# Nonlinear dynamical systems: Time reversibility versus sensitivity to the
initial conditions

Constantino Tsallis1,2,3<EMAIL_ADDRESS>Ernesto P. Borges4<EMAIL_ADDRESS>1Centro Brasileiro de Pesquisas Fisicas
and National Institute of Science and Technology for Complex Systems
Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
2 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
3 Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria
4Instituto de Fisica, Universidade Federal da Bahia
and National Institute of Science and Technology for Complex Systems
Salvador-BA 40170-115, Brazil

###### Abstract

Time reversal of vast classes of phenomena has direct implications with
predictability, causality and the second principle of thermodynamics. We
analyze in detail time reversibility of a paradigmatic dissipative nonlinear
dynamical system, namely the logistic map $x_{t+1}=1-ax_{t}^{2}$. A close
relation is revealed between time reversibility and the sensitivity to the
initial conditions. Indeed, depending on the initial condition and the size of
the time series, time reversal can enable the recovery, within a small error
bar, of past information when the Lyapunov exponent is non-positive, notably
at the Feigenbaum point (edge of chaos), where weak chaos is known to exist.
Past information is gradually lost for increasingly large Lyapunov exponent
(strong chaos), notably at $a=2$ where it attains a large value. These facts
open the door to diverse novel applications in physicochemical, astronomical,
medical, financial, and other time series.

###### pacs:

05.45.-a,05.45.Ac,05.45.Tp

The arrow of time has crucial implications in relevant issues related to
causality, validity of the second principle of thermodynamics, reconstruction
of a specific trajectory of a system, among others (see Zadunaisky1979  and
references therein). For example, conservative Newtonian systems are
deterministic and reversible with time. Indeed, the equation of motion
$\vec{F}(\vec{x})=m\frac{d^{2}\vec{x}}{dt^{2}}$ with given arbitrary initial
conditions $(\vec{x}(0),\dot{\vec{x}}(0))$ yields an unique solution which is
invariant through time reversal $t\to-t$. More precisely, assuming that all
calculations are performed with infinite precision, we may reverse time at any
moment including of course the sign change of all velocities $\dot{\vec{x}}$,
and we obtain backwards the same trajectory that we had followed forward.

A different behavior is observed for paradigmatic nonlinear dynamical systems
such as the dissipative logistic map

$x_{t+1}=1-ax_{t}^{2}\;\;(a\in[0,2];\,x_{t}\in[-1,1];\,t=0,1,2,\dots)\,.$ (1)

Similarly to the previous Newtonian system, this map is deterministic in the
sense that, once again assuming that we perform all calculations with infinite
precision, the trajectory within the interval $[1,-1]$ is unique given an
arbitrary initial condition $x_{0}$. However, Eq. (1) is not time-reversible.
Indeed, for any given $x_{t+1}$ we obtain two, and not one, values for
$x_{t}$.

For both classes of systems, the concept of sensitivity to the initial
conditions is, nevertheless, applicable. Indeed, the time evolution of a pair
of different, infinitely close, initial conditions might exhibit (i)
exponential divergence with time (i.e., positive Lyapunov exponents), thus
causing strong chaos; (ii) exponential convergence with time (i.e., negative
Lyapunov exponents), thus characterizing stable orbits; (iii) subexponential
evolution with time (i.e., zero Lyapunov exponents). The latter corresponds,
in many occasions, to the edge of chaos, thus reflecting weak chaos. The
corresponding subexponential divergence with time is present in many natural,
artificial and social complex systems and, for vast classes of phenomena, it
is of the power-law type.

In the present paper we reveal a — to the best of our knowledge — new
connection between time reversibility and sensitivity to the initial
conditions. We focus on the logistic map as an illustrative case. Let us start
with two paradigmatic values of the control parameter $a$, namely $a=2$
(occasionally referred to as the Ulam point), whose Lyapunov exponent
$\lambda$ is given by $\lambda=\ln 2>0$, thus corresponding to strong chaos,
and $a=a_{c}\equiv 1.40115518909205\dots$ (Feigenbaum-Coullet-Tresser point),
whose Lyapunov exponent $\lambda$ vanishes, corresponding to weak chaos.

For the strong chaos case, the Kolmogorov-Sinai-like entropy production rate
is characterized by the entropic functional introduced in the pioneering works
of Boltzmann and Gibbs Boltzmann1872 ; Boltzmann1877 ; Gibbs1901

$S_{BG}=-k\sum_{i=1}^{W}p_{i}\ln
p_{i}\;\;\Bigl{(}\sum_{i=1}^{W}p_{i}=1\Bigr{)}\,,$ (2)

$k$ being a conventional positive constant adopted once forever (in physics,
$k$ is chosen to be the Boltzmann constant $k_{B}$; in information theory and
computational sciences, it is frequently adopted $k=1$ and logarithm to the
base 2). Here we work with discrete random variables, but the arguments can
straightforwardly be translated for continuous cases.

In the simple case of equal probabilities, this entropic functional is given
by $S_{BG}=k\ln W$. Moreover, Eq. ( 2) is generically additive Penrose1970 .
Indeed, if $A$ and $B$ are two probabilistically independent systems (i.e.,
$p_{ij}^{A+B}=p_{i}^{A}p_{j}^{B}$), we straightforwardly verify that

$S_{BG}(A+B)=S_{BG}(A)+S_{BG}(B)\,.$ (3)

This celebrated entropic functional is consistent with thermodynamics for all
systems whose $N$ elements are either independent or weakly interacting in the
sense that only basically local (in space/time) correlations are involved. For
example, if we have equal probabilities (typical microcanonical ensemble) and
the system is such that the number of accessible microscopic configurations is
given by $W(N)\propto\mu^{N}\;(\mu>1;\,N\to\infty)$, then $S_{BG}(N)$ is
extensive as required by thermodynamics. Indeed $S_{BG}(N)=k\ln W(N)\sim
k(\ln\mu)N$. But if the correlations are nonlocal in space/time, $S_{BG}$ may
become thermodynamically inadmissible. Such is the case of equal probabilities
with say $W(N)\propto N^{\nu}\;(\nu>0;\,N\to\infty)$: it immediately follows
$S_{BG}(N)\propto\ln N$, which violates thermodynamical extensivity. To
satisfactorily approach cases such as this one, it was proposed in 1988
Tsallis1988 ; TsallisMendesPlastino1998 ; GellMannTsallis2004 ; Tsallis2009
to build a more general statistical mechanics based on the nonadditive
entropic functional

$S_{q}\equiv
k\frac{1-\sum_{i=1}^{W}p_{i}^{q}}{q-1}=k\sum_{i=1}^{W}p_{i}\ln_{q}\frac{1}{p_{i}}=-k\sum_{i=1}^{W}p_{i}^{q}\ln_{q}p_{i}=-k\sum_{i=1}^{W}p_{i}\ln_{2-q}p_{i}\;\;(q\in\mathbb{R};S_{1}=S_{BG})\,,$
(4)

with $\ln_{q}z\equiv\frac{z^{1-q}-1}{1-q}\;(\ln_{1}z=\ln z)$ and its inverse
$e_{q}^{z}\equiv[1+(1-q)z]_{+}^{1/(1-q)}$; $(e_{1}^{z}=e^{z}$; $[z]_{+}=z$ if
$z>0$ and vanishes otherwise); for $q<0$, it is necessary to exclude from the
sum the terms with vanishing $p_{i}$. We easily verify that equal
probabilities yield $S_{q}=k\ln_{q}W$, and that generically we have

$S_{q}(A+B)=S_{q}(A)+S_{q}(B)+\frac{1-q}{k}S_{q}(A)S_{q}(B)\,.$ (5)

Consequently, in the $(1-q)/k\to 0$ limit, we recover the $S_{BG}$ additivity.
For the anomalous class of systems mentioned above, namely $W(N)\propto
N^{\nu}$, we obtain, $\forall\nu$, the extensive entropy
$S_{1-1/\nu}(N)=k\ln_{1-1/\nu}W(N)\propto N$, as required by the Legendre
structure of thermodynamics (see Tsallis2009 ; TsallisCirto2013 ; Tsallis2022
and references therein).

Let us remind at this stage the connection between the entropy production rate
$K_{BG}$ and the Lyapunov exponent $\lambda_{1}$. For the $a=2$ logistic map,
the following Pesin-like identity holds LatoraBaranger1999

$K_{BG}\equiv\lim_{t\to\infty}\lim_{W\to\infty}\lim_{M\to\infty}\frac{\langle
S_{BG}\rangle(t)}{t}=\langle\lambda_{1}\rangle\,,$ (6)

where we have partitioned the phase space $x\in[-1,1]$ in $W\gg 1$ little
cells, and have started the map evolution by randomly putting $M\gg 1$
(typically $M=10W$) initial conditions within one of those cells, the
averaging being done over all $W$ cells; the Lyapunov exponent is defined
through the sensitivity to the initial conditions $\xi(t)\equiv\frac{\Delta
x(t)}{\Delta x(0)}=e^{\lambda_{1}\,t}$ .

Eq. (6) can be generalized as follows

$K_{q}\equiv\lim_{t\to\infty}\lim_{W\to\infty}\lim_{M\to\infty}\frac{\langle
S_{q}\rangle(t)}{t}=\langle\lambda_{q}\rangle\,,$ (7)

where $\lambda_{q}$ is defined through $\xi(t)=e_{q}^{\lambda_{q}\,t}[\sim
t^{1/(1-q)}$ if $q<1$] for $t\gg 1$ and $q\leq 1$. For the $a=a_{c}$ logistic
map, the following $q$-generalized Pesin-like identity holds
TsallisPlastinoZheng1997 ; LyraTsallis1998 ; SilvaCruzLyra1999 ;
LatoraBarangerRapisardaTsallis2000 ; BaldovinRobledo2002 ; AnanosTsallis2004 ;
BaldovinRobledo2004 :

$K_{q_{c}}=\langle\lambda_{q_{c}}\rangle\,,$ (8)

with $q_{c}=0.2445\dots$ ($10,026$ meaningful digits are presently known).

The finiteness of $K_{q}$ imposes a drastic change in the value of $q$ when we
contrast the $a=2$ case (for which we have a smooth $t\to\infty$ attractor
and, consistently, $q=1$) with the $a=a_{c}$ case (for which we have a
multifractal $t\to\infty$ attractor and, consistently, $q=q_{c}$). This fact
raises a crucial question, namely what happens with time reversibility? In
what follows we reveal a sensible difference between these two paradigmatic
cases.

We start with an arbitrarily chosen $x_{0}$ and iterate the map until arrival
to a final value $t_{f}$. We then consider

$x_{t}=\frac{x_{t}+x_{t_{f}-t}}{2}+\frac{x_{t}-x_{t_{f}-t}}{2}\equiv
S_{t}+A_{t}\,,$ (9)

where $S_{t}$ ($A_{t}$) is symmetric (antisymmetric) with regard to
$t\leftrightarrow(t_{f}-t)$. These quantities are depicted in Fig. 1, and we
easily appreciate the considerable difference of $A_{t}$ for $a=2$ and
$a=a_{c}$. Indeed, for $a=2$, the values of $A_{t}$ spread over the entire
phase space whereas, for $a=a_{c}$, $A_{t}$ can be strongly concentrated
around $A_{t}=0$ ! In other words, quasi-reversibility may occur for a
multifractal attractor, whereas by no means such remarkable property is
observed when the attractor includes smooth regions. The influence of $a$ on
various relevant quantities is depicted in Fig. 2.

Figure 1: (Color online) Top plots: Time dependence of $x_{t}$ and of
$x_{t_{f}-t}$ with $x_{0}=0.1$ and $t_{f}$=2000 for $a=2$ (black dots) and
$a=a_{c}\equiv 1.40115518909205\dots$ (red dots). Middle plots: Time
dependence of $S_{t}$ and $A_{t}$. Bottom plots: The value of $t_{f}$ has
relevant consequences, as illustrated here.

Figure 2: (Color online) Top: Influence, for $(x_{0},t_{f})=(0.1,2000)$, of
$a$ on $A_{t}$ [$A_{t}$ gradually (though non monotonically) spreads within
the entire phase space interval $[1,-1]$ for $a$ increasing from $a=a_{c}$ to
$a=2$]; Bottom: The influence of $a$ on $\langle S_{t}\rangle$ and $\langle
A_{t}\rangle$ for $(x_{0},t_{f})=(0.1,2000)$; for other values of
$(x_{0},t_{f})$, $\langle S_{t}\rangle$ and $\langle A_{t}\rangle$ remain
nearly the same.

Figure 3: (Color online) Influence of the initial condition on the standard
deviations $\sigma_{S}$ and $\sigma_{A}$ for fixed $t_{f}$.

Typical results of the influence of $(x_{0},t_{f})$ on the standard deviations
$(\sigma_{S},\sigma_{A})$ are depicted in Figs. 3 and 4.

Figure 4: (Color online) Top and Middle: Influence of $(t_{f},x_{0})$ on
$(\sigma_{S},\sigma_{A})$; for other values of $x_{0}$ the data follow similar
paths. Bottom: Illustration of the fact that, for exchanged values of $t_{f}$,
$\sigma_{S}$ and $\sigma_{A}$ can nearly coincide.

Let us conclude by reminding that classical mechanics is deterministic for
both time increasing forward or increasing backwards (see Zadunaisky1979  and
references therein). In contrast, other paradigmatic nonlinear dynamical
systems such as the dissipative logistic map are deterministic only for time
increasing forward, not so for time increasing backwards. We have here focused
on this map because of its relevance for many phenomena in natural sciences,
and because it is able to exhibit, in addition to stable orbits, both strong
(positive Lyapunov exponent) and weak chaos (vanishing Lyapunov exponent),
depending on the value of its external control parameter. These two classes of
chaos illustrate basic features of, respectively, Boltzmann-Gibbs statistical
mechanics (grounded on the additive entropic functional $S_{BG}$) and
nonextensive statistical mechanics (grounded on the nonadditive entropic
functional $S_{q}$). Consistently, the entropy production rate which appears
in the Pesin-like identity concerns $q=1$ for strong chaos (whose dynamical
attractor includes regions of smooth functions) and $q=q_{c}=0.2445\dots$ at
the Feigenbaum point (whose dynamical attractor is known to be a complex
multifractal). On the basis of these quite different dynamical behaviors, we
revealed here that quasi-reversibility with time, i.e. $x_{t_{f}-t}\simeq
x_{t}$, is possible when the Lyapunov exponent is nonpositive (at the
Feigenbaum point, for instance), whereas it is essentially lost when the
Lyapunov exponent is positive. This connection between a global time behavior,
namely reversibility, and the sensitivity to the initial conditions might be
very useful for analyzing issues such as predictability, causality and others
in physicochemical, astrophysical, economical and medical time series (e.g.,
EEG and ECG), among others. In fact, the practical usefulness of time-reversal
has been very recently exhibited by decreasing error bars in the predictions
of strong earthquakes Varotsosetal2023 . The study of how general this
phenomenon might be, and for what specific classes of low-dimensional and/or
high-dimensional nonlinear dynamical systems, certainly constitutes a deeply
interesting challenge, to be addressed in future efforts.

We acknowledge fruitful remarks by D.M. Abramov, E.M.F. Curado, H.S. Lima,
A.R. Plastino and R. Wedemann, as well as partial financial support by CNPq
and Faperj (Brazilian agencies).

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