Digit replacement: A generic map for nonlinear dynamical systems

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by  Vladimir García-Morales

Digit replacement: A generic map for nonlinear dynamical systems
Vladimir Garc´ıa-Morales1,a)
Departament de Termodina`mica, Universitat de Val`encia, E-46100 Burjassot,
Spain
(Dated: 8 September 2016)
A simple discontinuous map is proposed as a generic model for nonlinear dynamical systems. The orbit
of the map admits exact solutions for wide regions in parameter space and the method employed (digit
manipulation) allows the mathematical design of useful signals, such as regular or aperiodic oscillations with
specificwaveforms,theconstructionofcomplexattractorswithnontrivialpropertiesaswellasthecoexistence
of different basins of attraction in phase space with different qualitative properties. A detailed analysis of
the dynamical behavior of the map suggests how the latter can be used in the modeling of complex nonlinear
dynamicsincluding, e.g., aperiodicnonchaoticattractorsandthehierarchicaldepositionofgrainsofdifferent
sizes on a surface.
6
1 PACS numbers: 02.70.Bf, 05.45.Ac, 05.45.Df
0
2
p The quest for simple deterministic models that Some advantages of our formulation become evident, as
e exhibit complex and unpredictable temporal soon as it is realized that the mathematical method pre-
S evolution1–4 has been the subject of intense inter- sented allows each digit of any numerical quantity to be
7 est. Most remarkable examples are provided by individually addressed. As a consequence of this, the
systems of nonlinear ordinary differential equa- temporal evolution that constitutes the output of the

] tions leading to chaotic behavior, as the Lorenz5 map can be mathematically designed in detail a priori
D
or the R¨ossler6 systems, and by time-discrete dif- through its few control parameters. We find, for exam-
C ference equations, as the logistic map1,7 or the ple, that the specific shape of any regular oscillations
. Bernoulli shift3. All these systems involve dy- on an attracting limit cycle can be mathematically engi-
n
namical variables taking on numerical values that neered. The method also allows the design/modeling of
i
l can be expanded in terms of a (generally infi- time series with irregular or aperiodic waveforms.
n
[ nite) convergent series of rational numbers in a This article is organized as follows. In Section II
radix (base) p ∈ N (p > 1). The latter are each we introduce the mathematical tools, the digit function
3
v formed by an integer power of the radix p multi- and the replacement operator, in which our approach is
7 plied by an integer in the interval [0,p−1] called based. Then,weshowhowtoconvertanyarbitrarynon-
1 a digit. In this work, by directly addressing and autonomous map into the generic map that is the object
0 manipulating digits (by means of a digit function of our study (see Eq. (8) below). In Section III we dis-
3 that we have recently introduced8–11) we uncover cuss in detail its dynamics, providing explicit solutions
0 adigitreplacementoperatorthatallowsageneric fortheorbitsofthemap. Ourdiscussionculminatesina
.
1 map for nonlinear dynamical systems to be for- model for aperiodic nonchaotic attractors, showing how
0 mulated. Our model is exactly solvable in wide they can be constructed in phase space, all attractors
6 regions of parameter space and encompasses a sharing aspects of the same qualitative dynamics (even
1 wide variety of complex nonlinear dynamical be- whentheycanbemadedistinguishable). Weprovethat,
:
v havior from a new perspective. in spite of the dynamics on the attractors being aperi-
Xi odic, the Lyapunov exponent is zero.
r
a
I. INTRODUCTION
II. FORMULATION OF THE MAP
A dynamical system12,13 is a pair (M,φ), where M
is a compact metric space and φ a continuous map
φ : M → M. In this article we introduce a generic map Sinceourapproachisbasedonthepossibilitiesopened
for nonlinear dynamical systems φ : R → R where φ is by a digit function that we have recently discussed in
discontinuous. However, as we explicitly show, disconti- connection with fractals8,9,14 and cellular automata10,11
nuity can be seen here as arising from the discretization we first introduce this function. Let p > 1 be a natural
(shadowing) of any arbitrary specific continuous dynam- number called the radix. Any arbitrary real number x
ics as governed by an arbitrary non-autonomous map. can be represented in radix p as8
(cid:98)log |x|(cid:99)
p
(cid:88)
x=sign(x) pkd (k,|x|) (1)
p
a)[email protected]
k=−∞
2
where d (k,x) is the digit function, defined as8 To show how Eq. (8) can be derived starting from an
p
arbitrary non-autonomous map
(cid:22) (cid:23) (cid:22) (cid:23)
x x
d (k,x)= −p (2) x =F(x ,t) (9)
p pk pk+1 t+1 t
weapproximatex ateachinstantoftimetbytheleading
t
and where the brackets (cid:98)...(cid:99) denote the floor (lower part of x (i.e. we neglect the contribution to x of any
t t
closest integer) function. As an example, in the deci- digit that is not the most significant one). Therefore, we
mal radix p = 10, the number π ≡ 3.1415... has dig- have,intermsoftheleadingpartlp (x)functiondefined
p
its d10(0,π) = 3, d10(−1,π) = 1, d10(−2,π) = 4, in Eq. (7),
d (−3,π)=1, d (−4,π)=5 etc.
10 10 x ≈lp (x ) (10)
Trivial properties of the digit function that can be di- t rt t
rectlycheckedfromitsdefinitionarethepk+1 periodicity xt+1 ≈lpqt(xt+1)=lpqt(F(xt,t)) (11)
for any r ,q ≥ 2. The radix values r ,q ∈ N can be
d (k,x+pk+1)=d (k,x) (3) t t t t
p p conveniently chosen at each time so as to optimize the
approximation15. We demand here that they are equal
and the scaling property
tosomepowersrt ≡ph(t1),qt ≡ph(t2) ofanaturalnumber
dp(k,x)=dp(0,p−kx) (4) p ≥ 2, h(t1) and h(t2) being integers. We let p constant
and regard h(1) and h(2) as functions of time. We also
t t
Thereaderisreferredto8 forotherpropertiesofthedigit demand the following inequalities (which can always be
function and a detailed discussion/derivation of them. satisfied by suitably chosing p)
Inthispaperweshallconsidertanon-negativeinteger. (cid:40) (cid:41)
log x
In this case dp(0,t) = t mod p, i.e. the zeroth digit h(1) p t <1
function of an integer number is equal to that number t h(1)
t
modulo the radix p. We then observe that (cid:40) (cid:41)
log F(x ,t)
h(2) p t <1
d1(0,t)=0 (5) t h(2)
t
d (0,t)=t (6)
∞ where the brackets {...} denote the fractional part func-
tion. By subtracting Eqs. (10) and (11), we obtain
The number (cid:98)log |x|(cid:99) in Eq. (1) is the maximum ex-
p
ponent of the power of the radix with which x ∈ R is x ≈x +lp (F(x ,t))−lp (x )
represented in radix p.14 Thus, the function (cid:98)log |x|(cid:99) t+1 t qt t rt t
gives the order of magnitude of x when x is writtepn in =xt+qt(cid:98)logqtF(xt,t)(cid:99)dqt((cid:98)logqtF(xt,t)(cid:99),F(xt,t))−
radix p. The following function −r(cid:98)logrtxt(cid:99)d ((cid:98)log x (cid:99),x )
t rt rt t t
lpp(x)≡sign(x)p(cid:98)logp|x|(cid:99)dp((cid:98)logpx(cid:99),x) (7) =xt+ph(t2)(cid:98)logqtF(xt,t)(cid:99)dqt((cid:98)logqtF(xt,t)(cid:99),F(xt,t))−
egxivaemspthlee, lleeatdixng=pa6r.t02o2f·x1,0w23h,enthwenritltpen(ixn)ra=dix6·p.10F2o3r. −ph(t1)(cid:98)logrtxt(cid:99)drt((cid:98)logrtxt(cid:99),xt) (12)
10 We now note that,
As another example, we have lp (204) = 2 · 102 but
10
l(p121(0200141)00=) 1a·n2d7tshiunsce202404=i1s·w27ri+tte1n·2i6n+ra0d·i2x5p+=0·224a+s drt((cid:98)logrtxt(cid:99),xt)=dph(t1)((cid:98)logrtxt(cid:99),xt)
1·23+1·222+0·21+0·20. =dp(h(t1)(cid:98)logrtxt(cid:99),xt) (13)
The map studied in this article has the general form as well as d ((cid:98)log F(x ,t)(cid:99),F(x ,t)) =
qt qt t t
xt+1 =xt+pjt(dp(kt,µt)−dp(jt,xt)) (8) dp(h(t2)(cid:98)logqtF(xt,t)(cid:99),F(xt,t)). Thus, if we take
h(1) and h(2) at each time so that
t t
where x ∈ R and µ ∈ R are both non-negative real-
t t (cid:36) (cid:37) (cid:36) (cid:37)
valued functions at discrete time t, xt being the dynami- h(2) logpF(xt,t) −h(1) logpxt =0 (14)
cal variable and µt being the ‘objective signal’ ruling the t h(2) t h(1)
t t
instantaneous dynamics and j and k are integer-valued
t t
functions that also evolve with time. The parameter p, we see that, if we identify
called the radix, is a natural number p≥2. (cid:36) (cid:37)
log F(x ,t)
The map Eq. (8) has the following meaning: At time k ≡h(2) p t (15)
t+1, the value of the dynamical variable xt+1 is given t t h(t2)
bofyµxtt wbuhtenrepbloatchinngutmhbeejrts-tahrediwgirtitotfexntibnyrtahdeixktp-.thTdhiugsit, jt ≡h(t1)(cid:36)lohg(p1)xt(cid:37) (16)
the map replaces a single digit of the numerical value of t
x at each time step by a single digit of µ . µ ≡F(x ,t) (17)
t t t t
3
by using Eqs. (13) and (14) and by replacing all these A. Relaxation to a fixed point
quantitiesinEq. (12)wefinallyobtainEq. (8). Wenote
that there is considerable freedom above to choose h(1), We can now consider j and k as a digit replacement
t t t
h(t2) andµt. Thisfreedomisrelatedtoascalingproperty dynamicsgoingfromthemostsignificantdigitsofxt and
ofthereplacementdynamicsthatwepointoutbelow,see µ to the less ones
Eq. (20). (cid:4) (cid:5)
j ≡ log x −t (23)
We thus conclude that the dynamics of any arbitrary t p t
(cid:4) (cid:5)
non-autonomous map, Eq. (9), can always be approxi- kt ≡ logpµ −t (24)
mated by a digit replacement dynamics given by Eq. (8).
so that Eq. (19) reduces to the simple form
The contrary, however, is not necessarily true, i.e. there
are choices for jt, kt and µt in Eq. (8) such that the xt+1 =Rp(cid:0)(cid:4)logpxt(cid:5)−t,(cid:4)logpµ(cid:5)−t;xt,µ(cid:1) (25)
latter equation cannot be approximated by Eq. (9).
Since the digit replacement dynamics, Eq. (8), is This simple map can exactly be solved for the orbit for
more general than the dynamics provided by a non- any initial condition x0 as
autonomousmapoftheformEq. (9),theaimoftherest
of this article is to study the digit replacement dynam- x =t−1−(cid:88)(cid:98)logpx0(cid:99)p−jd ((cid:4)log µ(cid:5)−(cid:4)log x (cid:5)−j,µ)+
ics for the simplest input functions jt and kt; the latter t p p p 0
being constant, linear, or simply periodic functions of t. j=−(cid:98)logpx0(cid:99)
We shall take µt =µ constant throughout. (cid:88)∞
We define the replacement operator, R (j,k;a,b) as + p−jd (−j,x ) (26)
p p 0
j=t−(cid:98)logpx0(cid:99)
R (j,k;a,b)≡a+sign(a)pj(d (k,|b|)−d (j,|a|)) (18)
p p p
The proof of this result is trivial by induction. At t=0
we have
This operator replaces the j-th digit of a by the k-th
digit of b (when both are written in radix p). Thus, the ∞
(cid:88)
digit replacement dynamics of map Eq. (8) is concisely xt=0 = p−jdp(−j,x0)=x0 (27)
written in terms of this operator as j=−(cid:98)logpx0(cid:99)
x =R (j ,k ;x ,µ ) (19) Let us assume Eq. (26) valid at time t. Then, at time
t+1 p t t t t
t+1
We observe that, for any integers h and j, because of (cid:0)(cid:4) (cid:5) (cid:4) (cid:5) (cid:1)
x =R log x −t, log µ −t;x ,µ
the scaling property of the digit function, Eq. (4), the t+1 p p t p t
following important relationship holds
t−1−(cid:98)logpx0(cid:99)
R (cid:0)h+j,i+k;pha,pib(cid:1)=phR (j,k;a,b) (20) = (cid:88) p−jd ((cid:4)log µ(cid:5)−(cid:4)log x (cid:5)−j,µ)+
p p p p p 0
j=−(cid:98)logpx0(cid:99)
∞
(cid:88)
III. ANALYSIS OF THE DYNAMICS + p−jdp(−j,x0)+
j=t−(cid:98)logpx0(cid:99)
is Tbehteterricehxnpelsosreodfatnhde udnyndaermsticoaoldbsetheapvbioyrsotfepEsqt.ar(t1in9g) +p(cid:98)logpxt(cid:99)−t(cid:0)dp((cid:4)logpµ(cid:5)−t,µ)−dp((cid:4)logpxt(cid:5)−t,x0)(cid:1)
finrogmtothmemoroestcotrmivpilaelxinbsetahnavceiosro.fjItfajnd=kt(cid:4)alnodgpxro(cid:5)graesnsd- = t−(cid:98)l(cid:88)ogpx0(cid:99) p−jd ((cid:4)log µ(cid:5)−(cid:4)log x (cid:5)−j,µ)+
t p t p p p 0
(cid:4) (cid:5)
kt = logpµ , with µ constant, Eq. (19) reduces to j=−(cid:98)logpx0(cid:99)
∞
x =R (cid:0)(cid:4)log x (cid:5),(cid:4)log µ(cid:5);x ,µ(cid:1) (21) + (cid:88) p−jd (−j,x ) (28)
t+1 p p t p t p 0
j=t+1−(cid:98)logpx0(cid:99)
which trivially replaces the most significant digit of x
t
by the most significant one of µ and does nothing else. where we have used the obvious fact that, in this case
(cid:4) (cid:5) (cid:4) (cid:5)
Thus, already on the first time step, a fixed point x∞ is logpxt = logpx0 ∀t. This proves the validity of Eq.
so reached, with value (26). Thus, the map converges to the fixed point
∞
x∞ =x1 =Rp(cid:0)(cid:4)logpx0(cid:5),(cid:4)logpµ(cid:5);x0,µ(cid:1) (22) x∞ = (cid:88) p−jdp((cid:4)logpµ(cid:5)−(cid:4)logpx0(cid:5)−j,µ)
In the following, we shall, of course, consider more inter- j=−(cid:98)logpx0(cid:99)
esting and increasingly complex dynamics. =µp(cid:98)logpx0(cid:99)−(cid:98)logpµ(cid:99) (29)
4
FIG. 1. FirsteightiteratesofthemapEq. (25)forµ=π startingfrominitialconditionsx0∈[1,9900]andforp=2(left)andp=10(right).
The fixed point at µ (=π in this case) is always present, independently of the value of p, although its basin of attraction is wider for p larger.
Notethelogarithmicscaleonthext axis.
√
We observe that there is indeed an infinite number of nonlinear manner. For x = 2 the successive iterates
0
isolated fixed points separated in value by powers of of the map for µ=π and p=10 are
p. The fixed point x = µ is always present regard-
∞
x =1.414213... x =3.414213...
less of the value of p, although its basin of attraction 0 1
(which corresponds to all those initial conditions for x =3.114213... x =3.144213...
2 3
(cid:4) (cid:5) (cid:4) (cid:5)
which logpx0 = logpµ ) is explicitly p-dependent. x4 =3.141213... x5 =3.141513...
Thus, if x is the fixed point corresponding to x , x
∞ 0 ∞ ... x = π
attracts all initial conditions in the semiopen interval ∞
[p(cid:98)logpx0(cid:99),p(cid:98)logpx0(cid:99)+1) but there is also a fixed point whereitisclearthattheconvergencetothefixedpointπ
pnx foranyn∈Zwhichattractsallinitialconditionsin is achieved by replacing at each time step t a less signifi-
∞
the interval [pn+(cid:98)logpx0(cid:99),pn+(cid:98)logpx0(cid:99)+1). The existence cantdigitofx0 bythecorrespondingoneofπ (thedigits
of these fixed points is a consequence of the scale invari- that have been replaced are written in bold above). For
ance implied by Eq. (20). FromEq. (29)wealsoobserve p = 2, we have, however, the sequence of iterates given
that the larger p, the wider the basins of attractions of by
the fixed points. We also note that for x = 0, x = 0
independently of µ since (cid:4)log 0(cid:5)=−∞.0 ∞ x0 =1.414213... x1 =1.914213...
p
x =1.664213... x =1.539213...
2 3
InFig. 1theevolutionofx isplottedforinitialcondi- x =1.601713... x =1.570463...
t 4 5
tions x ∈ [1,9900] as obtained from Eq. (26) for µ = π
0 ... x = π/2
∞
and for p = 2 (left) and p = 10 (right). We observe the
existence of fixed points for the values predicted by Eq. as predicted by the above development. Although it is
(29) and the observation made above that the larger p, not apparent that this latter evolution is a digit replace-
thewidertheirbasinsofattraction. Intherangeofinitial ment dynamics, this is only so because we are giving the
conditions considered, there are 13 fixed points for p=2 numbersinthedecimalradix. Hadwechosenthebinary
and four fixed points for p = 10. As we see, indepen- radix for the representation of the above numbers, we
dently of the value of p, µ=π is always a fixed point for wouldthenseethat,indeed,digitsofx (thatareall0or
t
(cid:4) (cid:5) (cid:4) (cid:5)
the range of initial conditions log x = log µ (the 1) are being gradually replaced by those of µ (from the
p 0 p
thickness of the basin of attraction of the fixed point be- most to the less significant places). Thus, although the
ingp-dependent)asobtainedfromthetheoryabove. We choice of the radix is immaterial for the numbers them-
also observe that the smaller p, the slower the conver- selves (a number is ‘always the same’ regardless of the
gencetothefixedpoints. Thisisclearfromthefactthat radix used) the radix p, in which the dynamics governed
for lower p and constant x , more digits are needed to by Eq. (25) takes place, constitutes the optimal repre-
0
render x up to a certain precision and the number of sentation15 to understand what is going on.
0
such significant digits is directly proportional to the du- Eq. (25) and the above mathematical development
rationofthetransient. Theevolutiontoµ,startingfrom finds a physical application in the problem of hierarchi-
an arbitrary x generally converges in a non-monotonic, cal deposition of debris on a surface, in which grains of
0
5
FIG. 2. Fourtimestepsofthespatiotemporalevolutionxt(u,v)obtainedfromEq. (31)forp=3R=6453u∈[0,240]andv∈[0,240]. Two
differentviewsofthedevelopingsurfaceareshown,startingfromacleansurfaceatt=0.
different sizes are deposited in an order related to their Then, in the next time step smaller grains with area
size, from larger to smaller grains16–19. Here, the depo- p(cid:98)logpbp(u,v;R)(cid:99)−1 × p(cid:98)logpbp(u,v;R)(cid:99)−1 are deposited.
sition takes place according to a certain rule in which The process goes on from the bigger to the smaller
grains of bigger size deposit first and a next genera- scales until convergence to the fractal object b (u,v;R)
p
tion of grains, which are smaller by a factor ∼ 1/p3, is achieved. More significant digits of b (u,v;R) have
p
deposit next. We first construct a model for µ, that a longer spatial variability as we prove in the coarse
characterizes the objective attractor. We have recently graining theorem recently reported in9.
found9, that if u and v denote spatial coordinates on In Figure 2 this process of fractal surface growth is
the plane R2 then, for R ∈ [0,pp2 −1] an integer and shown, asobtainedfromEq. (31)forforp=3R=6453
kmax ≡max{(cid:98)logpu(cid:99),(cid:98)logpv(cid:99)}, the function given by u ∈ [0,240] and v ∈ [0,240]. Four time steps of the spa-
tiotemporal evolution x (u,v) are shown (the time step
t
k(cid:88)max is indicated on each panel), giving two different views of
bp(u,v;R)= pkdp(dp(k,u)+pdp(k,v),R) (30) thesamesurface. Weobserveafractalsurfacedetermin-
k=−∞ isticallygrowingfromthecoarsertothefinerdetails. Al-
thoughwehaveconsideredadigitreplacementdynamics
is a surface with fractal self-affine properties9. Thus, if
from most to less significant digits, it is also possible to
westartfroma‘clean’surfacex (u,v)=0itisnowclear,
0 select at random the digit to be replaced and the attrac-
fromthediscussiongivenabove, thatthissurfacewillbe
tor will be the same, after a sufficiently large number of
an attractor of the dynamics given by the map time steps. In combination with our results in9 this pro-
(cid:0)(cid:4) (cid:5) videsanexplanationofwhyiteratedfunctionssystems20
x (u,v)=R log b (u,v;R) −t,
t+1 p p p generally lead to fractal attractors.
(cid:4) (cid:5)
log b (u,v;R) −t;
p p
x (u,v),b (u,v;R)) (31)
t p
B. Periodic orbits
which corresponds to Eq. (25) with µ = b (u,v;R) and
p
with x having now a spatial dependence x (u,v) on the
t t Let us now consider j and k given by the following
real coordinates u and v. We also fix, in Eq. (25), t t
(cid:4) (cid:5) (cid:4) (cid:5) (cid:4) (cid:5) simple expressions
log x = log x = log b (u,v;R) . Thus, from
p t p 0 p p
Eq. (29), we observe that if we start from a ‘clean’ sur- j ≡(cid:4)log x (cid:5) (33)
t p t
face x (u,v)=0 the map shall converge to
0 (cid:4) (cid:5)
k ≡ log µ −d (0,t) (34)
t p n
x (u,v)=b (u,v;R) (32)
∞ p in which case, Eq. (19), reduces to
for any point (u,v) on the plane. This convergence (cid:0)(cid:4) (cid:5) (cid:4) (cid:5) (cid:1)
x =R log x , log µ −d (0,t);x ,µ (35)
proceeds as follows. On the clean surface, in the t+1 p p t p n t
first iteration the first grains are deposited, with
At each time step the most significant digit of x is re-
area p(cid:98)logpbp(u,v;R)(cid:99) × p(cid:98)logpbp(u,v;R)(cid:99) and height placed by a digit of µ that is d (0,t) digits away (tto less
n
p(cid:98)logpbp(u,v;R)(cid:99)dp(dp(kmax,u)+pdp(kmax,v),R). significant places) from the most significant digit of µ,
6
FIG. 3. Evolution of xt obtained from the map Eq. (35) for p=9 (left) and p=10 (right), for µ=2359765 and n=7, starting from initial
conditionsx0∈[1,9900]. Notethelogarithmicscaleonthext axis.
(cid:4) (cid:5)
given by log µ . We note that, at every time, only one makingthetransformationt→t+nandobservingthat,
p
of the n most significant digits of µ are used to replace since d (0,t−1+n) = d (0,t−1), this implies, from
n n
the most significant digit of x . This is so because of Eq. (37) that x = x ∀t ≥ 1. Thus, for example, for
t t t+n √
thedigitfunctionbeingperiodic,asmadeexplicitbyEq. b = 27, p = 10 and starting from x = 2, Eq. (35)
0
(3), and one has produces, for p=10 the iterates
dn(0,t+n)=dn(0,t) (36) x0 =1.41... x1 =2.41... x2 =7.41...
x =2.41... x =7.41... ...
There are thus two possibilities: 1) any of the n most 3 4
significant digits of µ is equal to zero when written in
In Fig. 3, the evolution of the map Eq. (35) is plotted
radix p, in which case x converges to zero; 2) none of
t for p = 9 (left) and p = 10 (right), for µ = 2359765 and
the n most significant digits of µ when written in radix
n=7. InbothcasestheorbitshaveperiodT =n=7for
p is zero, in which case, because of Eq. (36) the orbit of
all initial conditions. Although the period is the same,
x has period n as well.
t the shape of the oscillations is affected by changing p.
The case 1) is easy to understand. For, if any of the
Note that for p=10 the digits of µ=2359765 from the
n most significant digits of µ in radix p is zero, after
most significant to the less significant obey: 2 < 3 <
n time steps the most significant digit of x has been
(cid:4) t(cid:5) 5 < 9, 9 > 7 > 6 > 5. As a consequence, the shape
once replaced by zero and as a result log x becomes
p t of the oscillation consist in each period of two different
more negative. This means that the leading part of x
t monotonicparts,anincreasingoneandadecreasingone.
decreases in a factor of p. If this process is repeated an
(cid:4) (cid:5) For p = 9, however, since µ = 2359765 is written as
infinite number of times log x = −∞ which means
p t µ=(4385881) in radix 9, we have that the shape of the
that the leading partis proportional to p−∞, i.e. that x 9
t oscillation consists of 5 different monotonic parts 4 > 3,
has converged to zero.
3 < 8, 8 > 5, 5 < 8 = 8, 8 > 1. In both cases we
We now concentrate on the more interesting case 2) in
observe the existence of an infinite number of periodic
which none of the n most significant digits of µ is zero.
attractors a distance h ∼ p(cid:98)logpx0(cid:99)(p − 1) apart, each
Theorbitcanagainbetriviallysolvedinthissimplecase,
consisting of a continuous band of periodic orbits which
and has for any t≥1 the form
is d ∼ p(cid:98)logpx0(cid:99) thick. The existence of such periodic
(cid:0)(cid:4) (cid:5) (cid:4) (cid:5) (cid:1)
xt =Rp logpx0 , logpµ −dn(0,t−1);x0,µ + attractors is a consequence of the scale invariance of the
replacement operator, made explicit by Eq. (20).
(cid:98)logpx0(cid:99)−1
(cid:88) Wecannowmoveastepfurtherandconsideraslightly
+ p−jd (−j,x ) (37)
p 0 moreinterestingdynamicsthatiseasilyunderstoodfrom
j=0
the above. Let m ≥ 1 and n ≥ 2 be both finite natural
since all time dependence affects only the most signifi- numbers. We take
cant digit of xt accompanying a power p(cid:98)logpx0(cid:99) of the j ≡(cid:4)log x (cid:5)−d (0,t) (38)
radix and all other digits are left invariant. This time t p t m
(cid:4) (cid:5)
dependence has period n for all t > 0 as can be seen by k ≡ log µ −d (0,t) (39)
t p n
7
FmI=G.1+4.|(cid:4)lToegmppxo0r(cid:5)a|l. eNvootleuttihoenloogfaxritthombticaisnceadleforonmthtehxetmaxapis.Eq. (19) for p = 9 (left) and p = 10 (right), for µ = 2359765, n = 7 and
Again, we have two cases depending on whether any of The thickness d calculated previously (see Fig. 3) is now
thenmostsignificantdigitsofµinradixpiszeroornot. equal to d∼p(cid:98)logpx0(cid:99)−m+1 =1.
The outcome of the former case is that x converges to
t In Fig. 4 the evolution of x as obtained from Eqs.
t
zero for a sufficiently long time, as before. We assume
(19), (40), (41) and (42) for p = 9 (left panel) and
thus the latter case in what follows. The resulting dy-
p = 10 (right panel), for µ = 2359765, n = 7 and
namical evolution depends now on two periods n and m (cid:4) (cid:5)
m=1+| log x | is shown. Besides the oscillations of
p t
which lead, after a transient, to a periodic orbit, where
period T = 7 for initial conditions x ∈ [1,p), which are
0
theperiodT =lcm(m,n)istheleastcommonmultipleof
similar to those in Fig. 3, oscillations with different pe-
m and n. Note that Eq. (19) is trivially invariant under
riodicitiesT =14,21,28, arenowfoundforsetsofinitial
the transformation t→t+lcm(m,n) in this case. Again conditionsx ∈[p,p2), x ∈[p2,p3)andx ∈[p3,p4), re-
0 0 0
wefindattractorswithasimilarqualitativedynamicsal-
spectively. Indeed, we have that T = lcm(m,n) for the
beit with period T =lcm(m,n), as a consequence of the set of initial conditions x ∈ [pm,pm+1) and these ob-
0
scale invariance, Eq. (20).
served periodicities correspond to the fact that m is now
dependent on the initial condition through x . Differ-
t
entcolorshavebeenchoseninFig. 4toindicatethenow
C. Distinct attractors in phase space with same qualitativelydifferentbasinsofattractionandthebehav-
qualitative dynamics ior of the initial conditions contained in them, thus em-
phasizing the breaking of the scale invariance. Although
Letusnowbreaktheabovementionedscaleinvariance the thickness d ∼ p(cid:98)logpx0(cid:99)−m+1 = 1 is independent of
of Eq. (20) so that the attractors become distinct. The the initial condition, the fact that d is finite and does no
moststraightforwardwaytoachievethisistomakemin longer scale with p(cid:98)logpx0(cid:99) is revealed in the logarithmic
Eq. (38) dependent on the order of magnitude of x i.e.
t scale of x in Fig. 4 at low powers of the radix.
(cid:4) (cid:5) t
on log x . Since m ≥ 1 the easiest way to break the
p t
invariance is to take
D. Aperiodic nonchaotic attractors
(cid:4) (cid:5)
j ≡ log x −d (0,t) (40)
t p t m
(cid:4) (cid:5)
kt ≡ logpµ −dn(0,t) (41) If n=∞, in Eq. (39), we have, by using Eq. (6)
m≡1+|(cid:4)logpxt(cid:5)| (42) jt ≡(cid:4)logpxt(cid:5)−dm(0,t) (43)
(cid:4) (cid:5)
We then note that if xt is finite and its evolution is kt ≡ logpµ −t (44)
boundedtoaninterval[pk,pk+1)thenm=kisconstant,
and Eq. (19) reduces to
and the orbit has periodicity lcm(m,n) = lcm(k,n).
(cid:0)(cid:4) (cid:5) (cid:4) (cid:5) (cid:1)
Therefore, since m depends on x we find that the dy- x =R log x −d (0,t), log µ −t;x ,µ
t t+1 p p t m p t
namics now has attractors with different periodicities. (45)
(cid:4) (cid:5)
We note that, since m=1+| log x |=1 for any ini- Wenowconsiderµanirrationalnumber(forµarational
p t
tial condition x ∈ [1,p), Eq. (19) reduces to Eq. (35). number, the dynamics can be shown to be equivalent to
0
8
theonedescribedinSectionB,sinceµhasthenaperiod the distance between close orbits within the attractor be-
of digits after the radix point that repeats infinitely). ing constant even when the orbit themselves are all ape-
We have to distinguish two cases: (1) if µ has N digits riodic. Thisissoby mathematical design. Theconstancy
equal to zero after the radix point when written in radix of the distance shall be rigorously proved below (see Eq.
p, x can decay up to a factor p−N in the course of its (56)).
t
trajectorysince,eachtimesuchazeroisencounteredand Before we give a detailed example of this behavior,
used to replace the most significant digit of x we have we mention that the recipe in Section IIIC to break the
t
(cid:4) (cid:5) (cid:4) (cid:5)
log x = log x −1, i.e. the position of the most scaling invariance also applies in this case if we make m
p t+1 p t
significant digit goes to the next less significant place; to depend on t so that the dynamics is now given by
(2) if µ does not contain any digit equal to zero when
(cid:4) (cid:5) (cid:4) (cid:5) (cid:0)(cid:4) (cid:5) (cid:4) (cid:5) (cid:1)
written in radix p, we have logpxt = logpx0 i.e., the xt+1 =Rp logpxt −dm(0,t), logpµ −t;xt,µ
position of the most significant digit of x is constant, (49)
t
the dynamics is aperiodic and bounded to the interval with
[p(cid:98)logpx0(cid:99),p(cid:98)logpx0(cid:99)+1)andthereexistaninfinitenumber (cid:4) (cid:5)
j ≡ log x −d (0,t) (50)
of different aperiodic attractors that are separated and t p t m
(cid:4) (cid:5)
are not sensitive to nearby initial conditions. This is kt ≡ logpµ −t (51)
easily understood by taking the limit n → ∞ from the (cid:4) (cid:5)
m≡1+| log x | (52)
p t
casediscussedaboveandillustratedinFig. 4. Theperiod
of the oscillations is given by T =lcm(m,n) and when n In this way, all initial conditions contained in each sepa-
tendsto∞sodoesT. Thedynamicsisasfollows. Them rate interval [ph,ph+1) with h∈Z converge to a distinct
mostsignificantdigitsofx arecyclicallyreplacedbyless separate attractor.
t
andlesssignificantdigitsofµ. Therefore,ifµisirrational Let us illustrate all above with a particular example.
and does not contain any nonzero digit this process does We focus first on the former case, where µ may contain
not end on a finite time and, since less significant digits zero digits when written in radix p. We take µ=π
ofµreplacethemostsignificantdigitsofx thetemporal
t
evolution of x is necessarily aperiodic. π =3.1415926535897932384626433832795028841971693
t
Tofindvaluesofµforwhichnodecaytozerohappens 993751058209749445923078...
and one is left with an evolution that is forever aperi-
and we see that for p = 10 there are digits equal to
odic, let α be any arbitrary irrational number. Then,
zero accompanying powers of the radix 10−32, 10−50,
the number µ(α) defined by
10−54, 10−64, etc. Thus, we expect abrupt decays of
those trajectories that satisfy d (0,t) = 0, at
µ(α)≡ p1+(cid:98)logpα(cid:99) +(cid:98)lo(cid:88)gpα(cid:99)d (j,α)pj (46) t=32,50,54,64,.... 1+|(cid:98)logpxt(cid:99)|
p−1 p−1 In Figure 5 (left), the evolution of x obtained from
j=−∞ t
Eq. (49) is plotted for the initial conditions in the set
(cid:4) (cid:5)
x = [1,9900], m = 1 + | log x | and µ = π. We
is generally irrational as well (for p > 2) and does not 0 p 0
see that, although aperiodic trajectories seem to attract
contain any digit equal to zero when written in radix p.
subsets of the initial conditions, some of these trajec-
To see this note that
tories abruptly decay at t = 32,50,53 and 64 which
µ≡ p1+(cid:98)logpα(cid:99) +(cid:98)lo(cid:88)gpα(cid:99)d (j,α)pj (47) ajercetotrhioessectoilmoreesdprbelduiect(ewdhaobseovien.itFiaolrceoxnadmitpiolen,sthbeeltornag-
p−1 p−1 to the interval [103,104)) satisfy d (0,t) =
j=−∞ 1+|(cid:98)logpxt(cid:99)|
d (0,32) = 0 at t = 32 and, therefore, they decay by
(cid:98)log α(cid:99) (cid:98)log µ(cid:99) 4
p p
(cid:88) (cid:88) a factor p and at t = 33 meet the red curves in the
= (1+d (j,α))pj = d (j,µ)pj
p−1 p lower level. The latter remain unaffected since they
j=−∞ j=−∞
satisfy d (0,t) = d (0,32) = 2. At a later
1+|(cid:98)logpxt(cid:99)| 3
wherewehaveusedthat p1+(cid:98)logpα(cid:99) =(cid:80)(cid:98)logpα(cid:99)pj. Thus, time, t = 54, both blue and red trajectories decay to-
we find that µ has digits p−1 j=−∞ gether since now d1+|(cid:98)logpxt(cid:99)|(0,t) = d3(0,54) = 0. All
crossings and complex transitions can thus be explained
(cid:4) (cid:5) through the theory presented above. Since π may surely
d (j,µ)=1+d (j,α) j ∈(−∞, log µ ] (48)
p p−1 p contain an infinite number of digits equal to zero (not
regularly spaced), we can safely say that all trajectories
and since d (j,α) is an integer which obeys 0 ≤
p−1 decay to zero after a sufficiently long time.
d (j,α)≤p−2 then 1≤d (j,µ)≤p−1, and, there-
p−1 p Let us now construct an irrational number by means
fore, no digit of µ is equal to zero when written in radix
of Eq. (47) such that we can explore the case (2) above
p. That value of µ leads Eq. (45) to produce an aperiodic
in which the trajectories do not decay to zero but x
sequence of real numbers which is bounded from above (cid:104) (cid:17)t
and from below and which is contained in an attractor, staysboundedwithinaninterval p(cid:98)logpx0(cid:99),p(cid:98)logpx0(cid:99)+1
9
FIG. 5. First100iteratesofEq. (49)forp=10andµ=π(left)andµ=µ(π)(right),thelattercalculatedfromEq. (47)forα=πandinitial
conditionsx0∈[1,9900].
forever. From Eq. (47) by taking α = π we find the with s sufficiently small. Here we have used Eq. (1)
irrational number to expand |x −x(cid:48)| in radix p thus making clear that
0 0
the digits of the initial conditions x and x(cid:48) are equal
0 0
up to s significant digits after the radix point. Since
m is equal for both trajectories that are infinitesimally
µ(π)=4.25261376469181434957375449438161399521827
separated because m=1+|(cid:4)log x (cid:5)|=1+|(cid:4)log x(cid:48)(cid:5)|
14114862169311851556134189... (53) p 0 p 0
and since m is finite, positive, and non-vanishing, it is
enough to take s>m. The dynamics only affects the m
whichdoesnotcontainanyzerodigitforp=10. Wenow most significant digits of x(cid:48) and x and leaves all other
t t
consider the temporal evolution of xt taking this latter unaffected. Furthermore, and crucially, each single digit
number µ(π) as parameter. In Figure 5 (right) xt ob- of xt and x(cid:48)t that is being replaced at each time t is at
tained from Eq. (49) is plotted for the initial conditions the same position relative to the radix point for both
(cid:4) (cid:5)
in the set x0 = [1,9900], m = 1+| logpx0 |, p = 10 numbers, and it is replaced by the same digit of µ (i.e.
and µ=µ(π). Stable aperiodic oscillations are shown to the k -th digit of µ, as given by Eq. (51) at each t).
t
attract connected sets of initial conditions. The attrac- Consequently, the difference of the contributions coming
tors are all qualitatively distinct, by virtue of the scaling from these m digits to |x −x(cid:48)| cancel out and we are
t t
symmetry breaking, and each displays a distinct aperi- only left with
odic behavior. The trajectory exhibits more abrupt and
sudden changes for m = 1+|(cid:4)logpx0(cid:5)| close to unity ∆ ≡|x −x(cid:48)|= −(cid:88)s−1 pkd (k,|x −x(cid:48)|)
(i.e. for x0 ∈ [1,p)) and is coarser for x0 larger. We t t t p t t
observe that there is no exponential sensitivity to initial k=−∞
conditions even when the resulting dynamics is forever −s−1
(cid:88)
aperiodic: all close initial conditions within an interval = pkdp(k,|x0−x(cid:48)0|)=∆0 (55)
(cid:104) (cid:17)
p(cid:98)logpx0(cid:99),p(cid:98)logpx0(cid:99)+1 are attracted to a thin band of k=−∞
aperiodic trajectories of thickness d∼1 where they keep at every time. Therefore, the Lyapunov exponent
a cWonestcaannt,diinstdaenecde,.rigorously prove that the distance λ= lim 1ln ∆t = lim 1ln(cid:80)k−=s−−1∞pkdp(k,|x0−x(cid:48)0|)
of trajectories starting infinitesimally close remains con- t→∞ t ∆0 t→∞ t (cid:80)k−=s−−1∞pkdp(k,|x0−x(cid:48)0|)
stant: The Lyapunov exponent of the map is zero for =0 (56)
every possible value of m in Eq. (52) as we proceed to
show. Letx andx(cid:48) betwodifferenttrajectoriesstarting which means that the attractor, in spite of being ape-
t t
from initial conditions x and x(cid:48) that are infinitesimally riodic, is not chaotic. The above proof is equally valid
0 0
for the dynamics described in Sections B and C. For the
close. In practice, this means that we can take
dynamics in Section A we, trivially, find λ = −∞, since
∆ → 0 as t → ∞ as a consequence of the existence of
−s−1 t
(cid:88)
∆ ≡|x −x(cid:48)|= pkd (k,|x −x(cid:48)|)<p−s (54) a stable fixed point given by Eq. (29) which is the same
0 0 0 p 0 0
for both trajectories.
k=−∞
10
We must emphasize the importance of µ being irra- subsystem within a composite system, the aperiodic at-
tional for the dynamics above being aperiodic. Because tractors are synchronized very much like in reference27,
ifµwererational,thenitwouldhaveaninfinitelyrepeat- where synchronization of SNAs is discussed.
ingperiodof,say,a digitsaftertheradixpoint,andthe Although we must conclude here our discussion of Eq.
0
dynamics would trivially reduce to the ones described in (19)itshouldnowbeclearthatifµisnolongeraparam-
Sections B and C (with n=a finite). eter but a function of x as well, we can have chaotic at-
0 t
Let us summarize the results of the above analysis, all tractorsasaresult. Alreadyifwesimplyreplaceαbyx
t
trivial consequences of our construction: inEq. (47)andµbyµ(x )inEq. (45)or(49),theresult
t
is an aperiodic dynamics that is exponentially sensitive
• The dynamics is aperiodic on the attractor. to initial conditions: The closer two different initial con-
ditions x and x(cid:48) are, the longer the trajectories evolve
0 0
• The Lyapunov exponent is zero.
together. However, as soon as the corresponding digits
of x and x(cid:48) that are being used to replace the m most
• This behavior is possible only if µ is an irrational t t
significant digits of x and x(cid:48) begin to differ, the trajec-
number. If µ is rational, the dynamics is periodic. t t
tories begin to depart aperiodically from each other and
one can show that the Lyapunov exponent is positive in
Althoughwearenotestablishingherethefractalchar-
this case. In the long term, the value of x(cid:48) cannot be
acter of the aperiodic attractors, we must remark the t
estimatedfromthevalueofx evenwhenbothx andx(cid:48)
analogy of the behavior that we have described with t t t
were initially close.
the one found in so-called strange nonchaotic attractors
(SNAs)21–29, which are driven by a forcing consisting of
anincommesurateratiooffrequencies(thisratiobeingan
irrationalnumberthatplaysananalogousroletoµhere). IV. CONCLUSIONS
ASNAisafractalstructureattractingthetrajectoriesin
phasespacesothatthedynamicsofthesystemisnotex- In this work we have applied methods of digital
ponentially sensitive to initial conditions (its Lyapunov mathematics8,9,14,15toconstructagenericnonlinearmap
exponent being non-positive). The existence of SNAs whichoperatesbyreplacingdigitsofthenumericalvalue
was first pointed out in a seminal paper by Grebogi, of a dynamical variable x by those of a control pa-
t
Ott, Pelikan and Yorke21 and have been widely studied rameter (or signal) µ. By considering increasingly com-
since22–29. Experimentally,SNAswerefirstdiscoveredin plex dynamics, we have shown how our map can be
thebucklingofamagnetoelasticribbondrivenquasiperi- used to mathematically design aperiodic non-chaotic at-
odically by two incommensurate frequencies with the tractors. We have explicitly shown how this map can
golden ratio relationship30. Since then they have also be understood as a discretization of any arbitrary non-
been reported in other laboratory experiments involving autonomous map by working out in detail a mathemati-
electrochemical cells31, electronic circuits32, and a neon cal method to achieve this connection in general. Then,
glow discharge33. Recently, based on the unprecedented we have investigated the most simple cases of digit re-
light curves of the Kepler space telescope, SNAs have placement dynamics. Although the notation and the
also been observed for the first time out of the labora- functions introduced here may seem unfamiliar, we be-
toryinthedynamicsoftheRRcLyraestarKIC5520878, lieve that the nonlinear behavior obtained from the map
a blue-white star 16 000 light years from Earth in the issimplertounderstandthanthecomplexbehavioraris-
constellation Lyra34. ing from polynomial normal forms and maps involving
The temporal dynamics exhibited by the trajectories transcendental functions. We have explicitly shown how
in Fig. 5 (right) on any of the attractors (differently limit cycle oscillations and aperiodic behavior can be
colored) seems very similar to the time series found in mathematicallydesigned, showingtheadvantagesofthis
SNAs, as for example the ones obtained (in continuous formulation.
time) with a modified driven pendulum equation that The theoretical results presented in the manuscript
has been used to model a driven SQUID with inertia may be reproducible in experiments with nonlinear elec-
and damping32, see also Fig 1c in28. We must however tric circuits. The digit function can be obtained in
remark that the dynamics is here time-discrete and that practice from a certain superposition of square waves or
the Lyapunov exponent is zero. from a sawtooth signal appropriately quantized. Thus,
If we set m constant, independent of x (i.e. x ) in a circuit element performing the replacement operator
t 0
Eq. (52), we recover back the scaling invariance and the may be constructed and the electric current through it
above model reduces to Eqs. (43), (44) and (45) and all may be shown to reproduce the results presented in the
aperiodic attractors in phase space become synchronized manuscript.
inphase(whichinthiscontextmeansthat, althoughthe Although all physical numbers are radix independent,
signalsx havedifferentvalue,alltheirlocalmaximaand they are all always represented in a specific radix (e.g.
t
minima are locked and synchronous in time). Thus, in the usual decimal radix). We have here thus constructed
this situation, if one thinks on each separate aperiodic a new class of dynamical systems by regarding the pos-
attractor as reflecting the evolution of an independent sibility that the digits of the numbers are being replaced