We study the propagation of dissipative structures in inhomogeneous media with a focus on pinning and depinning transitions. We model spatial complexity in the medium as generated by dynamical systems. We are thus able to capture transitions from periodic to quasiperiodic, to homoclinic and heteroclinic, and to chaotic media. Depinning bifurcations exhibit universal laws depending on extreme value statistics that are encoded in the dimension of ergodic measures, only. A key condition limiting this approach bounds spatial Lyapunov exponents in terms of interface localization, and we explore the breakdown of smoothness and universality when this condition is violated and fluctuations in the medium occur on length scales shorter than a typical interface width.

We analyze front propagation in a simple bistable lattice differential equation in an inhomogeneous medium. When the space dependence of the medium is modeled over a smooth manifold and is stationary with respect to lattice translations, we argue that a simple reduced dynamical system captures all key features of the dynamics. In particular, we predict universal depinning exponents in terms of statistical properties of a generating diffeomorphism acting on the model manifold. We present computational tools that exploit this reduced description to predict locations and scaling laws of depinning transitions. Finally, we show both in chaotic media and in a simple toy heteroclinic medium how results break down when Lyapunov exponents of the generating diffeomorphism are comparable to the exponential localization of the front interface.

## I. INTRODUCTION

Dynamics in spatially extended bistable systems are often mediated by the propagation of an interface. For gradient flows, the propagation of the interface dissipates energy as the energetically favorable of the two stable states spreads in the medium. The relation between the speed and the difference in energies of the two stable states gives a kinetic relation that captures the essential feature of such a spatially extended system. Front dynamics in inhomogeneous media can be significantly more complex. If one defines an average energy difference $\mu $ between the two stable states and measures average speeds in terms of this average difference, speeds turn out to be not strictly monotone, even in very simple systems. The prime reason for these more complicated kinetic relations is the possibility of pinning, where a “local” energy difference may be negative and therefore prevent front propagation, although the average energy difference is positive. In periodic media and in lattices, this phenomenon is fairly well understood,^{1–3} with applications including signal propagation in myocardial tissue,^{4,5} martensitic phase transitions,^{6,7} or chemical reaction fronts.^{8} In these situations, depinning, that is, small non-zero speed past a non-zero critical value of the average energy difference occurs when two pinned states disappear in a saddle-node bifurcation. Speeds $s$ near this depinning transition scale as $s\u223c(\delta \mu )1/2$, due to the standard scaling of passage times near a saddle-node.^{1} Our interest here is in pinning and depinning in more complex media, striving to recover some universality in asymptotics of speeds near depinning, and studying the breakdown of these universal laws.

To fix ideas, we consider the Nagumo (or Allen-Cahn) equation in an inhomogeneous lattice

with initial conditions $uj\u21921$, $j\u2192\u2212\u221e$ and $uj\u21920$, $j\u2192\u221e$. For $aj\u22610$, pinning regions $|\mu |<\mu c(d)$ decrease exponentially in $d$ for large $d$; see Fig. 1 for kinetic relations, pinning relations, and pinned regions.

Our goal is to demonstrate that in the framework of “dynamically generated media,” one can systematically study kinetic relations and pinning relations, and, in particular, uncover universality in depinning asymptotics. The class of dynamically generated media that we focus on is modeled through actions of diffeomorphisms $\psi $ on smooth manifolds $M$ as

In this sense, each point $m\u2208M$ is associated with a medium ${aj}j\u2208Z$, and we refer to $m$ as a realization of the medium.

The first examples of interest are quasiperiodic media $M=TK=RK/ZK$ with rationally independent frequencies $(\omega j)1\u2264j\u2264K$, e.g.,

From here, we can then envision increasing the complexity in the medium through changes in the diffeomorphism $\psi $. The second example we consider here is a chaotic medium modeled over $T2$ with $\psi $ being the cat map

We shall explain below how different choices of functions $A$ and choices of initial conditions that are “typical” for different ergodic measures can lead to a variety of statistical behaviors. Depinning asymptotics are, however, governed by a simple statistical property of the medium, the dimension of the ergodic measure $\kappa $,

where $s\xaf$ is a suitably defined average speed of propagation. We compare these behaviors with some simple choices of random media, for which one can readily intuit depinning asymptotics.

These simple asymptotics break down when fluctuations in the medium occur on small spatial scales, compared to length scales associated with the front interface. We quantify this limit and demonstrate why and how our simple description breaks down near these boundaries. In the following, we first describe more precisely how we model inhomogeneous media and front dynamics in such media. We illustrate this point of view, first in the example of a quasi-periodic medium and then for a chaotic medium. We compare with different types of random media and finally explore limits of this point of view.

## II. DYNAMICALLY GENERATED MEDIA AND FRONT DYNAMICS ON A MEDIUM

The simplest description of front dynamics tracks the position of the front for a given realization of the medium $m$, that is, for coefficients generated through $aj=A\xb0\psi j(m)$. In the case of the lattice Nagumo equation, the position $\xi $ can be defined for instance through

as long as the front interface is near a location $j0$ with $j0\u2212J\u226b1$. For our purposes, equivalent definitions could rely on solving

for a strongly localized $S$ with $\u222bS=1$. We suppose that front dynamics in a given medium can effectively be reduced to dynamics of the position $\xi $,

and $s(\xi ;m)$ is a speed, depending, of course, on the specific definition of position $\xi (u)$. Choices of position function $\xi $ such that $\xi ({uj})=\xi ({uj\u22121})+1$ guarantee that dynamics at front position $\xi +1$ are the same as dynamics at position $\xi $, changing the realization of the medium to $\psi (m)$, that is,

Note also that average speeds, defined as times to propagate one lattice site, $s1:=(T1\u2212T0)\u22121$, where $\xi (T1)=\xi (T0)+1$, do not depend on the definition of $\xi $. We can therefore study front dynamics relative to the lattice (and the medium), considering (3) on

with identification $(1,m)\u223c(0,\psi (m))$; see Fig. 2. The gluing of $\xi =1$ to $\xi =0$ can sometimes be trivialized, when $\psi $ is homotopic to the identity through $\Psi t$,

We introduce new coordinates through $(\xi ,m)=(\xi ,\Psi \xi (m~))$, trivializing the identification $(1,m~)\u223c(0,m~)$ with the new differential equation

where $s~(\xi ;m~)=s(\xi ;\Psi \xi (m~))$, $h~(\xi ,m~)=h(\Psi \xi (m~),\xi )$. The new medium $m~$ can be thought of as the medium in a comoving frame; compare Fig. 2 for the two geometric viewpoints.

For quasiperiodic media, one chooses $\Psi \xi (m)=m+\omega \xi $, $h=h~=\omega $, and finds

The resulting vector field can be easily computed and we illustrate the result below for the systems with a periodic medium $K=0$ and with a quasi-periodic medium $K=1$.

Since the flow in both (3) and (5) is determined primarily by the scalar field $s$, one is first interested in zeros of $s$ which we shall refer to as pinned states. When we include the dependence of $s$ on the parameter $\mu $, we can typically solve $s(\xi ;m,\mu )=0$ for $\mu $ and obtain the pinning relations $\mu p(\xi ,m)$, that give the applied force $\mu $ that forces a pinned state at position $\xi $ in the medium $m$. The extrema of $\mu $ determine depinning transitions, at least when $\psi $ is transitive on $M$, that is, when trajectories of $\psi $ explore all possible values of $m$. We show pinning relations and their correspondence with average speeds and pinning regions in Fig. 1, and we give more examples below.

Pinning relations also give good estimates on barriers to propagation when noise is added to the system. Potential differences between stable and unstable pinned states, visible here as areas between the pinning relation and a $\mu \u2261const$ level set, are known as Peierls-Nabarro barriers and determine rates for jumps of the interface between lattice sites.

## III. ERGODICITY AND DEPINNING ASYMPTOTICS

In the framework of (3) on the quotient manifold (4), we can now study front dynamics from a dynamical systems point of view. Pinned front locations are equilibria. For $P$-periodic media, $M={m0}$, $P\u2208N$, we obtain dynamics on the circle $\xi \u2208[0,P]/\u223c$, $\xi \u2032=s(\xi )$. Dynamics are either periodic when $s\u22600$ or converge to pinned states. Average speeds are easily obtained as a harmonic average by integrating $1P\u222b0Ps(\zeta )\u22121d\zeta =s\xaf\u22121$. Near depinning, speeds are close to zero near a degenerate equilibrium, $\mu =\mu c+\mu ~$, $\xi =\xi c+\xi ~$,

and the harmonic average is at leading order given by the integral

In general, the dynamics of the medium can of course be arbitrarily complex. We then solve (3) by computing the time for $\xi $ to increase from $0$ to $1$, and then average times over the iterates $mj=\psi j(m)$ of the medium

If the initial condition is typical for an ergodic measure $d\nu (m)$ on $M$ associated with $\psi $ acting on $M$, this sum of $T1$ along orbits can be replaced by a phase space average using Birkhoff’s ergodic theorem

In quasi-periodic media, all realizations are typical for the Lebesgue measure, and the integral is simply the $K+1$-dimensional Lebesgue measure.

When $s(\xi ;m)$ almost vanishes for some near-critical parameter value $\mu =\mu c+\mu ~$ and near a non-degenerate minimum $\xi =\xi c+\xi ~$, $m=mc+m~$, the integral is at leading order

for some bilinear positive definite form $c2$. A simple scaling analysis shows that

where $\kappa $ is the dimension of the ergodic measure $\nu $, that is, the measure of a ball of size $r$ scales as $\nu (Br)\u223cr\kappa $. Similar asymptotics have been derived for the case of continuous media $x\u2208R$.^{9}

For quasi-periodic media with more than one frequency, $K>1$, we always see the first case, hard depinning, with discontinuous speeds near the transition. For quasiperiodic media with one frequency (irrational relative to the lattice periodicity), one can be slightly more precise and compute the coefficient at leading order

where $D=c211c222\u221214c212c212$ is the geometric (Gaussian) average of the curvatures of $s$ at the zero, where $c2(\xi ~,m~)=c211\xi ~2+c212\xi ~m~+c222m~2$.

We will demonstrate below that coefficients in these asymptotics can be readily obtained numerically with very high accuracy using direct Newton solvers.

## IV. COHERENT MEDIA

We apply and illustrate the analysis in the case of periodic media ($K=0$) and quasiperiodic media ($K=1$). We first consider $aj=0.1cos\u2061(\pi j/2)$ of period 4, with $d=0.1$ and $d=0.3$. We can consider the front dynamics on $\xi \u2208R/(4Z)$, with trivial medium $M1={0}$, or front dynamics on $\xi \u2208R/(Z)$, with medium $M4={0,1,2,3}$, $\psi (j)=j+1mod4$. Figure 3 shows pinning relations and effective dynamics on $\xi \u2208R/(4Z)$, with horizontal cuts showing how these dynamics would decompose into dynamics on $M4$. Pinning relations yield critical values $\mu c$ and depinning asymptotics according to (8) can be readily verified; see Fig. 1 for an illustration of the square root asymptotics for average speeds past the last pinned state.

In quasiperiodic media, we consider $aj=0.1cos\u2061[2\pi (grj+m)]$, $gr=(1+5)/2$. Computing the derivative of the position $\xi =\u2211j\u2265Juj$, we find the speed $s(\xi ;m)$, from (3) which can then be transformed to the speed in a comoving medium (6); see Fig. 4.

Equilibria in (3) can be calculated directly. We continued the branch of equilibria in $\mu $ and $m$, tracking the interface location $\xi $, and pinning relations, $\mu =\mu p(\xi ;m)$ as a function of $m$ and $\xi $; see Fig. 5.

We computed the extrema of $\mu p(\xi ;m)$ using a Newton method. We first define equations for equilibria and their position, with independent variables ${uj}|j|\u2264J,\xi ,m,\mu $,

with Neumann boundary conditions and where

We then solve these equations with the additional condition that derivatives of $\mu $ with respect to $m$ and $\xi $ vanish. From the resulting system, we directly obtain the principal curvatures at the extrema. Figure 6 illustrates the pinning relations showing cross sections through the surface of pinned states near the minimum. We can then use (11) to predict speed asymptotics near the depinning transition as

Figure 7 shows speeds measured in direct simulations using Matlab’s ode45 integrator for times $T\u223c107$ with appropriate shifts in a system of size $1000$, compared to our prediction. The techniques presented here can readily predict critical depinning parameter values $\mu c$ for quasiperiodic media with more than one additional frequency. Depinning is, however, hard, with discontinuous $s\xaf(\mu )$ at $\mu =\mu c$ such that asymptotics are not meaningful at leading order.

## V. CHAOTIC MEDIA

Chaotic dynamics of $\psi $ offer a rich set of examples of media as we shall demonstrate now, bridging the spectrum from simple periodic media over quasiperiodic media to chaotic, unstructured media. We shall focus here on a rather explicitly understood example of a chaotic map that will nevertheless serve as an example of a variety of phenomena. Consider therefore the cat map, an Anosov diffeomorphism of $T2$ given through

The map $\psi $ is area preserving and in fact ergodic with respect to the Lebesgue measure.^{10} On the other hand, all rational points $(mx,my)$ are periodic, with associated ergodic Dirac measures, and there are a plethora of explicit homoclinic and heteroclinic trajectories between periodic orbits, giving rise to shift dynamics and associated invariant Bernoulli measures.^{10} Regardless of the ergodic measure that captures the statistics of the trajectory associated with a particular medium, we can study dynamics of (3) on $[0,1]\xd7T2/\u223c$, with identification (4) given now by the cat map. Since the cat map is not isotopic to the identity (it acts nontrivially on the fundamental group), it is not possible to construct a comoving medium in the sense of (5).

We computed pinning relations $\mu p(\xi ,mx,my)$, for the choice $aj=\u03f5cos\u2061(2\pi mx,j)$; see Fig. 8. Note that the surfaces are periodic in $mx$ and $my$ but periodic in $\xi $ with the identification $(1,mx,my)\u223c(0,2mx+my,mx+my)$. We found a minimum of $\mu p(\xi ,mx,my)$ at $\mu c=0.3551$; see Fig. 9. Experimental speed measurements for a medium generated with initial conditions $mx=1/2,my=1/5$ over time intervals $T\u223c107$ confirm this depinning transition as a discontinuous transition with critical $\mu $ agreeing at $10\u22124$ accuracy; see Fig. 10.

Non-Lebesgue typical initial conditions lead to dramatically different depinning transitions. Initial conditions on the period-3 orbit $p0=(12,12)\u2192p1=(12,0)\u2192p2=(0,12)\u2192(12,12)$ are of course ergodic with respect to Dirac measures at the periodic orbit and sample the pinning relations along $\xi \u2208[0,1)$, $m=p0,p1,p2$, that is, along lines parallel to the $\xi $-axis with intercepts $pj$ at $\xi =0$ in Fig. 8. One finds depinning transitions at minimal $\mu c=0.3852$ and maximal $\mu c=0.6311$, with the expected square-root asymptotics of the speed.

More interesting are orbits of the cat map that are homoclinic to ${p0,p1,p2}$. Since stable and unstable foliations are explicitly given by the eigenvectors of $\psi $ on the covering space $R2$, we can explicitly determine a primary homoclinic point $q0=125(1+25,2+5)mod1$ such that

Figure 11 shows the homoclinic orbit on the torus, and Fig. 12 shows pinning relations relative to the trajectory of the medium.

We find that pinning is stronger near the periodic orbit and weaker during the excursion.

Orbits that stay in a small vicinity of the homoclinic orbit can be described by concatenations of homoclinic excursions and periodic loops. Hyperbolic shadowing guarantees that a formal infinite patching of sequences $\alpha ={\psi j(q0),\u22123\u2264j\u22642}$ and $\beta ={p0,p1,p2}$ actually corresponds to trajectories that follow the sequence itinerary quite closely.^{11} Media of the form $(mj)=(\u2026\alpha \alpha \beta \alpha \beta \beta \u2026)\u2208\Sigma 2={\alpha ,\beta}Z$ are therefore included in our cat map medium, for a variety of measures such as those induced by Bernoulli measures on two-symbol sequences.

We investigated depinning numerically for quasiperiodic sequences choosing the block $\alpha $ at the $\u2113$’th instance whenever $\u2113grmod1\u2264gr\u22121$, and the block $\beta $, otherwise. We found depinning at $\mu c=0.6313$, between the critical $\mu $ values of periodic and homoclinic excursion, as expected, with logarithmic depinning asymptotics. We also constructed sequences according to Bernoulli measures, choosing $\beta $ with probability $p$ and $\alpha $ otherwise; see Fig. 13. We found good fits $s\xaf\u223c\mu ~1/2$, which would be consistent with a zero-dimensional ergodic measure $K=0$. We shall give heuristics for these asymptotics in Sec. VI; compare (15).

## VI. RANDOM MEDIA

The horseshoe construction within the cat map illustrates how random dynamics occur naturally within the class of media we focus on here. Indeed, choosing characteristic functions for $A:M\u2192R$ on a Markov partition, we can realize quite general Markov chains. We therefore study directly media generated by Markov chains. As an example, we generated random sequences based on the Markov graph

that is, events $B$ are followed by event $A$ and $A$ follows $A$ with probability $p$. We define the medium $aj=0.05$ for event $A$ and $aj=0$, otherwise. We show measured speeds for $d=0.1$ in Fig. 14. A rough heuristic would assume that the speed is determined by the medium, locally, with passage times around lattice point $Tj\u223c|\mu \u2212\mu c|\u22121/2$. Noticing that points with almost pinning occur only in state $A$, which occurs with probability $(2\u2212p)\u22121$ in the stationary distribution, we obtain an expected average speed

see Fig. 14. An interesting, different class of random media is generated by a continuous distribution for the values $mj$. We shall explore here power law distributions that emphasize different extreme value statistics. We sample $m$ from a uniform distribution and distribute the $aj$ as $\rho \u223cm\gamma $, that is, according to a density $dm=1\gamma \rho \u22121+1/\gamma d\rho $.

If we assume that the front speed is determined by the nearest lattice point, only, by the usual square-root depinning laws, we find passage times near a lattice point and average speeds that scale as

The inverse average of the independent variables in the denominator converges to the expectation

for $\gamma >1$. The calculation is analogous to the calculation of average speeds in ergodic media and we find the effective corresponding dimension,

that is, the inverse sampling exponent gives the ergodic dimension of the associated measure. Figure 15 illustrates the depinning asymptotics, showing a general decrease in the speed with $\gamma $, consistent with the fact that large values of $\gamma $ increase the likelihood of almost-pinning events, and also an increase in the exponent of the power law. Figure 16 shows a comparison of the power laws with our prediction (16).

The same calculation shows that sampling $aj$ from a Bernoulli distribution, say $p(0)=q,p(0.1)=1\u2212q$, gives depinning asymptotics,

## VII. LIMITS OF SMOOTHNESS

We have shown thus far ample evidence that supports the assumption of a reduction to scalar positional dynamics, depending smoothly on the medium as in (3). In translation-invariant media with a weak inhomogeneity,

where $a(x;m)$ is generated by a continuous flow $\psi x(m)$ on a smooth manifold $M$, $a(x;m)=A\xb0\psi x(m)$, one can establish such a smooth description using perturbative methods.^{9,12} The analysis there relies, however, on an assumption of “Lyapunov dominance,” roughly requiring that the Lyapunov exponents of the flow $\psi $ generating the medium are bounded relative to the exponential decay rate of the front solution. This suggests qualitatively different behavior for lattice-differential equations on a medium with positive Lyapunov exponents (relative to the ergodic measure relevant for the statistics of the particular realization of the medium).

Indeed, exponential convergence rates of the front solution are

with large-$d$ asymptotics $\nu \u223c1+1/2d$. For the cat map, the unstable Lyapunov exponent is $1+gr$ and equals the unstable exponent of the medium for $d=0.5$; the $2$:$1$-resonance occurs at $d\u223c0.15$.

Practically, we observe this difficulty when computing pinning relations in (12) and continuing in the medium. Evaluating derivatives with respect to the medium in a Newton continuation algorithm becomes unfeasible when the derivative with respect to the medium at the front

is not spatially localized. We notice the competing effect of $uj(1\u2212uj)\u21920$ with the convergence rate of the front and of $\u2202mx\psi j(m)$ which grows with rate $1+gr$.^{13} Computations of the pinning relations reveal a steepening of isosurfaces as $d$ is increased, leading to irregular surfaces for larger $d$ that are difficult to resolve computationally; see Fig. 17 for the isosurfaces and Fig. 18 for a plot of $\mu $ along a section through the medium.

Beyond the structure of pinning surfaces, we tested the impact of this loss of smoothness on depinning asymptotics, in the example of random media. Figure 15 shows the change in depinning asymptotics when $d$ is increased from $d=0.1$ to $d=0.3$. Clearly, speeds are increased near the depinning transition and depinning asymptotics favor hard, discontinuous depinning transitions. Heuristically, this is caused by the fact that extreme events are less frequent when the front is wider and therefore interacts with multiple lattice sites at a time. Within the numerical resolution, here, depinning was discontinuous for all moderate values of $\gamma $.

More quantitatively, assuming non-smooth behavior near the extrema of the local speed in (9),

leads to power laws $s\xaf\u223c\mu ~1\u2212\kappa +1q$ with transition to hard depinning for the smaller dimension $\kappa =q\u22121$. For $q<1$, depinning is always hard.

In order to analytically explore this phenomenon, we studied what we believe to be the simplest toy example of a medium with nonzero Lyapunov exponents, a heteroclinic medium in continuous space $x\u2208R$, (17) with $\u03f5=0.2$ and $a(x;m0):=\psi x(m0)=m(x;m0)$, the solution to

For $m\u2208(0,1)$, $a(x;m)\u21920$, $x\u2192\u2212\u221e$, and $a(x;m)\u21921$, $x\u2192+\u221e$, with exponential rate $\tau \pi $. Thinking of $\u03f5=0.2$ as a small perturbation, one can compute an effective speed as a function of $x$ perturbatively, in particular near $|x|=\u221e$. Projecting the effect of the medium $\u2212\u03f5u(1\u2212u)a(x;m)$ on the Goldstone mode given by the derivative of the front $uf\u2032(x)\u223ce\u22122|x|$, we find

Since $m(x)\u223ce\u2212\tau \pi |x|$, we can express $s$ in terms of the medium as

In the latter case of large Lyapunov exponents in the medium, compared to decay rates of the front, leading order term is clearly not smooth. Figure 19 shows the associated dependence of pinning relations with a change to non-smooth behavior as $\tau $ increases, with a crossover to non-smoothness predicted at the $1$:$1$-resonance $\tau \u223c0.45$. We also measured instantaneous speeds in the medium, which show a similar crossover; see Fig. 19.

## VIII. DISCUSSION

We presented a study of pinning relations and depinning asymptotics in media parameterized over smooth manifolds. Depinning asymptotics depend on extreme-value statistics in a universal fashion, with power law exponent $1\u2212\kappa /2$, $\kappa $ the dimension of the ergodic measure. This complements findings in continuous media,^{9} where analogous power laws were found, with dimension reduced by 1 due to the lack of underlying lattice periodicity. Quite different power laws have been observed in media with nonlocal coupling,^{14} with power law exponents $3/2$ or higher, which corresponds to an ergodic dimension $\kappa \u2264\u22122$ in our setting. In many cases, the medium is not independent of the front and one refers to “self-pinning” when front speeds vanish for open parameter values. An extreme case of such an interaction is fronts between periodic states and trivial states observed for instance in the Swift-Hohenberg equation.^{3,15} We also presented limits to such a description, which turns out to be effective only when Lyapunov exponents related to the modeling of the medium are bounded in terms of localization of the front. We demonstrate how this leads to both non-smooth pinning relations and changed depinning asymptotics.

## ACKNOWLEDGMENTS

The authors gratefully acknowledge the support through the National Science Foundation (NSF) grant DMS-1612441.

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