Orbital carriers and inheritance in discrete-time quadratic dynamics
AAugust 5, 2020 0:18 WSPC/INSTRUCTION FILE inheritance˙arxiv-2020
International Journal of Modern Physics Cc (cid:13)
World Scientific Publishing Company
Orbital carriers and inheritance in discrete-time quadratic dynamics
Jason A.C. Gallas
Instituto de Altos Estudos da Para´ıba, Rua Silvino Lopes 419-2502,58039-190 Jo˜ao Pessoa, Brazil,Complexity Sciences Center, 9225 Collins Ave. Suite 1208, Surfside FL 33154, USA,Max-Planck-Institut f¨ur Physik komplexer Systeme, 01187 Dresden, [email protected]
Received 2 March 2020Accepted 24 March 2020Published 22June 2020https://doi.org/10.1142/S0129183120501004Explicit formulas for orbital carriers of periods 4, 5, and 6 are reported for discrete-time quadratic dy-namics. A systematic investigation of orbital inheritance for periods as high as k ≤
12 is also reported.Inheritance means that unknown orbits may be obtained by nonlinear transformations of known orbits.Such nested orbit within orbit stratification shows orbits not to be necessarily independent of eachother as generally assumed. Orbital stratification is potentially significant to rearrange trajectoriessums in trace formulas underlying modern semiclassical interpretations of atomic physics spectra. Thestratification seems to dominate as the orbital period grows.
Keywords : Orbital carriers; Orbital inheritance; Quadratic dynamics; Symbolic computation.PACS Nos.: 02.70.Wz, 02.10.De, 03.65.Fd
1. Introduction
Applied problems in physics normally require solving equations of motion, frequently expressedeither as differential equations involving continuous-time derivatives, or discrete-time maps.Since the advent of modern computers, solving equations of motion essentially boils down tonumber crunching using special-purpose numerical methods. For a representative selection ofmethods and applications see, e.g., Refs. , , .Numerical methods revealed much of what is presently known about the time-evolutionof complex systems. However, there are certain peculiarities that are totally out of reach toapproximate numerical methods and that have not yet been addressed as they could. Forinstance, consider cascades of periodic motions, which are among the most prominent featuresfound in systems governed by differential equations or by maps. Although such cascades cannotbe followed analytically for differential equations, they are accessible in systems governed bymaps with algebraic equations of motion, particularly in one-dimensional dissipative maps , , .Consider a popular class of models, namely one-dimensional maps governed by algebraicequations of motion. To delimit analytically their stability windows one needs to solve polyno-mials containing physical parameters. Although parameters may vary freely, conditions imposedon the stability boundaries greatly reduce such freedom as well as the complexity of the nu-merical values defining boundaries. For example, for the paradigmatic quadratic and H´enonmaps, the polynomial coefficients at intersections are simply given by integers or by algebraicnumbers , , , , .The self-similar regularities recorded for cascades of periodic motions in parameterized maps a r X i v : . [ n li n . C D ] A ug ugust 5, 2020 0:18 WSPC/INSTRUCTION FILE inheritance˙arxiv-2020 Jason A.C. Gallas pose a natural question regarding the generic arithmetic nature of the numbers delimitingadjacent windows of stability as parameters are varied. Knowledge of the arithmetical unfoldingof such cascades should provide insight into the analytical clockwork mechanism underlying thisforever repeating process. Such information cannot be inferred from approximate computationsbut could be eventually won using exact algebraic analysis.The purpose of this paper is to report a systematic investigation of orbital carriers and orbital inheritance in discrete-time quadratic dynamics, in the so-called partition generatinglimit , , whose equation of motion is x t +1 = 2 − x t . More specifically, we extend previous work to include carriers for orbits with periods 4, 5 and 6, and inheritance for periods k ≤
12 of themap. Such computations are quite strenuous. The latter limit is set by the capability of thehardware and software at our disposal to generate and to factor large polynomials of degrees noless than 4020, with exceedingly large numerical coefficients and discriminants. As discussed andillustrated below, carriers are polynomials encoding simultaneously all possible orbits of a givenperiod . Inheritance means that known periodic orbits reveal unknown orbits. New orbits areobtained through simple nonlinear transformations from known orbits , , . Inherited orbits are clones that share an arithmetic ancestry. Arithmetic interdependencies among periodic orbitsare hard, not to say impossible, to recognize in numerical simulations, where only approximatenumbers are considered.The starting point to investigate the arithmetic nature of equations of motion is the ring Z of integers, namely solving polynomials with integer coefficients. Key properties which facilitatethe study of polynomials with integer coefficients are the Euclidean algorithm and the uniquefactorization of integers (the ‘fundamental theorem of arithmetic’). Such properties no longeralways hold for rings of integers of higher algebraic number fields, involving polynomials with agood deal more complicated coefficients and which are the framework where algebraic equationsof motions must be considered.The first coherent discussion of complex integers a + ib with rational integral a and b waspresented by Gauss as far back as 1831-32, in his second paper on biquadratic reciprocity.Subsequently, the theory of quadratic algebraic numbers was essentially completed during thenineteen century by Kummer, Dirichlet, Dedekind, Hilbert and others . However, the corre-sponding knowledge regarding numbers as simple as cubics and relative cubics is by far lesscomplete, despite more than two centuries of work . The main difficulty comes from the well-known fact that irreducible cubics with three real roots, the so-called casus irreducibilis , cannothave their roots expressed in terms of real radicals. The equations of motion discussed here areattractive in that they require investigating towers of such cubic fields. We consider periods k ≤
12, and provide explicit solutions for polynomials of degrees as high as 18 and 24, involvingnested cubic roots.With respect to applications beyond the scope of dynamical systems, we mention briefly thatthe concept of inheritance is potentially attractive for atomic physics, where it seems to implythe interesting and unsuspected possibility of rearranging certain orbit-dependent contributionsin cycle expansions and semiclassical sums needed for calculating energy spectra and densityof states using, e.g., Gutzwiller’s trace formula , , , , , , , .
2. Orbital carriers for periods 4, 5 and 6
A recent work has shown that classical equations of motion of algebraic origin may be allconveniently extracted from just a single mathematical object, a polynomial called an orbitalcarrier . All possible orbits may be encoded simultaneously by a single carrier, with individualorbits parameterized by σ , the sum of their orbital points . In Ref. , such parameterization wasugust 5, 2020 0:18 WSPC/INSTRUCTION FILE inheritance˙arxiv-2020 Carriers and inheritance in quadratic dynamics established for period-three orbits using standard textbook knowledge of the theory of algebraicequations. Essentially, one uses certain functions of the roots of the equation of motion, the elementary symmetric functions , which may be expressed in a general manner by means ofthe coefficients of the equation of motion, without the equation itself being resolved. This factshifts the traditional study of orbital points to a new level, to the study of orbital equations ofmotion.Here, we extend the aforementioned orbital parameterization to include explicit expressionsfor carriers of periods 4, 5 and 6. Results for periods four and six may be obtained asparticular cases of general expressions obtained previously for the two-parameter H´enon map,( x, y ) (cid:55)→ ( a − x + by, x ). For arbitrary values of a , carriers for the quadratic map x t +1 = a − x t are obtained setting b = 0 in the expressions of the H´enon map. For the partition generatinglimit discussed here, set ( a, b ) = (2 , The period four carrier
Essentially, for a given period k , all period- k orbits may be encoded simultaneously by twopolynomials, as described in a recent open access paper : A σ -parameterized polynomial ψ k ( x ),called the carrier, and an auxiliary polynomial, S k ( σ ), which fixes the values of the parameter σ for each individual orbit. The parameter σ is just the sum of the orbital points. The degreeof the polynomial S k ( σ ) informs the total number of possible k -periodic orbits in the system.When substituted into ψ k ( x ), each individual root of S k ( σ ) = 0 “projects” ψ k ( x ) into the σ -selected individual orbit. Normally, S k ( σ ) is a reducible polynomial over the integers: nonlinearfactors of degree ∂ k correspond to orbital clusters , namely to irreducible polynomial aggregatescommingling together a total of ∂ k orbits. Linear factors correspond to non-clustered single orbits of degree k .For period-four there are three possible orbits, all encoded simultaneously by the doublet: ψ ( x ) = x − σx + ( σ + σ − x − ( σ + 3 σ − σ + 2) x + ( σ − σ + 9 σ − σ −
16) (1) S ( σ ) = ( σ + 1)( σ − σ − . (2) Substituting σ = − ψ ( x ) we obtain the orbit o , ( x ), while for (1 − √ / √ /
2, roots of the quadratic factor, we get o , ( x ) and o , ( x ), respectively: o , ( x ) = x + x − x − x + 1 , (3) o , ( x ) = x − (1 − √ x − (3 + √ x − (2 + √ x − , (4) o , ( x ) = x − (1 + √ x − (3 − √ x − (2 − √ x − . (5) When multiplied together, o , ( x ) and o , ( x ) produce the orbital cluster, or aggregate: c , ( x ) = o , ( x ) · o , ( x ) = x − x − x + 6 x + 15 x − x − x + 4 x + 1 , (6)a cluster that may be obtained directly by eliminating σ between ψ ( x ) and σ − σ − o , ( x ) and o , ( x ), which have algebraic coefficients, resulted in a clusterwith integer coefficients, a generic characteristic. Technically, o , ( x ) and o , ( x ) are defined by relativequadratic equations of motion . Manifestly, c , ( x ) decomposes over the field Q ( √ o , ( x )has always integer coefficients and is always an exact representation for the orbit. In sharp contrast,when projected onto the real axis, o , ( x ) and o , ( x ) will have necessarily approximate numericalcoefficients. Thus, the symmetries clearly visible between Eqs. (4) and (5) will be totally obliterated. ugust 5, 2020 0:18 WSPC/INSTRUCTION FILE inheritance˙arxiv-2020 Jason A.C. Gallas
This unambiguous dichotomic distinction between orbits remains valid for other periods and neatlydisplays the enhanced insight obtained by working with exact equations of motion.Doublets like Eqs. (1) and (2) may be determined for arbitrary periods. Expressions for arbitraryvalues of a of the quadratic map and arbitrary ( a, b ) of the H´enon map are available , . The period five carrier
For period-five there are six possible orbits, all encoded simultaneously by the doublet: ψ ( x ) = (360 σ − σ − x − σ (3 σ − σ − x + 60( σ + σ − σ − σ − x − (60 σ + 90 σ − σ + 1710 σ + 1860 σ − x +(15 σ + 45 σ − σ + 375 σ + 3480 σ − σ − x − σ − σ + 192 σ + 30 σ − σ + 1446 σ + 4248 σ − , (7) S ( σ ) = ( σ − σ + σ − σ − σ − σ + 8) . (8)Eliminating σ between ψ ( x ) and, successively, σ − σ + σ −
8, and σ − σ − σ + 8, we get, apartfrom multiplicative constants used to eliminate denominators in ψ ( x ), o , ( x ) = x − x − x + 3 x + 3 x − , (9) c , ( x ) = x + x − x − x + 34 x + 34 x − x − x + 12 x + 12 x + 1 , (10) c , ( x ) = x − x − x + 13 x + 78 x − x − x + 165 x + 330 x − x − x + 126 x + 84 x − x − x + 1 . (11)The clusters factor into quintics over Q ( √
33) and Q (cid:0) (cid:112) −
62 + 95 √− (cid:1) , respectively, thereby provid-ing explicit expressions for the remaining five period-five orbits. As before, clustered orbits involverelative quadratic and cubic equations, with algebraic (non-integer) coefficients which, in numericalcomputations cannot be determined exactly. The period six carrier
For period-six there are nine possible orbits, all encoded simultaneously by the doublet: ψ ( x ) = 160 ϕ σ ( x − σ x ) + 80 σ ( σ + 4)( σ − ϕ x − σ ϕ (cid:0) σ + σ − σ + 126 σ + 358 σ − σ − σ + 56 (cid:1) x +20 σ ϕ (cid:0) σ + 3 σ − σ + 48 σ + 679 σ − σ − σ + 1048 (cid:1) x − σ ϕ (cid:0) σ + 6 σ − σ − σ + 1693 σ − σ − σ + 5496 σ + 2508 σ − (cid:1) x +2 σ + 14 σ − σ − σ + 7984 σ − σ − σ + 157668 σ +184938 σ − σ + 88965 σ + 373032 σ − σ − σ + 15680 , (12) S ( σ ) = ( σ + 1)( σ − σ − σ + 28)( σ + σ − σ − σ + 16) . (13)where ϕ ≡ σ − σ − σ + 13. Apart from multiplicative constants used to eliminate denominatorsin ψ ( x ), by selecting σ = 1 and σ = − o , ( x ) = x − x − x + 4 x + 6 x − x − , ∆ , = 13 = 371293 , (14) o , ( x ) = x + x − x − x + 8 x + 8 x + 1 , ∆ , = 3 · = 453789 . (15)Again, for roots of the cubic and quartic factors in Eq. (13), the resulting coefficients in Eq. (12) aremore complicated algebraic numbers, not integers. When all orbits arising from the same σ − factor aremultiplied together one obtains a cluster, a polynomial aggregate with integer coefficients and degree ∂ = mk , multiple of the period k , where m > c , ( x ) = x − x + x + 135 x − x − x + 90 x + 1287 x − x − x + 459 x + 1385 x − x − x + 170 x + 72 x − x + 1 , (16) c , ( x ) = x + x − x − x + 252 x + 229 x − x − x + 5832 x +4540 x − x − x + 25284 x + 15001 x − x − x ugust 5, 2020 0:18 WSPC/INSTRUCTION FILE inheritance˙arxiv-2020 Carriers and inheritance in quadratic dynamics o ,j of the map x t +1 = 2 − x t . Here, σ ,j = (cid:80) x j is the sumof the orbital points. The triad and quartet of σ ,j values are roots of the cubic and quartic factorsin Eq. (13), respectively.Orbit x x x x x x σ ,j o , -1.770912051306 -1.1361 0.7093 1.4969 -0.2407 1.9421 1 o , -1.911145611572 -1.6523 -0.7301 1.4670 -0.1521 1.9769 -1 o , -1.990061550730 -1.9605 -1.8436 -1.3989 0.0431 1.9981 -5.142457360 o , -1.756443146740 -1.0849 0.8230 1.3227 0.2505 1.9372 1.491252188 o , -0.912421314706 1.1675 0.6369 1.5944 -0.5421 1.7061 3.651205171 o , -1.990663269435 -1.9629 -1.8530 -1.4336 -0.0552 1.9970 -5.287613777 o , -1.916491658218 -1.6730 -0.7989 1.3618 0.1455 1.9788 -0.902246984 o , -1.559348708126 -0.4314 1.8139 -1.2902 0.3354 1.8875 0.756484903 o , -0.971966826485 1.0553 0.8863 1.2145 0.5250 1.7244 4.433375858 -2-1.5-1-0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.77091205120 -2-1.5-1-0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.91114556789 -2-1.5-1-0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.99006152153 -2-1.5-1-0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.75644314289 -2-1.5-1-0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -0.91242128611 -2-1.5-1-0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.99066329002 -2-1.5-1-0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.91649162769 -2-1.5-1-0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.55934870243 -2-1.5-1-0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -0.97196680307 Fig. 1. Return maps x t × x t +1 for the nine period-6 orbits. Numbers refer to the leftmost orbital coordinate.Some of the orbits are topologically identical, despite their very distinct algebraic character. +20886 x + 7168 x − x − x + 2085 x + 101 x − x + 12 x + 1 . (17)Independently from ψ ( x ), the Maple driver given in Appendix A exemplifies how to extract o , ( x ), o , ( x ), c , ( x ), and c , ( x ) directly from the quadratic equation of motion.The orbital points for all nine period-six orbits are collected in Table 1, together with the sums σ ,j .Return maps for all nine orbits are illustrated in Fig. 1. Numbers inside panels identify the leftmostorbital point. From Fig. 1 one sees that some orbits are topologically identical despite the very distinct ugust 5, 2020 0:18 WSPC/INSTRUCTION FILE inheritance˙arxiv-2020 Jason A.C. Gallas nature of the algebraic numbers underlying them.It is interesting to mention that Eqs. (14) and (15) imply a novel twist in the current understandingof polynomial interdependence. As individual orbits, they are obtained as two “projections” arising froma common mathematical origin, the carrier ψ ( x ). Therefore, rather than independent orbits, they arein a certain sense a kind of “conjugated” orbits. Furthermore, since all nine orbits arise from the samecarrier ψ ( x ), the pair of orbits is also conjugated to the remaining seven orbits. This illustrates theexistence of a complex and subtle arithmetical interdependence lurking among such orbits, apparentlyrather different from the usual field isomorphisms familiar from Galois theory of equations. Orbitalcarriers allow conjugated orbits to have coefficients from very distinct number fields, a concept aliento the standard theory.As for the remaining polynomials, o , ( x ) factors into a pair of cubics over Q ( √ c , ( x ) factors into two equations of degree nine over Q ( √ c , ( x ), six cubics are obtained over Q ( √ α ), where α = − √− − √− √
21. For c , ( x ), eight cubics are obtained over Q ( √ β ),where β = 65 − √ √ − √
65. These factorizations provide explicit and exact solutions forall period-six orbits. Note that the factors of S k ( σ ) reveal how orbits are distributed into clusters andsingle orbits, if any.
3. Orbital inheritance
A simple example allows one to grasp easily what inheritance means , , . To this end, we apply thenonlinear transformation x − x to o , ( x ), obtaining the identity: c , ( x ) = o , ( x − x ) . (18)This identity shows that, as soon as the roots z i of o , ( x ) = 0 are determined, three new orbits followfrom the zeros of the six cubics x − x − z i = 0 . (19)Therefore, since o , ( x ) factors into a pair of cubics over Q ( √ o , ( x ) = (cid:0) x + (1 − √ x − (1 + √ x + (5 + √ (cid:1) × (cid:0) x + (1 + √ x − (1 − √ x + (5 − √ (cid:1) , (20)their roots provide exact analytical solutions in terms of radicals for all orbital points ofthe 18th-degree cluster c , ( x ) = 0. Such exact solutions are simple cascades, towers, of rela-tive cubic irrationalities . Incidentally, Maple surprisingly fails to solve the sextic o , ( x ) using aux:=solve(o62,x); convert(aux[1],radical); But it correctly breaks o , ( x ) into a pair of cu-bics when adding input from the sextic discriminant: factor(o62,21^(1/2)); What about the inherent character of the irrationalities underlying period-six orbital point? Thisquestion is particularly interesting because, while compositions of relative quadratic irrationalities arelong known , the considerably more complicated structures arising from nested cyclic cubic irrational-ities remains essentially open . Thus, present day computer algebra systems still have to grapple withdifficulties to simplify expressions containing cubic and higher roots , , , . By way of illustration,consider to reassemble the cubics in Eq. (20) starting from the exact expressions of their three roots.In this case, we get the leftmost number below as the second coefficient in the topmost equation, notits most simplified version:119 − √− − √− √ −
77 + 21 √− √− − √
21 = −
49 + 11 √ √
21) = (1 − √ (cid:39) − . . (21)Similarly garbled expressions are obtained for all other coefficients in Eq. (20). The good news is thatsuch expressions provide clues regarding the subfield structure underlying the solutions. Clues may bealso obtained from the algebraic numbers solving the factors in S k ( σ ).
4. Inheritance systematics up to periods k ≤ Periods k ≤ Using a slightly adapted version of the ad-hoc
Maple driver given in appendix A, we computed sys-tematically all genuine factors defining orbits with period k ≤
12. A summary of the relevant dataobtained for k ≤
11 is given in Table 2. This table reveals a number of interesting facts and trends: ugust 5, 2020 0:18 WSPC/INSTRUCTION FILE inheritance˙arxiv-2020
Carriers and inheritance in quadratic dynamics k . Type refers eitherto orbits o k,j or orbital clusters c k,j , ∂ is the degree of the corresponding polynomial, D = ∆are the standard polynomial and field discriminants, and L is the length, the number of digits ofthe discriminants. For a given k , highlighted cells indicate discriminants arising from identicalprime numbers (see text). k Type ∂ D = ∆
L k
Type ∂ D = ∆ L o , o , o , o , o , · c ,
18 3 · c , c ,
36 73 o , c ,
54 3 · c ,
10 3 · c ,
162 3 · c ,
15 31 c ,
216 7 · o , o ,
10 5 o , · c ,
20 41 c ,
18 3 · c ,
30 3 · c ,
24 5 · c ,
80 5 · c ,
21 43 c ,
150 11 · c ,
42 3 · c ,
300 3 · · c ,
63 127 c ,
400 5 · c ,
16 3 ·
23 11 o ,
11 23 c ,
32 5 · c ,
44 89 c ,
64 3 · · c ,
341 683 c ,
128 257 c ,
682 3 · c ,
968 23 · (1) The growth of the number of single orbits is much smaller than cluster growth.(2) Periods k = 7 and k = 8 contain only orbital clusters, no single orbits.(3) Orbits and clusters are all monogenic , i.e. the discriminant D of their minimal polynomialcoincides with their field discriminant ∆. Therefore, orbits and clusters admit power integralbases. For details, see Ref. .(4) The degree of single orbits and clusters is always a multiple of the period k .(5) As indicated by the length L giving the number of digits in the discriminants, D and ∆ growfast with the period. However, they contain powers of relatively small prime numbers.(6) The discriminants of, e.g., c , contain 3124 digits. It would be computationally hard to factorit if it was not a simple product of powers of a few identical and small primes, 23 and 89.(7) The highlighted values of D = ∆ for k = 6 ,
9, and 10 summarize all cases of inheritance foundfor k ≤ k = 6 the ratio of the polynomial degrees are ∂ ( c , ) /∂ ( o , ) = 3. Similarly, for k = 9 theratios are ∂ ( c , ) /∂ ( c , ) = ∂ ( c , ) /∂ ( c , ) = 3. Inheritance among these orbits involves theaforementioned cubic transformation: c , ( x ) ≡ c , ( x − x ) and c , ( x ) ≡ c , ( x − x ).(9) In contrast, for k = 10 the ratio is ∂ ( c , ) /∂ ( o , ) = 5, implying inheritance involving aquintic nonlinear transformation . In this case, we have c , ( x ) ≡ o , ( x − x + 5 x ).(10) For a given period k , the discriminants D and ∆ involve certain combinations of a small setof primes. We were not able to find interconnections between orbits with discriminants arisingfrom powers of distinct primes, although we see no reason to rule out the possibility of intricateinterconnections yet to be discovered.(11) From Table 2, it seems reasonable to conjecture inheritance to exist among polynomials withdiscriminants composed by powers of the same primes. Period k = 12 Table 3 summarizes data obtained for the sixteen individual factors resulting from the computationand factorization of the 4020th degree polynomial which contains all genuine period twelve orbits andclusters. These factors corroborate the properties listed above for k ≤
11. Note the fast increase in thenumber of digits of the discriminants, which for c , ( x ) contains no less than 6770 digits. In order tofactor arbitrary numbers of this size, computers need to check numbers of the order of the size of thesquare-root of the number to be factored, in the present case roughly 10 . ugust 5, 2020 0:18 WSPC/INSTRUCTION FILE inheritance˙arxiv-2020 Jason A.C. Gallas
Table 3. Individual factors of the 4020th degree poly-nomial containing all period twelve orbits and orbitalclusters. Here, ∂ refers to the degree of individual fac-tors, while length is the number of digits contained inthe discriminants D = ∆. Similar highlighting is usedfor discriminants defined by identical prime numbers. Nomore than pairs of interdependent orbits are observed. (cid:96) Degree ∂ D = ∆ Length1 12 5 ·
152 12 3 ·
153 12 3 ·
164 24 3 · ·
365 36 3 ·
636 36 7 ·
637 48 3 · ·
868 72 3 · · · · · · · · · · · · · · · · · N k of periodic orbits, as a function of the period k .The number of orbits roughly doubles as k increases. For simple equations and Mapleimplementations to obtain arbitrary values of N k see Refs. , . k
12 13 14 15 16 17 18 19 20 N k
335 630 1161 2182 4080 7710 14532 27594 52377 N k /N k − Numbers with 6770 digits are well beyond the capabilities of factorization, and also well beyondthe numbers currently used in data encryption. For instance, consider that the lifetime of the universe,currently estimated to be some 13.8 billion years, roughly 10 seconds, a number with 19 digits.Assuming a computer able to test one million factorizations per second, during the lifetime of theuniverse it would be able to check some 10 possibilities. However, for 6770 digits, roughly 10 ,one would need to check 10 possibilities, meaning that the time to do this amounts to roughly10 − = 10 times the lifetime of the universe! Fortunately, however, the very big numbers inTable 3 involve products of just a few and small primes, allowing them to be factored, as indicatedin the Table. The passage here is exceedingly narrow. Slight changes in the coefficients may precludefactorization.The most conspicuous difference when comparing the numbers in Table 3 with analogous resultsfor the lower periods in Table 2 is the surprising increase of the number of polynomials display-ing inheritance. For instance, abbreviating X = x − x , we find the following five nonlinear in-terconnections among polynomials of quite high degrees: c , ( x ) ≡ o , ( X ), c , ( x ) ≡ c , ( X ), c , ( x ) ≡ c , ( X ), c , ( x ) ≡ c , ( X ), and c , ( x ) ≡ c , ( X ). The verification of these identi-ties requires ad-hoc handling because of recurring Maple warnings “stack limit reached”.Table 4 illustrates how fast the number of orbits grows as a function of the period k . A simple andexplicit formula and its Maple implementation to compute such growth is available in the literature , .It would be interesting to extend the present calculations and check inheritance for the promising cases k = 14 ,
15, 16 and 18, something that should be feasible already by someone with access to morepowerful resources than available to us.Two additional aspects are worth mentioning: First, periodic orbits may be found by studyingpreperiodic points . Such procedure involves just straightforward but somewhat tedious computations,due to the large number of factors and orbits involved. Fortunately, the procedure involving preperiodicpoints may be programmed to run automatically. Second, by a process of reverse engineering and by ugust 5, 2020 0:18 WSPC/INSTRUCTION FILE inheritance˙arxiv-2020 Carriers and inheritance in quadratic dynamics suitably summing orbital points, one may recover the several individual factors arising in the S k ( σ )polynomials. For instance, in Appendix B we compute explicitly the three factors composing S ( σ ).For single orbits the factors are very simple to find. For instance, the single period-twelve orbits are o , ( x ) = x + x − x − x + 54 x + 43 x − x − x +110 x + 46 x − x − x + 1 , (22) o , ( x ) = x − x + x + 54 x − x − x + 27 x + 105 x − x − x + 12 x + 1 , (23) o , ( x ) = x + x − x − x + 53 x + 53 x − x − x +79 x + 79 x − x − x + 1 , (24)and we immediately recognize that s ( s + 1) are the linear factors of S ( σ ), a curious degeneratemultiplicity situation which seems to foretell that ψ ( x ) will be a reducible polynomial. Analogously,linear factors of S k ( σ ) may be read directly from the coefficients of the orbits: o , ( x ) = x − x + 27 x − x + 9 x − , (25) o , ( x ) = x − x − x + 7 x + 21 x − x − x + 10 x + 5 x − , (26) o , ( x ) = x − x + 35 x − x − x + 5 x + 25 x − x − , (27) o , ( x ) = x − x − x + 9 x + 36 x − x − x + 35 x + 35 x − x − x + 1 . (28)It is quite challenging to decompose orbital clusters combining more than two orbits, particularlythose combining an odd number of orbits. However, the coefficients of such decompositions hide thesecretest truth and most interesting relations among numbers which fix orbital individuality.
5. Conclusions and outlook
This paper presented explicit expressions for orbital carriers of periods 4, 5, and 6. In addition, thesystematics of orbital inheritance was considered for all periods k ≤
12. Evidence was found thatinheritance becomes more abundant as the period increases. Useful insight was obtained from theexact properties of equations of motion, instead of orbital points. An interesting open challenge is tocompute the distinct factors arising for orbits of periods k = 14 ,
15, 16 and 18, and to check if theyalso display inheritance and relations with orbits of lower periods, if any. A much harder problemseems to be to find out if orbits not displaying inheritance may nevertheless display some other type ofinterdependence. If found, this would certainly reveal unanticipated interconnections among familiesof algebraic numbers.As it is visible from Tables 2 and 3, the growth of the polynomial degrees ∂ k as a function of k and their partition into proper divisors of k are interesting open combinatorial questions. What is themechanism behind the decomposition of the number N k of periodic orbits into the several degrees ∂ k ofthe polynomial set defining k − periodic orbits? For instance, the 4020th degree polynomial of period-12orbits is partitioned into sixteen factors recorded in Table 3. What would be, say, the correspondingpartition for the 16254th degree polynomial corresponding to period-14 orbits and clusters? Or the32730th degree polynomial for period-15? Or the 65280th degree polynomial for period-16? Note thatthe partitions listed in Table 2 are not unique: for k = 6, instead of 6 + 6 + 18 + 24, we could equallywell have 12 + 18 + 24, 12 + 12 + 30, etc. Such alternative partitions, however, are never observed in thepresent context. It is clear that the partition sets have many elements, and an interesting combinatorialchallenge is to count them all and to predict partitions that may be observed for a given period of agiven map.Finally, for applications in physics and dynamical systems, it is of interest to mention that inalgebraic number theory one knows that every cyclotomic field is an Abelian extension of the rationalnumbers Q . In this context, an important discovery is the so-called Kronecker-Weber theorem, statingthat every finite Abelian extension of Q can be generated by roots of unity, i.e. Abelian extensionsare contained within some cyclotomic field. Equivalently, every algebraic integer whose Galois groupis Abelian can be expressed as a sum of roots of unity with rational coefficients. For details see, e.g.,Edwards . The study of the partition generating limit of the quadratic map x t +1 = a − x t seems to ugust 5, 2020 0:18 WSPC/INSTRUCTION FILE inheritance˙arxiv-2020 Jason A.C. Gallas lend hope that for a = 2 the map may also share an analogous correspondence with Abelian equationsas the one embodied in the Kronecker-Weber theorem , which is intrinsically related to the cyclotomicpolynomials generated by the map when a = 0, whose dynamics, unbeknownst to him, was studied byGauss in Sectio Septima of his
Disquisitiones Arithmeticæ . Such enticing possibility of correspondencedeserves to be further investigated.
Acknowledgments
This work was started during a visit to the Max-Planck Institute for the Physics of Complex Systems,Dresden, gratefully supported by an Advanced Study Group on
Forecasting with Lyapunov vectors .The author was partially supported by CNPq, Brazil, grant 304719/2015-3.
Appendix A. Maple driver to generate period six orbits and clusters a := 2:x[1]:= a - x*x: x[2]:= a - x[1]*x[1]: x[3]:= a - x[2]*x[2]:x[4]:= a - x[3]*x[3]: x[5]:= a - x[4]*x[4]: x[6]:= a - x[5]*x[5]:aux := factor(x-x[6]);
The above assignments are correct under Maple 2014, but are easy to adjust if emerging differently.Manifestly, the driver above may be easily adapted to generate equations for other periods.
Appendix B. Determination of the three factors composing S ( σ ) Here, in contrast to the arithmetic work done so far, we resort to numerically computed orbital pointsto illustrate how to find exact representations for the individual factors composing S ( σ ). The threeclusters whose roots give all period-seven orbital points may be easily generated by slightly adaptingthe Maple driver given in Appendix A. Such clusters read as follows: c , ( x ) = x − x − x + 19 x + 171 x − x − x + 680 x +2380 x − x − x + 3003 x + 5005 x − x − x + 1716 x + 1287 x − x − x + 55 x + 11 x − , (B.1) c , ( x ) = x + x − x − x + · · · − x − x + 44 x + 44 x + 1 , (B.2) c , ( x ) = x − x − x + 61 x + · · · + 40920 x + 5456 x − x − x + 1 . (B.3)From them, we extract the (21 + 42 + 63) / S ( σ ), all with degree multiple of three: (cid:89) j =1 ( σ − σ ,j ) = σ − σ − σ − , (B.4) (cid:89) j =4 ( σ − σ ,j ) = σ + σ − σ + 63 σ + 110 σ − σ − , (B.5) (cid:89) j =10 ( σ − σ ,j ) = σ − σ − σ + 118 σ + 573 σ − σ − σ + 2700 σ + 1576 σ − . Even though an expression for the period-seven carrier pair is still unknown, we were neverthelessable to extract S ( σ ). Its factors corroborate the three aggregates c ,m ( x ), m = 1 , , a priori knowledge of the three “brute-force factors” in Table ugust 5, 2020 0:18 WSPC/INSTRUCTION FILE inheritance˙arxiv-2020 Carriers and inheritance in quadratic dynamics σ ,j of all points. The remaining orbital points follow by iterating x t +1 = 2 − x t .Orbit x σ ,j o , -1.97868673615022039558470712622 -2.88823600884341144649527347953 o , -1.81089647498629323144358511206 -0.61507162581156506493032243917 o , -1.04188068097586057235256289907 4.50330763465497651142559591867 o , -1.99762811048164609235032811636 -7.24813219626988235042435866316 o , -1.88487616566742883844738056630 -1.22906022702843317616182868886 o , -1.94098358831481064644663464031 -0.774908002370389501560130448853 o , -1.61228898351072554484668557445 1.84413185283999824109215112803 o , -1.71974598668362014848475373830 2.74482456161490583899876274448 o , -1.20298163000374078956875931348 3.66314401121380094805540392837 o , -1.99755283242852256525640526121 -7.17543506383968793122213507689 o , -1.97801140897626144475812107452 -2.93599431271533079530605723303 o , -1.88125825920768775450635179933 -1.60237082080316266127006810916 o , -1.93911972959649314295081201674 -0.525572883742883886011679753072 o , -1.80499303815485256128035783817 0.0196507480462068262052865602836 o , -1.60040839696003410107405022267 2.19795804752238429773060428492 o , -1.71107014481703192827696436581 2.36473479389711276652833789748 o , -1.17956942634103896113267847206 3.29077832666324426013918210422 o , -1.01424772773954618184418426842 5.36625116497211712320652932523 c ,m ( x ). However, using preperiodic points generated by an infinitefamily Q (cid:96) ( x ) of polynomials , the same three groups my be discovered independently, directly fromnumerically approximated orbital equations. How to accomplish this will be presented in a forthcomingpublication. References
1. J. Argyris, G. Faust, M. Haase, and R. Friedrich, An Exploration of Dynamical Systems and Chaos,Second Edition (Springer, Berlin, 2015).2. M. Cencini, F. Cecconi, and A. Vulpiani, Chaos - From Simple Models to Complex Systems (WorldScientific, Singapore, 2010).3. M. Ausloos, M. Dirickx (eds.), The Logistic Map: Map and the Route to Chaos: From the Beginningto Modern Applications, Proceedings of the ”Verhulst 200 on Chaos”, Brussels, Belgium (Springer,Heidelberg, 2005).4. J.A.C. Gallas, Units: remarkable points in dynamical systems, Physica A , 125-151 (1995).5. J.A.C. Gallas, Lasers, stability, and numbers, Physica Scripta , 014003 (2019).6. J.A.C. Gallas, Nonlinear dependencies between sets of periodic orbits, Europhys. Lett. , 649-655(1999).7. J.A.C. Gallas, Infinite hierarchies of nonlinearly dependent periodic orbits, Phys. Rev. E , 016216(2000).8. J.A.C. Gallas, Method for extracting arbitrarily large orbital equations of the Pincherle map,Results in Physics , 561-567 (2016).9. J. Sommer, Vorlesungen ¨uber Zahlentheorie (Teubner, Leipzig, 1907).10. B.N. Delone and D.K. Faddeev, The Theory of Irrationalities of the Third Degree (AmericanMathematical Society, Providence, 1964).11. S. Landau, How to tangle with a nested radical, Math. Intelligencer , 49-55 (1994).12. S. Landau, Computations with Algebraic Numbers, in J. Grabmeier, E. Kaltofen, and V. Weispfen-ning (eds.), Computer Algebra Handbook (Springer, Berlin, 2003), pp. 18-19.13. B.C. Berndt, H.H. Chan, and L.C. Zhang, Ramanujan’s association with radicals in India, Am.Math. Month. , 905-911 (1997).14. B.C. Berndt, H.H. Chan, and L.C. Zhang, Radicals and units in Ramanujan’s work, Acta Arithm. , 145-158 (1998).15. F. Haake, Quantum Signatures of Chaos, third edition (Springer, Berlin, 2010).16. J. Marklow, Arithmetic quantum chaos, in Encyclopedia of Mathematical Physics , vol. 1, edited byJ.P. Fran¸coise, G.L. Naber, and T.S. Tsun (Academic Press/Elsevier, Oxford, 2006), pp. 212-220. ugust 5, 2020 0:18 WSPC/INSTRUCTION FILE inheritance˙arxiv-2020 Jason A.C. Gallas
17. P. Bleher, Trace formula for quantum integrable systems, lattice-point problem, and small divisors,in