# Evolving disorder and chaos induces acceleration of elastic waves

©M. Ahumada  Departamento de Física, Universidad Técnica Federico Santa María,
Casilla 110 V, Valparaíso, Chile.  L. Trujillo  Université de Lyon, Inria,
ECL, INSA Lyon, Université Claude Bernard Lyon 1, Université Lumière Lyon 2,
CNRS, LIRIS UMR 5205, F-69622, France  J. F. Marín<EMAIL_ADDRESS>Departamento de Física, Facultad de Ciencias Naturales, Matemática y del Medio
Ambiente, Universidad Tecnológica Metropolitana, Las Palmeras 3360, Ñuñoa
780-0003, Santiago, Chile.

###### Abstract

Static or frozen disorder, characterised by spatial heterogeneities,
influences diverse complex systems, encompassing many-body systems,
equilibrium and nonequilibrium states of matter, intricate network topologies,
biological systems, and wave-matter interactions. While static disorder has
been thoroughly examined, delving into evolving disorder brings increased
intricacy to the issue. An example of this complexity is the observation of
stochastic acceleration of electromagnetic waves in evolving media, where
noisy fluctuations in the propagation medium transfer effective momentum to
the wave. Here, we investigate elastic wave propagation in a one-dimensional
heterogeneous medium with diagonal disorder. We examine two types of complex
elastic materials: one with static disorder, where mass density randomly
varies in space, and the other with evolving disorder, featuring random
variations in both space and time. Our results indicate that evolving disorder
enhances the propagation speed of Gaussian pulses compared to static disorder.
Additionally, we demonstrate that the acceleration effect also occurs when the
medium evolves chaotically rather than randomly over time. The latter
establishes that evolving randomness is not a unique prerequisite for
observing wavefront acceleration, introducing the concept of chaotic
acceleration in complex media.

##  I Introduction

A heterogeneous medium, characterised by spatial variations in physical
properties, stands in contrast to homogeneous counterparts. These variations
originate from localised changes in the medium’s properties and can be
effectively modelled using continuous, piecewise, or random spatial functions
Sahimi (2003); Bahraminasab  _et al._ (2007). Found extensively in nature,
materials exhibiting such inherent heterogeneity, including granular
materials, porous media, glassy systems, the atmosphere, and colloidal
suspensions, play crucial roles in technological and industrial applications
influenced by human manipulation de Gennes (1999). Efforts to model complex
materials have led to stochastic descriptions of processes within these
systems Buryachenko (2007); Christakos (2012); Bahraminasab  _et al._ (2007).
These spatial random field models find applications in various real-world
phenomena, such as transport processes in porous media Quintard and Whitaker
(1993), oil reservoir characterisation parameters Sahimi (1992); Wu  _et al._
(2006), and the spatial distribution of elastic moduli in polycrystals
Hamanaka and Onuki (2006), granular materials Aoki and Ito (1996); Luding and
Straub (2001); Trujillo  _et al._ (2010), and composites Mohanty  _et al._
(1982); Torquato (1985); Shalaev (1999). Numerical modelling has gained
significant interest for practical applications, with acoustic and elastic
wave propagation analysis serving as a key method for non-invasive tomography
and seismic wave source characterisation in geophysics Boore (1972); Kelly
_et al._ (1976); Frankel and Clayton (1984, 1986); Dablain (1986); Kneib and
Kerner (1993).

Wave localisation, a robust phenomenon observed in various disordered systems
with spatially dependent randomness, i.e. frozen disorder Lagendijk  _et al._
(2009), has been a subject of significant exploration He and Maynard (1986);
Maynard (2001); Sahimi  _et al._ (2009). Phillip Anderson’s seminal work in
1958 demonstrated that frozen disorder can transform a conductor into an
insulator, halting transport due to multiple scattering Anderson (1958). The
exploration of waves in disordered media and localisation constitutes a highly
intricate field, unveiling hitherto unknown phenomena such as Lévy flight
Barthelemy  _et al._ (2008) and hyper-transport Levi  _et al._ (2012).
Significantly, the dynamic evolution of disorder introduces nontrivial
effects, as illustrated by Levi et al. Levi  _et al._ (2012) in a paraxial
optical context. Their findings revealed that an evolving random medium
induces stochastic acceleration, facilitating the momentum exchange between
the wave packet and the disorder. This process transcends diffusion,
culminating in the attainment of ballistic transport Levi  _et al._ (2012).
The universality of this phenomenon extends its relevance to diverse wave
systems featuring disorder.

Motivated by these observations, our investigation explores the propagation of
elastic waves within a one-dimensional random medium featuring both frozen and
evolving disorders in the mass-density distribution (diagonal disorder).
Through extensive numerical simulations, we provide the first demonstration
that evolving diagonal disorder induces stochastic acceleration of elastic
waves. Our investigation considers two types of random fluctuations: one
following a uniform distribution and another following a delta-correlated
zero-mean Gaussian distribution. Additionally, we explore the effects of
chaotic fluctuations in the mass-density of the propagating medium. We show
that the system exhibits an equivalent acceleration of elastic waves under
chaotic dynamics. Thus, evolving randomness is not a necessary condition for
the acceleration of wavefronts in evolving media, an observation that leads us
to coin the term _chaotic acceleration in complex media_. Moreover, we show
that contrary to the case of stochastic acceleration, where acceleration is
enhanced as the noise strength increases, there is a band of optimal levels of
chaos where the system reaches an acceleration plateau. Thus, highly evolved
chaos does not always give larger acceleration effects.

The remaining of the article is organised as follows. In Section II, we
introduce our model for elastic wave propagation in one-dimensional
heterogeneous materials with diagonal disorder. We also introduce here the
models of disorder used for random frozen disorder, stochastic evolving
disorder, and chaotic evolving disorder. In Section III, we show a summary of
our results from numerical simulations. We first study in Section III.1 the
effects of complexity through randomness (for frozen and evolving media), and
in Section III.2 the effects of introducing complexity through chaos. Finally,
we conclude and summarise our main results in Section IV.

##  II Elastic waves in heterogeneous media with diagonal disorder

###  II.1 The model

We will model a random heterogeneous medium that is, in some sense, close to a
deterministic homogeneous reference medium Ostoja-Starzewski (2007). More
abstractly, a random (complex) material is represented by a random medium
$\mathcal{M}$ member of a family of random media $\mathcal{M}(\zeta)$, where
$\zeta$ is a point in the space of events $\Lambda$ (sample space) Ostoja-
Starzewski (2007); Gardiner (2009). In probability theory, $\Lambda$ consists
of all possible outcomes $\zeta$ of an experiment or observation performed on
a random system. The statistical description of the system is completed by
providing a probability density function over the members of
$\mathcal{M}(\zeta)$ Ostoja-Starzewski (2007); Gardiner (2009). The system’s
physical properties are obtained through an ensemble average over multiple
observations of the system so that the final results are statistically
independent and could be related to experimental results. However, most
heterogeneous materials are nonequilibrium, quenched disordered media for
which no true statistical ensemble exists. Thus, our description of a
heterogeneous system as a random medium corresponds to a heuristic model for
the local random spatial variations of physical properties, limiting our study
to simple models in which each member of the family $\mathcal{M}(\zeta)$
differs slightly from a homogeneous reference medium $\mathcal{M}^{0}$. In
particular, we will consider random fluctuations introduced in the mass
density $\rho$ of the medium, i.e.,

$\rho(x,t)=\rho_{0}\left[1+\epsilon\eta(x,t)\right],$ (1)

where $\eta(x,t)$ is a stochastic process which can also be random in space,
$\rho_{0}$ is the mass density of the homogeneous reference medium, and
$\epsilon$ is the noise strength. Here, we coined the term random frozen
disorder for the case of a variable $\eta=\eta(x)$ that only depends on space.
Alternatively, we shall use the term random evolving disorder for the case of
a spatiotemporal variable $\eta=\eta(x,t)$. The random variable $\eta$ can
take values from -1 to 1 and parameter $\epsilon$ can be regarded as a measure
of the medium’s departure from homogeneity. Equation (1) describes what is
known as _diagonal disorder_ Ziman (1979) since it captures from a microscopic
point of view fluctuations of a one-cite parameter: the mass of particles
constituting the propagation medium.

Elastic waves propagating in a one-dimensional heterogeneous medium, whose
linear mass-density $\rho$ varies randomly in space, are governed by the
following hyperbolic wave equation,

$\partial_{tt}\phi-c^{2}(x,t)\partial_{xx}\phi=0,$ (2)

where $\phi$ is the dimensionless perturbation, and
$c(x,t):=\sqrt{K_{0}/\rho(x,t)}$ is the phase velocity of waves. We assume
that $K_{0}$ represents the bulk modulus of the homogeneous reference medium.
In Appendix A, we provide a brief derivation of Eq. (2) from a microscopic
model of elastic waves in a one-dimensional system with diagonal disorder. In
the same spirit of the models studied in Refs. Bahraminasab  _et al._ (2007);
Shahbazi  _et al._ (2005); Allaei and Sahimi (2006); Allaei  _et al._ (2008),
Eq. (2) can be regarded as a modified model based on elastic wave propagation
in homogeneous media, where we have introduced spatial variations in the phase
velocity to account for micro-structural density disorder Marín (2013);
Alcântara  _et al._ (2018); Discacciati  _et al._ (2022, 2023).

###  II.2 Models of disorder

We consider three types of statistics for the disorder in this work:

  * •

The first and most simple type corresponds to a random and uniform
distribution of uncorrelated values for the variable $\eta$. We generate such
noisy variable $\eta(x,t)$ in numerical simulations using standard algorithms
for pseudo-random number generation at each space and time step.

  * •

The second type of disorder assumes that statistical fluctuations of $\eta$
obey a Gaussian-type delta-correlated distribution in space with zero mean,
i.e.,

$\langle\eta(x,t)\rangle=0,$ (3a)
$C(x,t):=\left\langle\eta(x^{\prime},t^{\prime})\,\eta(x+x^{\prime},t+t^{\prime})\right\rangle_{x^{\prime},t^{\prime}}=\sigma^{2}\delta(x)\delta(t),$
(3b)

where $\sigma$, $\delta$, and $\langle\cdot\rangle$ denote the variance, the
Dirac delta function, and the ensemble average with respect to the underlying
Gaussian probability distribution, respectively. In our numerical simulations,
we employed the Ziggurat method Marsaglia and Tsang (1984) to generate
statistics of this kind. This method is widely employed in the literature to
model various noisy sources in nature Gardiner (2009). Notice that from Eqs.
(1) and (3) we have $\langle\rho\rangle=\rho_{0}$.

  * •

We introduce a third type of evolving disorder modeled from a complex but
deterministic source. In particular, at $t=0$, we considered a random
distribution of uncorrelated values uniformly distributed in space for the
field $\eta(x,t=0)$. Then, for each coordinate $x=x^{*}$, we introduced
chaotic evolving disorder through time series generated from the well-known
logistic map Strogatz (2018) 111It is important to remark that there is no
coupling between spatial nodes in the spatiotemporal dynamics of $\eta$, and
thus the chaotic evolution is independent of the spatial disorder.,

$\eta_{n+1}(x^{*})=r\eta_{n}(x^{*})\left[1-\eta_{n}(x^{*})\right],$ (4)

famously known for exhibiting intermittency and a period-doubling route to
chaos as the bifurcation parameter $r$ increases within the interval $[0,4]$.
Evolving chaos is obtained for $r>r_{c}\simeq 3.56995$, except for the
isolated ranges corresponding to the periodic windows of the logistic map. One
obtains completely developed chaos for $r=4$.

Figure 1: Typical behaviour of waves in random complex media with an
underlying uniform probability distribution. Cases are shown for a single
disorder realisation with disorder strength $\epsilon=0.3$. The colour scale
indicates the field of the elastic wave $\phi$. (a) Evolution under frozen
randomness. The upper panel shows the random fluctuations of the phase
velocity $c$ in space for a single realisation. (b) Evolution under evolving
randomness. The inset shows a realisation of the random disorder in the phase
speed. The upper panel shows the random fluctuations of the phase velocity $c$
in space at $t=0$ for various disorder realisations, along with the time and
ensemble-averaged speed over 500 realisations (black dashed-dotted line).

##  III Results

We simulate numerically the time evolution of an initially-at-rest Gaussian
wave packet of width $\sigma_{p}$, given by

$\phi(x,t=0)=\exp\left(-\frac{(x-x_{0})^{2}}{2\sigma_{p}^{2}}\right),$ (5)

governed by Eq. (2) considering free-end (Neumann) boundary conditions.
Hereon, we shall consider only dimensionless variables where the space $x$ is
measured in units of $\sigma_{p}$, and time $t$ is measured in units of
$\sigma_{p}/c_{0}$, where $c_{0}=\sqrt{K_{0}/\rho_{0}}$ is the phase speed of
elastic waves in the homogeneous reference medium. At $t=0$, the wavepacket is
stationary and centred at $x_{0}=L/2$ (the centre of the space domain), with
$L=60$. We used 800 discrete nodes in space and finite differences of second-
order accuracy for the space derivatives with step $\Delta x=0.075$. For the
time integration, we used the Dormand-Prince algorithm as an explicit Runge-
Kutta of orders 4 and 5 with an adaptive time step Shampine and Reichelt
(1997). We calculated 500 statistical realisations of the disorder for each
combination of parameters using a parallel code written in Matlab ©, and we
have verified that the ensemble average of the speed and the amplitude fields
becomes statistically invariant.

###  III.1 Complexity through randomness: frozen and evolving media

Figure 1 shows typical instances of wave propagation in a random media
obtained under frozen randomness [Fig. 1(a)] and evolving randomness [Fig.
1(b)]. We considered uniform probability density distributions with
$\epsilon=0.3$ in both cases. In the trivial homogeneous case where the phase
speed $c$ is constant, it is well known that under Eq. (2), an initially at-
rest Gaussian wavepacket breaks into two counterpropagating pulses with half
the amplitude of the initial pulse. In Figure 1(a), the initially at-rest
Gaussian wavepacket also breaks into two counterpropagating pulses in our
heterogeneous material. These two pulses propagate at nearly constant speed,
and multiple backscattering due to the spatial disorder within the density of
the medium produces a small amplitude interference pattern observed in the
region between them. The upper panel in Fig. 1(a) shows a typical statistical
realisation of the frozen disorder induced in the phase speed in the
heterogeneous medium, which is indeed random in space with mean value
$c_{0}=0.5$.

For the case of evolving randomness, Fig. 1(b) shows that the multiple
backscattering produces a larger amplitude interference pattern than the case
of frozen randomness. These patterns are also more complex and emerge from the
spatiotemporal complexity of the phase velocity (see inset in Fig. 1(b)). The
upper panel in Fig. 1(b) shows several statistical realisations of the phase
velocity at $t=0$. The ensemble average of the phase speed after 500 disorder
realisations gives an almost constant phase speed $\langle
c\rangle=c_{0}=0.5$, as shown with a black dashed-dotted line in the upper
panel of Fig. 1(b). We have noticed that the counterpropagating pulses travel
faster under random evolving disorder than under random frozen disorder. As
indicated with double arrows in Fig. 1, during the same time interval $\Delta
t=40$, the pulse travels a distance $\Delta x\simeq 18.8$ under random frozen
disorder and $\Delta x\simeq 20.2$ under random evolving disorder. This
observation suggests an acceleration process occurs in the evolving medium
compared to the frozen case. To inquire about this observation, we computed
the ensemble average of the pulse speed for different values of $\epsilon$ and
compared the observed behaviour between frozen and evolving cases. Figure 2(a)
reveals that as the strength of frozen randomness increases, the speed of
travelling pulses decreases, and thus the wave tends to localise. Indeed,
spatial randomness is expected to induce a strong tendency for waves to
localise in one-dimensional systems. Surprisingly, Fig. 2(a) also reveals that
the speed slightly increases as the strength of the evolving randomness
increases. Thus, increasing the disorder strength in evolving media, rather
than promoting wave localisation, enhances wave transport. Consequently, for a
fixed value of $\epsilon$, waves travel faster under a random evolving
disorder than under a random frozen disorder.

Figure 2: Stochastic-induced acceleration of elastic waves: Ensemble average
speed $\langle v\rangle$ as a function of disorder strength $\epsilon$ for (a)
uniform distribution, and (b) delta-correlated normal distribution with zero
mean and variance $\sigma=50$. Error bars are too small to be shown in the
scale. (c) Dependence of the ensemble average speed as a function of the
variance of the underlying normal distribution. The speed of the wave under
evolving randomness remains almost invariant under changes in the variance and
noise strength.

Thus, we have discovered a kind of stochastic-induced acceleration in the
propagation of elastic waves. This phenomenon is very similar to that observed
by Levi et al. in Ref. Levi  _et al._ (2012), where hyper-transport of light
by evolving disorder is reported numerically and experimentally using an
effective 2+1-dimensional paraxial optical setting. Here, we report a similar
phenomenon for elastic waves in an evolving 1+1-dimensional randomly
disordered medium.

We determined that the observed phenomenon is robust against changes in the
underlying probability distributions. In Fig. 2(b), we considered a delta-
correlated normal distribution with $\sigma=50$, following Eq. (3), and
obtained the same qualitative behaviour. In this case, random frozen disorder
enhances a weaker localisation than the case of frozen uniform randomness [see
Fig. 2(a)]. This is because samples of a random variable under a Gaussian
probability distribution predominantly accumulate near its mean value, and
extreme events corresponding to the tails of the Gaussian distribution are
less probable to occur. This frozen randomness with accumulated values near
the mean induces a weaker wave localisation in contrast with a uniform
distribution, where samples away from the mean are equally probable to occur
than values near the mean. Similarly, evolving randomness following a Gaussian
distribution leads to a weaker enhancement of wave transport when compared to
the case of uniform distributions. However, as evidenced in Fig. 2(b), the
evolving randomness counteracts wave localisation, sustaining the speed at a
nearly constant value as the disorder strength increases.

Following the previous observations, it is natural to wonder about the effect
of a Gaussian probability distribution with a higher dispersion of sample
points. Thus, we repeated our analysis with a variance $\sigma=100$, i.e.
twice as that in Fig. 2(b). Figure 2(c) shows minimal differences in the
outcome from the ensemble average speed. Thus, although extreme events are
more probable for $\sigma=100$ than for $\sigma=50$, they are still much less
probable than events near the mean of the stochastic process. Thus, the effect
of enhanced transport under evolving randomness is qualitatively the same for
both variance values.

###  III.2 Complexity through chaos: chaotic acceleration in complex media

Figure 3: Chaos-induced acceleration of elastic waves: (a) Ensemble average
speed $\langle v\rangle$ as a function of disorder strength $\epsilon$ for
different values of the bifurcation parameter $r$. (b) Maximum speed of
elastic waves as a function of the bifurcation parameter. The maximum values
are obtained within a broad bandwidth of $r$. The inset shows the Lyapunov
exponent $\mathcal{L}$ as a function of $r$ along with three sample points
used to characterise the speed dependence on the chaos level (black, blue and
red dots). (c) Phase space representation, indicating regions in the parameter
space $(\epsilon,r)$ where localisation (blue regions) and chaos-induced
acceleration (red regions) are observed. (d) Measured scaling laws of the
chaos-induced acceleration with the disorder strength. Error bars are too
small to be shown in the scale.

Following our previous observations, we wondered if the evolving nature of the
media must be strictly random to induce the acceleration of elastic waves.
Thus, we explored the outcome of the system if the medium evolves in a
deterministic but complex way, following the logistic map shown in Eq. (4). We
have discovered that chaos also induces the same qualitative acceleration
effect, as summarised in Fig. 3. Our explorations lead us to coin the term
_chaotic acceleration in complex media_.

In Fig. 3(a), we show the ensemble-averaged speed of elastic waves as a
function of the disorder strength $\epsilon$ for different bifurcation
parameter $r$ values. For $r=3.52$, the logistic map exhibits a periodic orbit
with period $4$. We observe that the wave tends to localise as $\epsilon$
increases. In this case, the elastic waves undergo multiple scattering at
localised heterogeneities periodically modulated in time. Thus, the wave tends
to localise as the amplitude of such modulations becomes larger.

However, we notice that for $r=3.80$, a value of the bifurcation parameter
where the system evolves chaotically, there is a remarkable increase of the
wave speed as the disorder strength $\epsilon$ increases, as shown in Fig.
3(a). Indeed, $\langle v\rangle$ increases monotonically with $\epsilon$.
Importantly, comparing Fig. 2(a) and Fig. 2(b) with Fig. 3(a), we notice that
chaotic dynamics in the evolving medium induces a larger increase of the wave
speed than in the stochastic case.

We have systematically studied the ensemble-averaged speed of waves for
increasing bifurcation parameter values. For $r=4$, the value at which the
medium exhibits completely evolved chaos, one might expect that the effect of
increased transport will reach its stronger regime. However, we have found
that the accelerating effect is more moderate than for $r=3.80$, as shown in
Fig. 3(a). The latter evidences an optimal value of $r$ (and thus, an optimal
level of chaos), where the chaotically induced transport is stronger.

We have computed the maximum value of $\langle v\rangle_{\max}$ in terms of
the bifurcation parameter $r$ in Fig. 3(b). We observe that the maximum
ensemble-averaged speed reaches a maximum plateau for intermediate values of
$r$ within the interval $[r_{c},\,4]$, where $r_{c}\simeq 3.56995$ is the
critical value for the onset of chaos in the logistic map. A black point in
Fig. 3(b) indicates the case $r=3.52$, which corresponds to the non-chaotic
dynamics of the system. The blue point in Fig. 3(b) indicates the case where
we have detected the largest value of $\langle v\rangle$, corresponding to
$r=3.80$. The red point Fig. 3(b) corresponds to the case of completely
evolved chaos for $r=4$.

Notice in Fig. 3(b) that there are ranges for the breakdown of the chaotic
acceleration effect in the middle of the chaotic acceleration plateau. These
breakdown ranges of $r$ correspond to sub-ranges of $[r_{c},\,4]$ where the
speed decays abruptly due to the emergence of the periodic windows of the
logistic map. Indeed, in the inset of Fig. 3(b), we show the Lyapunov exponent
$\mathcal{L}$ of the logistic map, evidencing that the broadest range of
acceleration breakdown corresponds to the wider periodic window of the
logistic map. For each periodic window of the logistic map, which is fractally
distributed within the range $[r_{c},\,4]$, there will be a corresponding
breakdown of the chaotic acceleration effect. Also, in the inset of Fig. 3(b),
we show the cases corresponding to $r=3.52$, $r=3.80$ and $r=4$, with black,
blue and red dots, respectively, to emphasise that highly evolved chaos do not
imply higher acceleration effects: an optimum value is somewhere in between
the points $r=r_{c}$ and $r=4$.

Figure 3(c) shows a diagram in the parameter space $(\epsilon,r)$ of the
chaotic system. Here, the colour scale indicates the value of $\partial\langle
v\rangle/\partial\epsilon$. Thus, red regions indicate combinations of
parameters for chaotic acceleration, whereas white and blue regions are for
localisation. Notice that the breakdowns of chaotic acceleration emerge as
white stripes within red regions in parameter space.

Finally, in Fig. 3(d), we study the scaling laws of the ensemble-averaged
speed jumps, $\Delta\langle v\rangle$, with the disorder strength $\epsilon$.
We observe that, although $\langle v\rangle$ can reach large values for
$r=3.67$ than for $r=4$, it grows faster with $\epsilon$ for completely
evolved chaos ($r=4$) than for moderate chaos ($r=3.67$). Indeed, it is clear
that $\partial\langle v\rangle/\partial\epsilon\sim\epsilon^{1.288}$ for
$r=3.67$, and $\partial\langle v\rangle/\partial\epsilon\sim\epsilon^{1.9962}$
for $r=4$. Thus, although the largest ensemble-averaged speed occurs for
moderate chaos, the ensemble-averaged speed increases faster with $\epsilon$
for completely evolved chaos.

##  IV Conclusions

In summary, we have shown that evolving disorder, random or chaotic, enhances
the acceleration of elastic waves in one-dimensional systems under diagonal
disorder. Using as a model a hyperbolic partial differential equation with
space-time variations in the phase velocity, we have characterised the
acceleration effect considering three types of statistics for the disorder:
uniform (random), Gaussian (random), and chaotic (deterministic)
distributions. Under frozen disorder, elastic waves localise as the disorder
strength increases. However, we have shown that evolving random disorder tends
to increase the speed of wavefronts as the disorder’s strength increases. We
have discussed similarities between the observed phenomenon and the hyper-
transport of light observed by other researchers in random evolving media Levi
_et al._ (2012). Furthermore, we have discovered that chaotic (deterministic)
evolving disorder induces the same acceleration effect. Thus, randomness is
not a unique prerequisite to observing enhanced transport. We have
characterised the chaotic acceleration effect considering chaotic signals
generated from a logistic map and show that the maximum value of the ensemble-
averaged speed reaches a maximum plateau within an interval of the bifurcation
parameter of the logistic map. We provide evidence of the breakdown of chaotic
acceleration at intervals of the bifurcation parameter corresponding to the
periodic windows of the logistic map, thus supporting the hypothesis that
acceleration in such deterministic systems is purely enhanced by chaos and not
by periodic signals composed of a large number of frequencies.

## Acknowledgments

The work on “evolving random disorder” was started when L.T. and J.F.M. were
associated scientists at Instituto Venezolano de Investigaciones Científicas
(IVIC), Venezuela. The idea of “chaotic disorder” was pursued at Universidad
Tecnológica Metropolitana and Universidad Técnica Federico Santa María, Chile.
L.T. thanks the ICTP, the Institut National des Sciences Appliquées de Lyon
(INSA-Lyon) as well as the Laboratoire d’InfoRmatique en Image et Systèmes
d’information (LIRIS) for hospitality while part of this research was done.
M.A. thanks Agencia Nacional de Investigación y Desarrollo (ANID) for the
financial support through the FONDECYT POSTDOCTORADO grant No. 3240443. L.T.
was partially supported by the INSA Visiting Professor Fellowship.

##  Appendix A Derivation of the wave equation under diagonal disorder

Elastic wave propagation in a one-dimensional medium can be described
microscopically by a system of $n$ consecutive point-like particles of mass
$m_{j}$ coupled by springs with a Hooke’s constant $k_{j}$ ($j=1,\ldots,n-1$).
Diagonal disorder corresponds to the situation where $k_{j}=k$ is constant and
deterministic, whereas $m_{j}$ for $j=1,\ldots,n$ is given by a stochastic
process. Let $\phi_{j}(\omega,t)$ be the displacement from equilibrium of
particle $j$ at time $t$ in a realisation $\omega(t)$ of the random evolving
medium. If such displacements are small compared to the inter-particle
distance $a$, the system exhibits a linear response to external perturbations
and the stochastic equation of motion for the $j$-th mass within the bulk of
the material ($j=2,\dots,n-1$) reads

$a\rho_{j}(\omega,t)\frac{d^{2}\phi_{j}}{dt^{2}}=-\frac{2K_{0}}{a}\phi_{j}+\frac{K_{0}}{a}(\phi_{j+1}+\phi_{j-1}),$
(6)

where $\rho_{j}(\omega,t):=m_{j}(\omega(t))/a$ and $K_{0}:=ak$ are the
microscopic mass density and stiffness of the medium, respectively. We can use
some coarse-graining strategy to obtain a continuous (macroscopic) wave
equation from Eq. (6). The use of integral representations through weighted
space-time averages of particle quantities is one such alternative Murdoch
(2012), which can be achieved by multiplying both sides of Eq. (6) by a
smoothing kernel of radius $h$, namely $f(x-x_{j},h)$, and integrating over
$x_{j}$ Murdoch and Bedeaux (1994); Glasser and Goldhirsch (2001); Goldhirsch
and Goldenberg (2002). This type of coarse-graining formalism is known to
solve problems related to discontinuities and the lack of derivatives in many
stochastic processes, such as the case of fractional Brownian motions (as
shown in Mandelbrot’s seminal work on Ref. Mandelbrot and Ness (1968)). Here,
we assume that the kernel $f$ has a compact support domain $\mathcal{S}(x)$ of
radius $h$ around $x_{j}$, i.e. the function $f$ vanishes if $|x-x_{j}|\geq h$
Marín  _et al._ (2014); Petit  _et al._ (2014). Thus, the left-hand side of
Eq. (6) becomes

$a\int_{\mathcal{S}(x)}\rho_{j}(\omega,t)\frac{d^{2}\phi_{j}}{dt^{2}}f(x-x_{j},h)dx_{j}=a\langle\rho(x,t)\partial_{tt}\phi(x,t)\rangle_{h},$
(7)

where $\langle\cdot\rangle_{h}$ denotes the continuum function obtained after
coarse-graining with a smoothing function of radius $h$. Similarly, the right-
hand side of Eq. (6) after coarse-graining and using a Taylor expansion for
$\phi_{j\pm 1}-\phi_{j}\gg a\forall j$, becomes

$\frac{K_{0}}{a}\int_{\mathcal{S}(x)}(\phi_{j+1}-2\phi_{j}-\phi_{j-1})f(x-x_{j},h)dx_{j}=aK_{0}\langle\partial_{xx}\phi(x,t)\rangle_{h}.$
(8)

Equating Eqs. (7) and (8), setting $c(x,t):=\sqrt{K_{0}/\rho(x,t)}$, we obtain
a hyperbolic partial differential equation of the same form as Eq. (2).

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