg/dl. To obtain a two-sided confidence limit of 2% we need a sample size of 1225 evaluable cases. As the number of drop outs is difficult to estimate and the sample size has to be sufficient also for the longitudinal part of the study we will recruit 2000 subjects for the total sample.

(2) Construction and validation of the new estimating formula for kidney function:

We need two subsamples, one sample for development (= learning sample) and one sample for validation of the new formula. Sample size estimation refers to the validation sample. The sample in which the formula will be developed should be of the same size.

The plan is to start with the same variables that were also used in the MDRDequation. Alternative models will be built including further variables such as diabetes, hypertension, and variables linked to muscle mass (BMI, WHR), as well as laboratory variables such as urea, urine-or serum-albumin.

Precision of the new formula will be quantified according to square of the Pearson correlation. We expect a value of 81%, which corresponds to r = 0.9. We want to estimate two-sided 95% limits of confidence for this precision of 5%. Thus a sample of evaluable subjects of n = 214 is necessary. To adjust for dropouts of 25%, the validation sample should consist of n = 285 subjects, we want to recruit 300 subjects.

The development sample will be of the same size, thus we will have to determine GFR in 600 participants.

(3) Estimation of incidences and determination of risk factors:

In the longitudinal analysis -taking into account 25% of dropouts with respect to the initial 2000 subjects -we expect n = 1500 evaluable subjects. We have poor prior knowledge about the observed number of events during the follow-up of two years.

Depending on the size of observed rates we expect a precision of between 0.8% and 2.3% (two-sided 95% confidence limits for rates of 2% to 30%). Accordingly we can detect hazard ratios of between 2.8 (30 events) and 1.3 (456 events) (level of significance = 0.05 two-sided, 80% power in a significance test, prevalence of risk factors 0 50%). For very rare risk factors, the detectable hazard ratios will be smaller.

Statistical analysis

(1) Construction and validation of the new estimating formula for kidney function:

Methods similar to those used by Levey et al [START_REF] Levey | A new equation to estimate glomerular filtration rate[END_REF] will be applied in the subsample of 600 subjects undergoing Iohexol measurements. Learning and validation samples will be used to obtain unbiased estimates for the precision of the formula.

(2) Estimation of kidney function:

The formula developed in the subsample of n=600 subjects will be used to estimate the kidney function in the total sample of n = 2000 subjects. This analysis will be done adjusted for the oversampling of elderly subjects. We do not expect a severe bias from including the 600 subjects used for the development of the formula in the overall estimation of kidney function.

(3) Estimation of incidences and determination of risk factors:

Cumulative incidences of new CKD, dialysis, and heart failure will be determined based on the follow-up after 2 years using standard calculating cumulative incidences for competing risks. Simple and multiple Cox regression will be used to obtain hazard ratios including 95% limits of Confidence for the relevant events based on the followup data. Finally, applying variable selection techniques a multivariate risk index will be developed. All analysis will be done using the newest releases of standard software (SPSS for Windows, R).

Quality assurance

Data entry will be web based via computers with internet access at the sites of the participants' examination. For data entry an AJAX (Asynchrone Java-Script and XML) -based 21 CFR 11 conform data management system (SAPPHIRE; MEDEORA GmbH, Köln, Germany) will be used. The system possesses an encoded datamanagement, -transfer, and an audit trail as well as the classical standard components for clinical studies such as support for monitoring and query-handling.

The participants' data will be transferred in an encoded fashion and with the help of an ISO 25237 conform pseudonymisation-service pseudonymized (conversion of the participant identifier [PID] to a pseudonym [PSN]). Data will be stored in a central data base. An independent center (University of Wuerzburg) has been commissioned for monitoring and quality assurance. This center has been featured with an own administrative access to the central database and several automated analyses and plausibility checks. Figure 2 clarifies data flow and quality assurance.

From the central database a core data set will be transferred to a core database once a year. This core data base is in custodial custody of the "KfH-foundation of preventive medicine". Data from several projects converge in this joint data base and are available for future research projects. Individual data will be controlled using ranges during data entry. A continuous monitoring of data referring to completeness, plausibility, observer effects and distributional characteristics will be implemented by the biostatistical center. These analyses will not change the design or the hypotheses of our study, thus no adjustment for interim analysis will be necessary.

DISCUSSION

CKD is one of the large and growing public health problems that aging populations will have to face to an increasing degree. This increase in CKD may not only be agerelated but also attributed to the increasing incidences of diseases associated with kidney disease, including diabetes mellitus, hypertension and other cardiovascular diseases. Clinical assessment of kidney function to define the CKD stage is essential when practicing medicine but difficult in the elderly. The level of GFR and its magnitude of change over time are crucial to the detection of CKD and for making decisions about diagnosis, prognosis and treatment. None of the existing equations to date to estimate GFR are validated in this age group since elderly people tend to be generally underrepresented [START_REF] Rule | Using serum creatinine to estimate glomerular filtration rate: accuracy in good health and in chronic kidney disease[END_REF] and they tend to underestimate GFR. The newest development, the CKD-EPI equation arose from cross-sectional pooled data sets of 26 studies [START_REF] Levey | A new equation to estimate glomerular filtration rate[END_REF]. Even though a very big sample of several thousands of participants it again included only few elderly persons.

By applying Iohexol plasma clearance, which should be unbiased by urinary clearance of the marker substance and thus is a practical method in an aging population we will develop a new equation from original data exclusively for persons ≥70 years of age. Apart from developing a new estimation equation to accomplish more exact numbers of prevalence also co-morbidities and risk factors being associated with or contributing to the progression of CKD in the elderly will be assessed longitudinally.

The BIS has potential limitations that should be mentioned. They include the problem of a presumably low response rate especially at that age-group leading to a certain degree of selection bias despite contacting the participants randomly. The fact that the recruitment includes participants from one insurance company only may also introduce selection bias although this company is one of the biggest insurance companies and insures more than half of the target audience covering people with different socioeconomic status. Besides we are aware of the problem that there could be a higher response rate in those patients who have not yet renal insufficiency but for some reasons are aware of a potential risk. On the other hand there might be a lower response rate in subjects with manifest renal insufficiency. A second shortcoming we will have to face will be losses to follow-up, especially in this rather old population. Starting at the age of 70 years and above we expect a presumably higher mortality rate compared to a younger cohort. In this context the rather short follow-up of two years regarding evaluating the progression of CKD was chosen.

While this period may be viewed as insufficient, previous studies have demonstrated that this duration is adequate to determine progression in kidney function [START_REF] Hemmelgarn | Progression of kidney dysfunction in the community-dwelling elderly[END_REF][START_REF] Levey | Assessing the progression of renal disease in clinical studies: effects of duration of follow-up and regression to the mean. Modification of Diet in Renal Disease (MDRD) Study Group[END_REF]. A third potential limitation would be the fact that not every participant selected for Iohexol measurement will agree to participate. In this case we would replace subjects who participate in the baseline examination but refuse measurement with Iohexol by subjects matched for age, sex, and kidney function according to our routine measurement. Last, in case of a newly diagnosed kidney disease we will have to face the possibility of influencing the course of disease by recommending check-ups and even treatment between baseline and follow up visit.

A validated equation estimating GFR will have implications concerning referral to the nephrologist, pharmacological treatment and dosing of potentially nephrotoxic drugs and preparing for invasive diagnostic or therapeutic procedures. Furthermore it may help to increase the pool of much needed potential living kidney donors for a growing population of elderly patients with end stage renal disease. Longitudinal assessment of incident cases of kidney function, as well as progression of CKD, co-morbidities and associations with risk factors will hopefully answer questions of the aging-kidney phenomenon.

The BIS seeks to contribute to fill these gaps by combining a cross-sectional and longitudinal approach.

Legends Table 1: inclusion/exclusion criteria longitudinal analysis

Inclusion criteria:

age ≥ 70 years -AOK insurance status informed consent

Exclusion criteria:

dialysis treatment planned or current kidney transplantation

4 . 4 .

 44 To assess associations between risk factors and kidney function / requirement of dialysis and between kidney function and cardiovascular events Cardiovascular events are defined as non fatal or fatal myocardial infarction or cardiac death; non fatal or fatal stroke (ischemic/hemorrhagic); revascularisation, including angioplasty or bypass of the coronary arteries or other arteries, and non traumatizing limb amputation. Cross-sectional design 1. To develop a new equation to predict GFR in a subset of the selected study population (development or learning cohort) 2. To validate the new equation in another subset of the selected study population (validation cohort) 3. To assess the prevalence of CKD in people ≥70 years of age applying the classical MDRD equation and the by then newly developed Berlin Initiative Study (BIS) equation To assess risk factors that correlate with prevalence of CKD

ACKNOWLEDGEMENTS

The authors thank Andrej Woehrmann for his assistance in the quality assurance process and the manuscript preparation. Grant support: This work was supported by the KfH Foundation of Preventive Medicine (www.kfh-stiftung-praeventivmedizin.de), a non-profit organisation supporting science and research in the field of prevention. Address: KfH-Stiftung Praeventivmedizin, Martin-Behaim-Strasse 20, 63263 Neu-Isenburg

Participant recruitment

The aim is to enrol a community based sample of 2000 participants. In Berlin the insurance company called "Berliner Allgemeine Ortskrankenkasse" (AOK) gives insurance coverage to almost 50% of the persons ≥70 years of age. Of approximately 370.000 (28) ≥70 year olds residing in Berlin 166.500 have AOK insurance.

To avoid selection bias the AOK will randomly pick a sample of participants fulfilling the inclusion/exclusion criteria named below (table 1). From another ongoing population study of elderly persons (subproject of autonomy despite multimorbidity in old age (AMA) consortium, funded by Federal Ministry of Education and Research, http://www.ama-consortium.de/f-omaha.html) it is known that overall response rates are expected to lie between 10-20% only displaying decreasing response rates with increasing age. Thus stratified oversampling for higher ages will be performed. The AOK will contact between 10.000 and 20.000 persons. In a fractionated manner invitation letters will be sent to the potential participants offering them a free kidney check-up at one of the 13 study sites. The participants are allowed to pick the study site that is most convenient for them to reach. Study subjects will be enrolled at one of the 13 study sites. Informed consent will be obtained from every participant before enrolment. Every participant will be informed about his/her laboratory results per mail. We think out of ethical reasons we are obliged to give a recommendation to have all elevated creatinine values or pathologic urine-results re-checked and evaluated for causes.

Depending on serum creatinine values the following statements will be given on the reports based on the "Deutsche Hochdruckliga" (German Hypertension League): Creatinine (male / female): ≤1.17 mg/dl / ≤0.95 mg/dl: normal (laboratory threshold)

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