F kσ. (x) (φ) x. φ x. Fixed point

Transcription

1 ÙÖ ½ F kσ () Ψ,, l τ (φ) Fied point φ

2 ÙÖ ¾ µ Z 1 B z + v e,2 B 2 1+ v Z e,1 2 z1 Z 1 B 2+ v Z 2 c,2 B 3+ Z1 v c,1 Z 2 µ Ψ lτ (φ) 2 π 3/2 π π π/2 φ π/2 π φ min φ ma B 2+ B 1+ B + B 3+ µ Ψ lτ (φ) 2 π 3/2 π π π/ φ π/2 π φ min (3+ 3+) (1+ 1+) φ ma 2+ +

3 ÙÖ µ v e B 1 Z 2 B v e v c Z 2 v c Z 1 Z 1 B 2 v c Z 1 v B c 3 Z 2ve Z 2 v e Z 1 µ Ψ lτ (φ) 2 π B 3 µ Ψ lτ (φ) 2 π 3 3/2 π B 2 3/2 π 1 2 π B 1 π 1 π/2 B π/2 3 2 φ π/2 π φ min φ ma φ π/2 π φ min φ ma

4 ÙÖ µ B + v e,2 Z 1 B 1+ Z 2 Z 2 Z 1 v e,1 B 2+ B 3+ Z 2 v c,2 Z 2 Z 1 Z 1 v c,1 µ Ψ lτ (φ) 2 π 3/2 π π π/2 B 3+ B 2+ B 1+ µ Ψ lτ (φ) 2 π 3/2 π π π/2 (2+2+) (1+1+) (3+3+) (2+2+) (++) φ π/2 π φ min B + φ ma φ π/2 π φ min (++) φ ma

5 ÙÖ µ Z 1 B B 1 Z 2 v e,1 v c Z 2 Z 1 v e,2 v e B 2 v c B 3 Z 2 v e v e Z 1 Z 1 Z 2 v c µ 2 π Ψ lτ (φ) B 2 µ Ψ lτ (φ) 2 π 2 3 3/2 π B 1 3/2 π 1 π B π 1 π/2 B 3 π/2 2 3 φ π/2 π φ min φ ma φ π/2 π φ min φma

6 ÙÖ d pairs of orbits

7 ÙÖ D(Λ) Ikeda D(Λ) Henon p=12 p=13 p=14 p= Λ 1 p= p= p= p= Λ

8 ÙÖ Λ 1.8 Λ p=1.4 p= ln(λ) Λ ln(λ) Λ p=12.4 p= ln(λ) Λ ln(λ) Λ p=14.4 p= ln(λ) ln(λ)

9 ÙÖ Λ (j) orb (p).8 Λ (j) orb (p) period p Ikeda Henon period p

10 Theor and Applications of the Sstematic Detection of Unstable Periodic Orbits in Dnamical Sstems Detlef Pingel and Peter Schmelcher Theoretical Chemistr, Institute for Phsical Chemistr, INF 229, Universit of Heidelberg, 6912 Heidelberg, German Fotis K. Diakonos Department of Phsics, Universit of Athens, GR Athens, Greece arxiv:nlin/611v1 [nlin.cd] 7 Jun 2 Ofer Biham Racah Institute of Phsics, The Hebrew Universit, Jerusalem 9194, Israel (Februar 17, 218) A topological approach and understanding to the detection of unstable periodic orbits based on a recentl proposed method (PRL 78, 4733 (1997)) is developed. This approach provides a classification of the set of transformations necessar for the finding of the orbits. Applications to the Ikeda and Hénon map are performed, allowing a stud of the distributions of Lapunov eponents for high periods. Particularl the properties of the least unstable orbits up to period 36 are investigated and discussed. I. INTRODUCTION Unstable periodic orbits (UPOs) represent a skeleton of comple chaotic sstems and allow the calculation of man characteristic quantities of the underling dnamics like Lapunov eponents, fractal dimensions and entropies of the attractors [1,2]. For dissipative sstems epansions in terms of periodic orbits are well established in the literature [1,3 6] and demonstrate the relevance of the ccles for the understanding of chaotic dnamics. Both low-dimensional model sstems such as discrete maps [1,2] as well as eperimental time series [7 1] have been studied. Furthermore the series epansion of semiclassical properties of classicall chaotic Hamiltonian sstems with respect to the length and stabilit coefficients of the periodic orbits is a fruitful and frequentl applied technique: it allows the investigation of the energ level densit as well as other quantum properties [11]. Much has been added to the importance of the UPOs b using them to control chaotic dnamical sstems ( see [12] and references therein ). More recentl, ccle epansion techniques have been invented and shown to converge well, especiall when the smbolic dnamics is well understood [4,13]. Series epansions over periodic orbits used for calculating dnamical averages are tpicall ordered according to the orbit length p [1,13 15]. Drawbacks of these epansions are the large number of orbits (increasing eponentiall with p), the required completeness of the set of ccles for given period and the slow convergence properties [14,15]. A promising proposal was made [16,17] saing that series epansions could converge better if the are truncated according to the stabilit of the orbits [18]. What is more, stabilit ordering does not rel on the knowledge of the sstems smbolic dnamics, which is unknown for generic dnamical sstems. Chaotic dnamics is intrinsic for man phsical sstems, which is wh periodic orbit theor is not restricted to specific areas of phsics. Much effort has been spent on developing efficient techniques for calculating UPOs of a given dnamical sstem. What makes this quest so difficult is the eponential proliferation of the number of ccles with increasing period and their increasing instabilit (in full chaotic sstems). Specificall the modified Newton Raphson method is an all purpose method, which does not require a special form of the underling equations of motion. However, it needs a good initial guess for the starting points. Therefore this algorithm rapidl becomes epensive and is limited to relativel short periods and low dimensional sstems. A variet of other methods have been developed which focus either on time series analsis or on finding UPOs for given equations of motions [2,4,14,19 21]. For a special class of sstems, a numerical technique for calculating arbitraril long UPOs to an desired accurac was introduced in [21] for the Hénon map [22] and later applied to a few other dnamical sstems [15,23,24]. This method allows the sstematic computation of all UPOs of an given order, each given b a unique binar smbol sequence. Recentl, we proposed an alternative method (in the following referred to as SD method) for the calculation of UPOs [25]. The basic idea is to transform the full chaotic sstem to a new dnamical sstem with the periodic orbits 1

11 keeping their positions but changing their stabilit properties: For a particular transformed sstem a certain fraction of the periodic ccles becomes stable and can be found b simpl iterating the transformed sstem. This fraction depends on a tuning parameter which represents an upper stabilit cutoff for the fied points (FPs) to be detected. In the following we will use the term stabilisation and stabilised fied points for the process and fact respectivel that the unstable FPs of the original chaotic dnamical sstem have become stable in the corresponding transformed sstem. The reader is kindl asked to distinguish this use of the term stabilisation from the one used in control theor of chaos via unstable periodic orbits. According to the above the SD method allows the sstematic calculation of the least unstable periodic orbits of an given order p [26]. The latter possibilit meets the requirements of the series epansions using stabilit ordering, since the allow to derive properties of a phsical sstem b eploring onl the least unstable period orbits up to a given stabilit cutoff. The technique of the SD method is highl fleible and can be applied in a straightforward manner to a great variet of discrete dnamical sstems of an dimension. Continuous dnamical sstems can thereb be treated using properl chosen Poincaré surfaces of section. Therefore, it might open the wa for emploing the proposal of [16,17] for a great variet of dnamical sstems. The present work has a twofold aim. The first goal is to enhance our understanding of and give insights into the SD method. To that end we provide a complete classification of the involved transformations. As a result we gain both topological as well as geometrical understanding and interpretation of the transformations. Corresponding invariant structures are thereb revealed and the FPs can be classified similarl to the stabilising transformations. This opens the future perspective to selectivel detect UPOs not onl with respect to their stabilit but also with respect to certain desired geometrical properties. We will thereb learn how simple global operations on the dnamical sstem change the stabilit properties of fied points. The second aim of this work is to elucidate and etend the work in ref. [25]. To this end we provide etensive results of applications of our approach to the Ikeda [28] and Hénon [22] maps. For the Ikeda map, we calculate the complete sets of FPs for periods up to p = 15. The number of orbits is large enough to investigate the distribution of Lapunov eponents. This distribution is compared with the corresponding one of the Hénon map according to [21]. Net the algorithm of [26] to determine the least unstable orbits is slightl modified and applied to the Ikeda and Hénon map. The ten most stable orbits of a given length up to period p = 36 are studied for both maps. Their Lapunov eponents var in a characteristic wa as a function of the period. In detail we proceed as follows: In Section 2.1 we briefl review the SD method for finding UPOs as described in [25]. The topological/geometrical classification and geometrical etension is presented and discussed in Sec In Sec. 3.1 we show the results of complete sets of orbits and we present and analse the distribution of Lapunov eponents in Sec The etended method to stabilise the least unstable ccles is given in Sec. 3.3 together with the corresponding results for the Hénon and Ikeda map. Sec. 4 concludes with a summar. II. THEORY OF THE STABILISATION OF FIXED POINTS A. Brief Review of the underling Method In order to be self contained and for our further theoretical investigation we recall in the following the ke ideas of the method developed in ref. [25] to detect UPOs. Consider a discrete chaotic dnamical sstem U : r i+1 = f(r i ) in n dimensions. The FPs of the pth iterate f (p) are points of the UPOs of period p. To find the FPs of U the following strateg is emploed: A set of transformations is specified which transforms U into new dnamical sstems {S kσ } with the FPs keeping their original locations in space. The set {S kσ } is chosen such that for each unstable FP α u of U there eists a specific transformed sstem S k σ of the set {S kσ} for which this FP has become dissipativel stable (i.e. Re(λ i ) < for continuous sstems, λ i < 1 for discrete sstems, with eigenvalues λ i ) and can therefore be detected b simpl iterating properl chosen starting points of the transformed sstem S k σ. For each S k σ a different set of FPs of U is stabilised. Let us denote b I min the minimal set of pairs (kσ) with the propert that there eists for each unstable FP α u at least one pair (k σ ) I min for which S k σ transforms α u into a stable FP α s. This set holds for arbitrar period p. The search for the FPs of U is then straightforward: A starting point chosen in the global neighbourhood (see below) of the FP α u iterated with the transformed dnamical sstem S k σ, (k σ ) I min converges, due to the stabilit of α s, to the position of α s which is equal to that of α u. Propagating a set {r i } of starting points and using all (kσ) I min we end up with a set of FP of U whose completeness can be ensured b successivel enlarging the set {r i }. Let us now specif the sstems S kσ : S kσ : r i+1 = r i +Λ kσ [ f(r i ) r i ] (1) S kσ are linear transformations of the original dnamical law U. Λ kσ are invertible constant n n matrices. The definition of S kσ and U clearl shows that their FPs are one to one and at the same positions. Eqn. (1) implies the following relation for the stabilit matrices T U and T Skσ of U and S kσ, respectivel: 2

12 T Skσ = 1+Λ kσ (T U 1) (2) In ref. [25] it was shown that Λ kσ can be cast in the form Λ k = λ C kσ with < λ < 1. The set of matrices {C kσ } contains all orthogonal matrices with onl one non-vanishing entr ±1 per row or column, i.e. the represent a group of special reflections and permutations. σ = ± indicates the sign of the matri determinant, and k is an additional label to uniquel specif the matrices. In two dimensions we have C + = C 2+ = ( 1 C 3 = ( 1 1 ) 1, C = C 2 = ( ( ) 1 1), C1+ = C 3+ = 1 1 and C 1 = ). The matrices {C kσ k =...3;σ = ±} form a group with {C k+ k =,...,3} (matrices with positive determinant) being a subgroup of order 4. Table I is the corresponding multiplication table. Obviousl, the product of two matrices is C k σ = C kσ C k σ with k = k +σ k mod 4 and σ = σσ (3) The notation introduced above is different from the one used in ref. [25] and will reveal its meaning in the course of the classification of all possible FPs (occurring in the original and transformed sstem), as provided later on. The arithmetics with respect to the first inde (k,k,k,..) has alwas to be taken modulo 4. We remark that the minimal set I min is significantl smaller than the set of pairs (kσ) belonging to the matrices {S kσ }. Given a certain unstable FP α u the above choice of linear transformations represented b the set {C kσ } of matrices allows to find a particular C k σ such that the real parts of the eigenvalues of C k σ (T U 1) are negative at the position of the FP. As a consequence (see Eqn. (2)), if λ is chosen sufficientl small, the magnitude of the real parts of the eigenvalues of the FP α s in the transformed sstem are smaller than one and we therefore encounter a stable FP which can be detected as described above. The critical value of λ, which just suffices to make the FP stable, can immediatel be read off from the quadratic equations relating the stabilit coefficients of the original sstem U and those of the transformed sstem S kσ [25]. The above procedure can easil be etended to higher iterates f (p) (r) of U (b replacing f with f (p) in Eq. (1)) allowing us to determine all order p ccles of U. The advantage of the SD method is clearl its global character in the sense that even points far from the linear neighbourhood of a FP are attracted close to the FP after a finite number of iterations of the transformed dnamical law. The basin of attraction of a single stabilised FP is a simpl connected area in phase space. The tpical number of starting points needed to obtain the UPOs of a given length on the attractor is onl slightl more than the epected number of ccle points themselves. The parameter λ is a ke quantit here. It is related to the stabilit of the desired ccle in the transformed sstem. With increasing period of the ccles, λ has to be reduced to achievestabilisation of all FPs. One ma, however, also be interested in the most stable periodic orbits of a given period p [26] which is one of the ke issues of the present work. In this contet λ operates as a filter allowing the selective stabilisation of onl those UPOs which possess Lapunov eponents smaller than a critical value. Therefore, starting the search for UPOs within a certain period p with a value λ = O(1 1 ) and graduall lowering λ we obtain the sequence of all unstable orbits of order p sorted with increasing values of their Lapunov eponents. In [26] it was shown that for the specific choice (kσ) = (+) I min the relation between λ and the stabilit coefficients of the FPs of the original sstem U is a strict monotonous one. Transformed dnamical sstems S kσ belonging to other pairs (k σ ) (+) do not obe such a strict behaviour but show a rough ordering of the sequence of stabilit eigenvalues of the FPs of U stabilised in the course of decreasing values for λ. B. Geometrical Interpretation of the Stabilising Transformations S kσ 1. Classification Scheme The stabilit properties of the FPs of the dnamical sstems S kσ have been investigated in refs. [25] eclusivel in the contet of their relation to the stabilit coefficients of the original sstem U. To gain a deeper insight into the geometrical meaning and the interpretation of the transformations S kσ which turn unstable FP into stable ones, one has to go beond the pure consideration of their eigenvalues at the positions of the FPs. In the following we develop a geometrical approach allowing us to classif the FPs which are stabilised b different matrices C kσ. We will hereb focus on sstems with two degrees of freedom. Epectedl there should, however, be no major obstacles with respect to the generalisation to arbitrar dimensions. When dealing with the stabilit transformations the natural problem arises: Restricting ourselves to the set of orthogonal C kσ matrices with eactl one non-vanishing entr (±1) in each row and column and to the linearised dnamics around a FP, what can we sa about the action of the matrices C kσ (see Eqn. (1)) on this simple dnamical sstem? To approach this problem consider the following set of equations: 3

13 ẋ = F(), F() = f (n) (), F() = (F1 (),F 2 ()) T (4) which describes a vector field around the FP located at f, where F( f ) = (the superscript T denotes the transposed). In the following sections we generall focus on the discussion of the sstem F( f ), unless noted differentl. Now we appl the transformation ẋ = F kσ () = C kσ F(), Fkσ () = (F kσ,1 (),F kσ,2 ()) T (5) It is important to note that the dnamical sstem S kσ in Eqn (1) represents a discretisation of the continuous sstem (5). Multiplication with C kσ intermingles the and coordinates of F(), which in general changes the eigenvalues and eigenvectors of the corresponding stabilit matri. One direction of the above problem is: Are there an points in the neighbourhood of f where this change of the dnamics is controllable? In fact let us consider the manifolds Z 1, Z 2 defined b [27] Z i = { F i () = }, i = 1,2 (6) In the linear neighbourhood of f these sets clearl define straight lines. In the more general case of a nonlinear sstem the are implicitl defined continuous curves in an area of phase space where the Jacobian of the map is regular. Their intersection is Z 1 Z 2 = { f }. With C kσ acting on Z 1, Z 2, the either sta the same if C kσ does not interchangethe coordinates, or the aretransferredone onto the other if C kσ does interchangethe coordinates. In this sense, the manifold Z = Z 1 Z 2 is invariant with respect to application of the set of matrices C kσ, i.e. C kσ (Z) = Z for all (kσ). In the following we derive a classification scheme for the linearised dnamics around a FP whose validit is however, due to the global character of our approach, not limited to the linear regime. In order to elucidate the action of the stabilising transformations and to distinguish between FPs with different stabilit properties, let us introduce two different was of classifing the matrices of a two dimensional dnamical sstem, each providing its own insights. The classifications are such that the reflect certain geometrical features of the flow around the FP. These features are the different invariant sets Z on the one hand and additional geometrical properties of the matrices which have a particular Z in common on the other hand. The first classification introduces classes consisting of matrices which have the manifold Z = Z 1 Z 2 in common. ThearelabelledC(φ min,φ ma ),whereφ min andφ ma aretheazimuthalanglesofthemanifoldsz 1 andz 2,respectivel, being sorted with increasing order. φ min and φ ma are related to the stabilit matri of the FP in the following wa: The linearised dnamics of eqn. (4) in the neighbourhood of a FP reads ẋ = B, where B = (a ij ) 1 i 2 is the 1 j 2 stabilit matri of F() at the FP and is the displacement with respect to the FP. It can be shown that φ min and φ ma are given b (φ min,φ ma ) = ( min {φ i },ma i i ) {φ i } ( with φ i = arctan a ) i1 a i2, i = 1,2 (7) If a stabilit matri B belongs to the class C(φ min,φ ma ) also its products {C kσ B} belong to this class. For the later on discussion we introduce here three sets of FPs each of which is an infinite unification of classes C(φ min,φ ma ) : C 1 = {C(φ min,φ ma ) < φ min,φ ma < π/2} C 2 = {C(φ min,φ ma ) < φ min < π/2 < φ ma < π} (8) C 3 = {C(φ min,φ ma ) π/2 < φ min,φ ma < π} A further classification of the matrices within each class C(φ min,φ ma ) is needed for a more detailed identification of the geometrical properties of the flu around a particular FP. To this aim, we assign a label (lτ) to each stabilit matri B lτ of FPs with the following meaning: τ = ±1 gives the sign of (detb lτ ). To illustrate the meaning of l, we write F() = (rcosψ,rsinψ) T in polar coordinates and consider the azimuthal angle ψ min of F() for = (cosφ min,sinφ min ) T. B construction, φ min can take the values mπ/2, m =,...,3. Now we define the inde l = m+τ 1 mod 4. Encircling the FP on a unit circle = (cosφ,sinφ) T, the normalised flu F()/ F() describes a circle in the local coordinate sstem centred in, too. τ gives the orientation of this circular rotation of the flu (τ = +1: anticlockwise, τ = 1: clockwise) whereas l is directl related to the phase of the flu at φ min. This naturall introduces a subset C lτ (φ min,φ ma ) of the class C(φ min,φ ma ): those members of C(φ min,φ ma ) belong to C lτ (φ min,φ ma ) which possess the indices (lτ), i.e. the sign τ for the rotation of the flu and the phase l of the flu at 4

14 φ min. We can now allow φ min,φ ma to var while keeping the indices (lτ) of the matrices of this set fied. For each (lτ) we thereb form a class A lτ = {C lτ (φ min,φ ma ) φ min,φ ma 2π}. For fied φ min,φ ma, multiplication b the matrices C kσ transfers one complete set A lτ into another set A l τ. The corresponding transitions are given in table II. The asterisks in the first three columns of table II indicate the sets C i and A lτ to which stabilit matrices of a FP of a two dimensional chaotic sstem can belong. This is of relevance when asking for the possible sets of matrices C kσ which stabilise all FPs of a dnamical sstem (see below). As one can read off table II, the matri C kσ necessar to transfer stabilit matrices of a class A lτ into matrices in another class A l τ is given b C kσ : A lτ A l τ with l = k +σ mod 4 and τ = σ/τ (9) It is important to note that table II holds not onl for the unified sets A lτ but particularl also for the individual subsets C lτ (φ min,φ ma ). It is a remarkable result that the multiplication table of the matrices C kσ, i.e. table I, considered as a transition table, has the same entries as the transition table II. The multiplication law eqn. (3) becomes a law for the transition of C kσ to C k σ via C k σ with the corresponding indices k = k σσ k and σ = σσ. This suggests that the matrices in a class A lτ are in a wa similar to the matrices C kσ with k = l and σ = τ as far as the rules for multiplication with the matrices C kσ are concerned. The class C lτ (φ min,φ ma ) contains still an infinite number of (stabilit) matrices. However, to gain relevant information on the stabilit properties of FPs it suffices, as we shall see in the following, to know to which of the sets {C i i = 1...3} and additionall A lτ the stabilit matri of the FP corresponds. 2. Properties of the Angular Functions ψ(φ) of the Flu and Eamples Let us consider an arbitrar but fied stabilit matri B lτ C(φ min,φ ma ), B lτ A lτ, and form the matrices C kσ B lτ, k =,...,3;σ = ±1. Each of these matrices is element i.e representative of a different class A l τ and we therefore have B l τ = C kσ B lτ with l = k +σl mod 4 and τ = σ/τ (1) In the following we will call the set {B lτ l =,...,3;τ = ±1} the famil of the matri B lτ. A central issue is to analse which members of the famil of a FP are stable. Once the information on the stabilit of the members of a famil becomes available this represents an important step towards the selective use of eqn. (1), i.e. the selective detection of FPs. In the following we stud the orientational properties of the flu for a famil of stabilit matrices with fied but arbitrar (φ min,φ ma ). For the matrices B lτ = (b (lτ) ij ) 1 i,j 2 A lτ we introduce the angular functions ψ lτ (φ) of the flu ẋ = B lτ at a point = (cosφ,sinφ) T ψ lτ (φ) = arctan (B ( lτ ) 2 3 +π (B lτ ) 1 ( b (lτ) 21 cosφ+b(lτ) 22 sinφ = arctan b (lτ) 11 cosφ+b(lτ) 12 sinφ 2 sign(b lτ ) 2 ) +π ) ( 3 ( ) ) 2 sign b (lτ) 21 cosφ+b(lτ) 22 sinφ ψ lτ (φ) is the azimuthal angle of the flu (given b F lτ ()) with respect to a local polar coordinate sstem centred at the displacement = (cosφ,sinφ) T. This is illustrated in Fig. 1. Let us derive relevant properties of the angular flu functions ψ lτ (φ). ψ lτ (φ) is a continuous function of φ with ψ lτ (φ) [,2π]. Due to smmetr reasons it is sufficient to consider the range < φ < π. Furthermore ψ lτ (φ) is defined mod 2π and ψ lτ () ψ lτ (π) = π. Its derivative reads with L lτ (φ) = ẋ = (11) ψ lτ(φ) = det(b lτ) L 2 lτ (φ) (12) (b (lτ) 11 cosφ+b(lτ) 12 sinφ)2 +(b (lτ) 21 cosφ+b(lτ) 22 sinφ)2 (13) being the velocit of the flu in. Since multiplication with an C kσ affects complete rows of B lτ b interchanging them or inverting their signs, L 2 lτ (φ) is invariant, i.e. the same for an of the resulting matrices B lτ. Two functions ψ lτ (φ) and ψ l τ(φ) therefore differ onl b a shift: ψ lτ (φ) ψ l τ(φ) = (l l mod 4) π 2. l can be interpreted (see also subsection 2.2.1) as the phase of the linearised flu around the FP: 5

15 ψ lτ (φ min ) = (l+1 τ) π 2 Accordingl we have det(c kσ B) = det(c kσ ) det(b), with det(c k+ ) = +1 and det(c k ) = 1 and therefore (14) ψ lτ(φ) = ψ l τ(φ) (15) which ields ψ lτ (φ) = 2 ψ lτ (φ min ) ψ l τ (φ) (16) τ = ±1 indicates the sign of det(b lτ ) and determines whether ψ lτ (φ) is rising or falling, according to eqn. (12). In the remaining part of this subsection we provide generic eamples and illustrations of the linearised dnamics around a FP and the corresponding angular functions of the flu for the different members of a famil {B lτ }. ) 4 The stabilit matrices B lτ in Figs. 2a) and Fig. 3a) are members of a famil of the matri M 1 = 5, corresponding to the class C(.24,1.1). Figs. 4a) and 5a) are obtained in the same wa, showing the famil of M 2 = ( ) , which is in the class C(.24,2.3). Each subfigure shows the linear neighbourhood of the corresponding FP. (,) are the coordinates of the displacement of a trajector with respect to the FP. The manifolds Z 1, Z 2 are displaed as long dashed lines with the direction of the flu on the Z 1, Z 2 lines being indicated with open arrows. The thick lines with filled arrows show the position and stabilit properties of the eigenvectors (For saddle points, v e, v c are indicated b outward and inward looking arrows corresponding to the epanding and contracting manifolds, respectivel. For sinks and sources the corresponding eigenvectors are labelled v e,1, v e,2 and v c,1, v c,2, respectivel). Additionall, some trajectories have been included to visualise the direction of the flow around the FP. In Figs. 2b) 5b) the corresponding angular functions ψ lτ (φ) are plotted. It is important to note that these particular functions as well as the phase portraits given in the corresponding figures are merel eamples to demonstrate the qualitative variation of the dnamics generated b members of a famil of stabilit matrices. There eist in general other families in the same class C(φ min,φ ma ) A lτ whose phase portraits and functions ψ lτ (φ) look different from those displaed in the figures. However, the functions ψ lτ (φ) of all these matrices in C(φ min,φ ma ) A lτ have the values of ψ lτ (φ) in φ = φ min and φ = φ ma and the sign of ψ (φ) lτ in common. The actual ψ lτ (φ) can var in between according to the definition (11). Nevertheless the above information about the stabilit matrices is in fact sufficient to determine their stabilit properties. Now, looking at the eamples in Figs. 2a) 5a) ecept the cases of spiral points B 1+ and B 3+, we are alreadin the position to suggest a criterion for the stabilit and the approimate position of the eigenvectors of the FP. Consider the direction of the flu in each of the sectors which are determined b the manifolds Z 1, Z 2 and the coordinate aes. The manifolds Z 1, Z 2 are b definition the lines for which the first, respectivel second component of the flu vanishes. Since the flu F() is a continuous vector function of the angle φ there have to be certain values φ e for which the flu F() is collinear with the position vector = (cosφ,sinφ) T, i.e. F() = ±c, c >. These angles φe obviousl are the polar angles of the eigenvectors of the corresponding stabilit matri. The intervals where these angles φ e are located are bounded b Z 1 and Z 2. The manifolds Z 1 and Z 2 represent the boundaries of sectors in which the angle of the flu varies b π/2. For reasons of continuit of ψ lτ (φ) ( see Figs. 2a) 5a)) one or even two eigenvectors are located within these sectors, which are shaded gre in the figures. In specific cases parts of the sectors defined b Z 1 and Z 2 are ecluded for the localisation of the eigenvectors since collinearit of and F() is not possible. In these cases the outer coordinate sstem (, ) represents the boundar of the sectors for the occurrence of collinearit. The fact that φ e is the polar angle of an eigenvector of the stabilit matri A is equivalent to ( 1 1 ψ lτ (φ e ) = { φe : Re(λ) >, unstable eigenvector φ e +π : Re(λ) <, stable eigenvector (17) This means that a crossing of ψ lτ (φ) with χ n (φ) = φ+n π, n =,1 (18) indicates an unstable or stable eigenvector for the corresponding value of φ, respectivel. 3. Stabilit Properties of the Classes and the Alternative Sets of Stabilisation Transformations We will show in the following that for fied φ min and φ ma two complete classes A l τ and A l,τ are related to stable FPs, i.e. the matrices in these classes have eigenvalues λ 1, λ 2 for which Re(λ 1 ),Re(λ 2 ) < holds. To these 6

16 two classes correspond two matrices C k σ and C k σ which transfer an original stabilit matri in a class A lτ into the desired stable matrices in A l τ and A l τ. This corresponds to a transformation of the original FP into the desired stable ones in the transformed dnamical sstems S k σ and S k σ. The two matrices C k σ and C k σ which accomplish this transformation can be read off the transition table II immediatel. It will in particular become evident that a minimal set of three C kσ matrices is sufficient to stabilise an FP of a two dimensional full chaotic sstem. What is more, this classification proves to be useful not onl for saddle points but also for sstems with repellors and spiral points, which can be transformed to stable FPs (sinks) via a certain matri C kσ (see below). We now discuss the properties of all possible stabilit matrices according to their ψ lτ (φ) diagrams. To do this it is sufficient to consider the assignment of the matrices to the classes C 1, C 2 and C 3 as introduced in (8) and to the classes A lτ. In the following we concentrate on the classes C 1 and C 2 since the argumentation for the class C 3 is analogous to that for C 1. For the same reason we restricted the eamples in Figs. 2 5 to the classes C 1 and C 2. In Figs. 2c) 5c) we show the areas in the ψ lτ (φ) diagrams where a crossing of ψ lτ (φ) and χ (φ) or χ 1 (φ) ma occur as gra shaded boes. The are derived b simple application of continuit arguments concerning ψ lτ (φ). The labels lτ in the boes are the labels of the corresponding class A lτ of matrices whose real eigenvectors have azimuthal angles in this range of φ. Two labels given in brackets indicate the possibilit of either two real eigenvalues with eigenvectors in this range (sink or source) or comple eigenvalues without real eigenvectors of the corresponding matri (spiral points). These two possibilities cannot be distinguished within our classification scheme of matrices. However, this fact does not affect the final selection of the minimal stabilising set of matrices C kσ. We begin our discussion with stabilit matrices B with negative determinants: Matrices B {A l l =...3} are stabilit matrices of saddle points. The stabilit properties of these FPs are eas to determine from the corresponding ψ lτ (φ) diagrams in Figs. 3b), 5b). It is obvious from the monotonicit and continuit of the ψ lτ (φ) curve that the intersect the unstable and the stable lines χ (φ), χ 1 (φ), respectivel, eactl once. The sectors where the corresponding eigenvectors are localised are shaded gre in the corresponding phase diagrams Figs. 3c), 5c). FPs with positive Jacobian B are a bit more subtle matter. The belong to the classes {A l+ l =...3} and include sinks and sources as well as spiral points for which the real part of the stabilit eigenvalues Re(λ 1 ) Re(λ 2 ) possess the same signs. Spiral FPs possess stabilit matrices B with eigenvalues Im(λ 1 ),Im(λ 2 ). This implies det(b) >. As pointed out earlier the classification of the stabilit matrices with C(φ min,φ ma ) and A lτ does neither specif the matri nor its eigenvalues completel. In particular for stabilit matrices whose famil contains spiral points an additional criterion is needed to analse their stabilit since the corresponding ψ lτ (φ) functions do not cross the lines χ (φ) or χ 1 (φ). For this analsis we suggest the following criterion (whose completeness we could not prove et). Consider the angles φ t, t = 1,2, for which ψ lτ (φ t) = 1, i.e. ψ lτ (φ) is tangential to χ n (φ), and define the distance to both lines χ n (φ) Now the stabilit of the spiral point is suggested b d n = min t=1,2 ψ lτ(φ t ) φ t nπ, n =,1 (19) d < d 1 = Re(λ e ) >, unstable spiral point (2) d > d 1 = Re(λ e ) <, stable spiral point (21) This implies that for spiral FPs the line χ n (φ) which is closest to ψ lτ (φ t ) determines the stabilit and this can be seen as a generalisation of the criterion of the crossing with χ n (φ) in the case of real eigenvalues. It is an interesting propert that the stabilit of matrices B l+ of a famil {B lτ l =...3,τ = ±} changes with l l + 2 mod 4, i.e. if B 1+ is stable, then B 3+ is unstable, if B 2+ is stable, then B 4+ is unstable and vice versa. If the eigenvalues are real the eigenvectors are not affected b the shift (onl their stabilit changes). This propert is obvious taking into account that the shift in l corresponds to a shift of ψ lτ (φ) b π, with π being also the shift between the lines χ (φ), χ 1 (φ). Phsicall, the shift b π is for an stabilit matri equivalent to a time reversal (which corresponds to a reversal of the flu F() F(). We now come back to the discussion of the stabilit properties of matrices B possessing positive determinant: Matrices B {A l+ l =...3} describe sinks and sources and spiral points. We first address the sinks and sources and second the spiral points. The corresponding curve ψ lτ (φ) crosses one of the lines χ (φ) (sink) or χ 1 (φ) (source) twice. It is obvious but nevertheless important to note that there cannot be more than two crossings, which can also formall be seen from eq. (12). Having three or more cuts with χ n (φ), n =,1 implies that ψ lτ (φ) has at least two turning points in an interval [φ d,φ u ] with φ d φ u = π/2. This means that L 2 lτ (φ) 7

17 has two or more etrema in [φ d,φ u ]. Differentiating ( ) L 2 lτ (φ) (see eqn. (13)) ields (for the sake of simplicit we omit the superscripts (lτ) in the entries of B) b (lτ) ij 1 i,j 2 d L 2 lτ (φ) d φ = (b b 2 11)2cosφsinφ+(b b 2 12)2sinφcosφ+ 2(cos 2 φ sin 2 φ)(b 21 b 22 +b 11 b 12 ) = (b b 2 12 b 2 21 b 2 11)sin(2φ)+ 2(cos 2 φ sin 2 φ)(b 21 b 22 +b 11 b 12 )cos(2φ) (22) d L 2 lτ (φ) d φ b 21 b 22 +b 11 b 12 = tan(2φ) = b b2 12 b2 21 b2 11 (23) which has eactl one solution for φ in an interval [φ d,φ u ] with φ d φ u = π/2, i.e. there is onl one turning point in this interval. The class C 1 : As can be read off directl from the diagrams Figs. 2b) and c), matrices in C 1 A 2+ and C 1 A 2 are sinks and sources, respectivel. One of the eigenvectors is in the interval [φ min,φ ma ], the other one is located in [π/2,π]. Matrices of the class C 1 A 1+ and C 1 A 3+ are either sinks and sources or spiral points. For real eigenvalues matrices of C 1 A 1+ are sinks whereas matrices in C 1 A 3+ are sources. The orientation of the eigenvectors of both stabilit matrices are within the interval [φ ma,π/2]. Consideringthelattercaseofspiralpoints, wecanatleastsathatwithinonefamil{b lτ l =...3,τ = ±1} of matrices either B 1+ is a stable spiral and B 3+ is an unstable spiral point or vice versa. The class C 2 : Here more cases are possible. Looking at the matrices {B l+ l =...3} of the famil {B lτ l =...3,τ = ±} there can be either no spiral point, but two sinks and two sources (analogous to C 1 ) two spiral points (one stable, one unstable), one sink and one source (analogous to in C 1 ) four spiral points (two stable, two unstable) Which of these cases occur is a question of the variation of ψ lτ (φ) φ and of the phase ψ lτ (φ min ). If real eigenvalues occur, the angles of the corresponding eigenvectors are within [π/2,φ ma ] for matrices in A 1+ and A 3+. The position eigenvectorsof stabilit matrices matrices in A 1+ and A 3+ is not determined within this classification scheme. To determine the minimal sets S i of matrices C kσ necessar for stabilisation of all FPs of a two dimensional chaotic dnamical sstem let us first consider the classesa lτ C i whose representativescorrespondto stable FPs. We herefore form pairs (lτ,l τ ) abbreviating the two classes A lτ and A l τ for an Ci. C 1 : (1+,2+) or (2+,3+) C 2 : (1+,2+) or (2+,3+) or (3+,+) (24) C 3 : (1+,2+) or (2+,3+) When looking for the minimal set of matrices C kσ necessar for stabilisation one has to take into account that onl certain kinds of FPs can occur in the sstem ẋ = F(), eqn. (4), derived from the original dnamical sstem i+1 = f( i ). In two dimensions f() can have saddle points onl and the FPs of F() are therefore either saddle points too or sinks. Let us discuss the set S saddle of matrices C kσ which stabilise saddle points first. Since the determinant of the stabilit matri is negative for a saddle point and positive for an stable sink or spiral point, the corresponding stabilising matri C kσ has the form C k. Since the saddle points of all classes C 1, C 2 and C 3 have to be stabilised b S we have to determine S such that an class A l of original matrices is transferred into at least one element in each of the pairs (1+,2+) or (2+,3+) or (3+,+), which is the union of the pairs of all three classes C 1, C 2 and C 3 in eqn. (24). The transition table II shows that there are just two possibilities of minimal sets: S saddle = {C,C 2 } or {C 1,C 3 }. Each of these sets has to be combined with sets S sink that stabilise the sinks of ẋ = F() to ield a possible set S. Since the latter are alread stable, the identit transformation S sink = {C + } is sufficient. Indeed, it is eas to see that no other C kσ is able to achieve the same: Sinks can occur in both C 1 and C 3 for the classes A 1+ 8

18 and A 2+ and in C 2 for the classes A 2+ and A 3+. We therefore can list these different classes of sinks and sets S sink of matrices C kσ necessar to stabilise them as follows: original classes stable classes sets of stabilising C kσ C 1 A 1+ : (1+,2+) {C + } or {C 1+ } C 1 A 2+ : (1+,2+) or (2+,3+) {C + } or {C 1+,C 3+ } C 2 A 2+ : (1+,2+) or (2+,3+) {C + } or {C 1+,C 3+ } C 2 A 3+ : (2+,3+) or (+,3+) {C + } or {C 1+,C 3+ } C 3 A 1+ : (1+,2+) {C + } or {C 1+ } C 3 A 2+ : (1+,2+) or (2+,3+) {C + } or {C 1+,C 3+ } (25) So the smallest set that stabilises an sink independent of the classes C 1, C 2 and C 3 is S sink = {C + }. If we now form the unions S saddle S sink to get a set of matrices C kσ that stabilise all FPs of a two dimensional full chaotic dnamical sstem we end up with two global minimal sets: S 1 = {C +,C,C 2 } and S 2 = {C +,C 1,C 3 } (26) There are other sets which also do the job, but the contain at least four matrices and are therefore not minimal. In previous numerical investigations it was observed [25,26] that the transformation belonging to the matri C + ields a particularl large number of stabilized fied points. From the above discussion this becomes now understandable since C + is responsible for the stabilization of the sinks in equation (5) which occur in man different classes. Let us remark that the above discussion includes also sinks of the original sstem f since the become sinks of F and can therefore be treated in the same wa, i.e. are stabilised using the same minimal sets S 1 or S 2. The concept of characterising FPs b sets of manifolds which are invariant with respect to the operations C kσ can be etended in a natural wa. One can consider manifolds which are composed b not onl two subsets Z 1 and Z 2 as above, but of four subsets Z 1...Z 4. These subsets are defined implicitl b demanding a given ratio of the two components of the flu. The correspond for the linear regime to four angles {φ 1,φ 2,φ 3,φ 4 }, which are a generalisation of the parameters {φ min,φ ma } in the discussion above. This larger set of parameters leads to a finer partition of the space of all 2 2 matrices, which in turn allows a complete assignment of stabilit properties to the different matrices. The definition of the famil of a given matri is analogous to the corresponding definition (1) and reflects the action of the group of matrices {C kσ }. However, finding the proper partition is not straightforward and is left to a future investigation. III. APPLICATIONS In this section we appl the SD method whose theoretical background has been discussed so far. Furthermore we will discuss several improvements of its original algorithmic implementation to locate the UPOs [25,26] With respect to the detection of the orbits our aim is twofold. First we are interested in complete sets of UPOs for higher periods of the above maps and in analsing them with respect to the distributions of the corresponding Lapunov eponents. Second we want to demonstrate the suitabilit of our method to detect the least unstable orbits up to high periods. The latter is an etension of the work given in ref. [26]. We remark that ver recentl an efficient algorithm for detecting UPOs in chaotic sstems based on a combination of the SD method and a Newton-Raphson like approach has been developed and successfull applied [29]. A. Finding the Fied Points Concerning the efficient algorithmic implementations of the SD method we face two main problems: 1) The completeness of the detected set of UPOs which is of conceptual character. 2) Separating closel neighboured UPOs which is an issue onl for our particular implementation of the SD method, i.e. characteristic for our numerical approach. Let us first address the completeness problem. Of course there is no eact proof of completeness for the detected UPOs within the SD method. However, a properl chosen sequence of sets of initial conditions which cover the phase space of the dnamical sstem as neat as possible can significantl lower the probabilit of missing an UPOs. Due 9

19 to the presence of length scales which differ b man orders of magnitude induced b the fractal structure of the corresponding strange attractor we proceed here as follows. We introduce a set of grids G i, i=1,2,... of starting points which are cumulative in the wa that the points of G i fill gaps on the attractor that are larger than a given size in the union of the preceding grids G 1 G 2... G i 1. In the particular case of the Ikeda map (see below) we generate a sequence of si grids G 1,...,G 6 with G 1 containing approimatel 45 points, while G i, i = 2,...,6 contain approimatel 15 points each. The starting points of each set G i are propagated with the transformed maps S +, S, and S 2, i.e. appling the matrices C +, C and C 2 according to (1). The propagation of a particular trajector is stopped at the point if δ = f (p) () < ǫ, where ǫ is the desired resolution of the FPs. For periodic orbits of the Ikeda map with p = 14 and 15, ǫ < 1 1 turned out to be sufficient to resolve the different ccles. Propagating the starting points of the grids G i with S +, S and S 2 ields the sets N i,+, N i, and N i,2, i = 1,...,6, respectivel, of points which to good accurac approimate the FPs of the Ikeda map. The set N i = N i,+ N i, N i,2 contains the FPs of the map found b propagating the points of G i. Then we consider the number n i,kσ of FPs that appear in a particular N i,kσ (and therefore in N i ) with (kσ) = (+), ( ), (2 ), but which are not contained in an other N j, j < i. If n i 1,kσ = n i,kσ =, i.e. no additional FPs have been found when propagating two subsequent grids G i 1, G i, the set of FPs of the transformed sstem S kσ is considered to be complete and the procedure of constructing subsequent grids is stopped. The second improvement concerns the separation of neighbouring UPOs. Using however an appropriatel defined distance d between two orbits of period p, i = ( 1i, 2i ) T, i = ( 1i, 2i ) T, i = 1,...,p: d = min k=,......,p 1 ( ji j(i+k mod p) ) 2 (27) 1 i p 1 j 2 In the following we consider a set of FPs that belong to different periodic orbits. The FPs obtained in the procedure of propagating the transformed sstems S +, S and S 2 as described above provide an eample for such a set. B forming all possible pairs of an two FPs of this set and looking at the resulting distribution of values of d (e.g. b plotting all d in a logarithmic scale) one can distinguish three subsets, separated b gaps differing b several orders of magnitude (Fig. 6). The set with the largest values of d contains all pairs formed with different orbits, whereas the second and third largest set is composed of pairs of identical orbits. A possible eplanation for the appearance of the gap between the second and the third zone is the following: A trajector of the transformed map approaches the FP on a curve which in the local but still nonlinear neighbourhood is close to the least stable of both stable manifolds of the FP. Two trajectories { i } and {z i } can therefore evolve towards a FP o along the same or opposite directions. The same holds for all other FPs of the orbit f (r) ( o ) with the trajectories { f (r) ( i )} and { f (r) (z i )}, r = 1,...,p 1. In the case of an antiparallel approach to the FPs of a given ccle the contributions to d are much larger compared to the case of a parallel approach, which finall ields the gap between the second (parallel approach ) and third (antiparallel approach) region in Fig. 6. If the first and the second zone happen to merge, the accurac of the FPs is not sufficient to distinguish different orbits and has to be refined b further propagation with the corresponding transformed maps. What is more, this distinction can be used to derive an estimate for the absolute accurac of the FPs derived b propagating the appropriatel transformed sstem: The maimal separation of two points belonging to the same ccle is given b the square root ofthe value of d at the upper edge of the lowestband in the distribution of values of d. With the above method we selected eactl one point of each periodic orbit we found. The other points areobtained b simpl propagating the selected point with the original map. B. Complete Sets of Orbits and Lapunov Distributions In the following we first focus on the Ikeda map [28]. It is given b n+1 = α+β( n cosw n n sinw n ), n+1 = δ β( n sinw n + n cosw n ), where w n = γ 1+. The attractor to be investigated appears for α = 1., β =.9, 2 n +2 n γ =.4 and δ = 6.. For the Ikeda map the periods p = 1,2,...,13 have alread been investigated in ref. [25]. In order to indicate the applicabilit of the previousl discussed algorithmic implementation of the SD method beond those periods we calculate complete sets of orbits for p = 14,15. A rough hint towards the completeness of the result is the convergence of the corresponding topological entrop h = lim p h p with h p = lnn(p)/p, n(p) being the number of FPs of all ccles of period p (see table III). h seems to be converged fairl well to a constant value. Our sets of orbits for the Ikeda map, together with the corresponding results for the Hénon map, allow us to stud the distributions of the Lapunov eponents of the UPOs for the two different sstems (see below). 1

20 Before entering into this discussion we comment on some observations made b appling the SD method to the Ikeda map. As eplained in Sec. 2.2, each point of an UPO is stabilised b two matrices, which both have either positive or negative determinant. Therefore, we can group the points of one UPO of period p into the sets S + (p) : sinks, stabilised b either C +, C 1+, C 2+, or C 3+ (28) S (p) : saddles, stabilised b either C, C 1, C 2, or C 3 (29) We found the surprising result that for all periods p = 1 14 both sets S + (p) and S (p) contain the same number of points, S + (p) and S (p), respectivel. The values are S ± (p) = 1,2,4,8,11,26,36,64,121,242,419,692,1262,2256 for p = Onl for p = 15 we observe a deviation from this fact: S + (15) = 449, S (15) = At the current state we cannot judge whether this indicates missing orbits or represents a violation of the observed rule for p = If we assume the validit of the equalit S + = S as a general law, this implies a smmetr relation between the orbits of the map: Each orbit with a positive determinant of its stabilit matri is related to eactl one orbit with negative determinant. In section 3, we will seek for the most stable orbits of the Ikeda map for higher periods and we will find a similar pairing of orbits, i.e. two orbits, one in S + and one in S, have nearl the same Lapunov eponent. This might suggest a smmetr of the underling Ikeda map. For all detected UPOs we calculate the Lapunov eponent Λ orb = log( ρ )/p, where ρ is the largest eigenvalue of the matri M = M p... M 2 M 1. Fig. 7 shows the normalised distributions D(Λ) of Lapunov eponents Λ of all orbits of order p = 24,...,27 for the Hénon map [22] given b n+1 = n +.3 n, n+1 = n, and for p = 12,...,15 of the Ikeda map (primitive orbits onl). For both maps the Lapunov eponents form a band and the distributions show a more and more pronounced peak as p increases. The peak of the distribution appears around Λ =.5 for the Hénon and around Λ =.68 for the Ikeda map. Around this peak, the distribution is epected generall to be well approimated b a Gaussian [12], [3]. However, globall the distributions of both maps clearl deviate from Gaussian behaviour in some respects: Unlike smmetric distributions, both distributions ehibit an enhancement for small values of the Lapunov eponent. This indicates that there are correlations within these orbits. A second feature is the occurrence of peaks in the main bulk of the distribution. A characteristic peak of Hénon map appears at Λ =.551 and a similar spike is visible for low values at Λ =.435. Probabl the Lapunov distribution for the ccles of the Ikeda map displas similar features, though the number of ccles for period p = 15 is not enough to allow for a sufficient resolution of the distribution. These features show that the Lapunov distributions contain interesting information on the sstems dnamics. C. Stabilit Ordering of Ccles Since the SD method is, b construction, changing the stabilit properties of a dnamical sstem, it is not surprising that the value of the tuning parameter λ is of relevance to the magnitude of the Lapunov eponents Λ (j) orb to be detected. The critical parameter λ kσ,i of a point r i of a ccle stabilised with C kσ is defined as the largest value of λ for which both eigenvalues of the transformed stabilit matri T Skσ in eqn. (2) have an absolute value less than unit, which marks the transition from instabilit to stabilit. As introduced in ref. [26], an approimatel monotonous relation between Λ (j) orb and the critical value λ(j) kσ,i can be observed. For the orbits of the Hénon map, which are stabilised b a specific C kσ matri, this monotonous relation can be clearl seen. However, when eamining other maps, e.g. the Ikeda map, one finds that this relation is obeed less strictl. However, a slightl different concept of ordering does the job: We consider all points r (j) i, i = 1..p of an orbit j of given period p, which in general are stabilised b different C kσ -matrices with different critical λ (j) kσ,i values. Incontrasttothe approachchosenin[26]wenowallowalleightc kσ -matricestobe usedasstabilisingtransformations. As eplained in section 2.2, each point of the orbit j is stabilised b two matrices C kσ, C k σ, with a particular λ(j) kσ,i. To each orbit of period p there belongs a set of 2, 3 or 4 C kσ -matrices stabilising different ccle points and a set of 2 n values λ (j) kσ,i. Now we ask for the largest λ c out of this set and call it the λ (j) orb of the corresponding orbit j. The corresponding plot for the Ikeda map is presented in Fig. 8 and shows a sufficient ordering of the stabilit coefficients of the detected UPOs with respect to the corresponding critical values λ (j) orb. The area in each subfigure which is shaded gra indicates the approimate position of the points Λ (j) orb (λ(j) orb ) for period p = 15. Obviousl the distributions become increasingl flatter with increasing p, which corresponds to a better stabilit ordering of the respective periodic orbits. This allows us to calculate the least unstable orbits of a map in a sstematic wa. We 11