Target Patterns in a 2-d Array of Oscillators with Nonlocal Coupling
We analyze the effect of adding a weak, localized, inhomogeneity to a two dimensional array of oscillators with nonlocal coupling. We propose and also justify a model for the phase dynamics in this system. Our model is a generalization of a viscous eikonal equation that is known to describe the phase modulation of traveling waves in reaction-diffusion systems. We show the existence of a branch of target pattern solutions that bifurcates from the spatially homogeneous state when , the strength of the inhomogeneity, is nonzero and we also show that these target patterns have an asymptotic wavenumber that is small beyond all orders in .
The strategy of our proof is to pose a good ansatz for an approximate form of the solution and use the implicit function theorem to prove the existence of a solution in its vicinity. The analysis presents two challenges. First, the linearization about the homogeneous state is a convolution operator of diffusive type and hence not invertible on the usual Sobolev spaces. Second, a regular perturbation expansion in does not provide a good ansatz for applying the implicit function theorem since the nonlinearities play a major role in determining the relevant approximation, which also needs to be “correct” to all orders in . We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refined ansatz for the approximate solution, which obtained using matched asymptotics.
Key words and phrases:Target patterns, Nonlocal Eikonal equation, Kondratiev space, Fredholm operators, Asymptotics beyond all orders
2010 Mathematics Subject Classification:primary: 47G20; secondary: 45M05, 35B36;
Reaction-Diffusion equations describe the evolution of quantities that are governed by “local” nonlinear dynamics, given by a reaction term , coupled with Fickian diffusion,
They are generic models for patterns forming systems and have applications to a wide range of phenomena from population biology [Fis37, KPP37], chemical reactions [Win73, KH81, Mer92, TF80a], fluid [NW69, Seg69] and granular [ER99, AT06] flow patterns, and in waves in neural [Fit61, NAY62, CH09] and in cardiac [GJP95] tissue.
The onset of self-organized patterns in reaction-diffusion models typically corresponds to a bifurcation where a steady equilibrium for the local dynamics loses stability either through a pitchfork bifurcation, giving a bistable medium, or through a Hopf bifurcation, giving an oscillatory medium. In the latter case, a defining feature is the occurrence of temporally periodic, spatially homogeneous states, and an analysis of the symmetries of the system show that they can generically give rise to traveling waves, spiral waves, and target patterns [GSS88, DSSS05]. A prototypical example of these behaviors is the Belousov-Zhabotinsky reaction where chemical oscillations are manifested as a change in the color of the solution. This system displays self-organized spiral waves, i.e. they form without any external forcing or perturbation, as well as target patterns, which in contrast form when an impurity is present at the center of the pattern [SM06, PV87, TF80a].
As already mentioned, these patterns are not specific to oscillating chemical reactions and in fact can be seen in any spatially extended oscillatory medium. In this more general setting, and for the particular case of target patterns, an impurity or defect constitutes a localized region where the system is oscillating at a slightly different frequency from the rest of the medium. Depending on the sign of the frequency shift, and for systems of dimensions , these defects can act as pacemakers and generate waves that propagate away from the impurity. In the case when , these waves are seen as concentric circular patterns that propagate away from the defect.
Showing the existence of these target pattern solutions in reaction-diffusion systems, and related amplitude equations, has been the subject of extensive research, see [KH81, TF80b, Hag81, Nag91, MM95, GKS00, SM06, KS07, Jar15] for some examples as well as the reference in [CH93]. Mathematically one can describe these patterns as modulated wave trains which correspond to solutions to (1) of the form , where is a periodic function of that depends on the wavenumber, [KH81, DSSS05, KS07]. In other words, these patterns correspond to periodic traveling waves, whose phase varies slowly in time and space. Using multiple scale analysis one can show that the evolution of the phase, , over long times is given by the viscous eikonal equation,
This equation has been studied extensively in the physics literature, starting with the work of Kuramoto [KT76]. More recently it was shown that it does indeed provide a valid approximation for the phase modulation of the patterns seen in oscillatory media [DSSS05].
Since oscillating chemical reaction can be thought of as a continuum of diffusively coupled oscillators, it is not surprising that an analogue of the above equation can also be derived as a description for the phase dynamics for an array of oscillators. Indeed, in Appendix A we formally show that the following integro-differential equation provides a phase approximation for a slow-time, O(1) in space, description of nonlocally coupled oscillators,
This equation will be the focus of our paper. Here the operators and are spatial kernels that depend on the underlying nonlocal coupling between the oscillators. In particular, the convolution kernel models the nonlocal coupling of these phase oscillators and can be thought of as an analog of . Similarly, the term represents nonlocal transport along diffused gradients and is a generalization of the quadratic nonlinearity of the viscous eikonal equation. Finally, the function represents an inhomogeneity that perturbs the “local” frequency of the oscillators. Notice that the viscous eikonal equation is recovered when and are the Laplacian and the identity operator, respectively. This framework also incorporates other models for spatio-temporal pattern formation, including the Kuramoto-Sivashinsky equation which corresponds to .
Our mathematical motivation for studying the above model comes from its nonlocal aspect and the resulting analytical challenges. The approach we propose for studying the operator is novel and could be adapted to study other problems which involve similar convolution operators. In particular, the type of linear operators that we will consider are a generalization of the kernels used in neural field models or continuum coupled models of granular flow. Just as in reaction-diffusion systems, these models exhibit spatio-temporal periodic patterns, bumps, and traveling waves, (see [Coo05, Erm98, PE01, KB10a] for the case of neural field models, and [UMS98, VO98, VO01, AT06] for the case of patterns in granular flows). In particular, among the examples of traveling waves seen in experiments and replicated in the neural field models are spirals and target patterns [HTY04, KB10b, FB04], which are of interest to us.
The challenge however is that, like our model (2), these systems are not amenable to methods from spatial dynamics, which are typically used to study these phenomena. So we look for a more functional analytic approach, e.g. using the implicit function theorem, for proving the existence of solutions. As with other integro-differential equations the difficulty comes from the linearization, which is a noncompact convolution operator, and in general not invertible when considered as a map between Sobolev spaces. This is a significant analytical challenge, and one way to overcome this difficulty is to use specific convolution kernels that allow for these integro-differential equations to be converted into PDEs via the Fourier Transform [LT03], or in the radially symmetric case use sums of modified Bessel functions as models for synaptic footprint to simplify the analysis [FB04].
The approach we consider in this paper is broader as it allows us to consider a larger class of convolution kernels by showing that these operators are Fredholm in appropriate weighted spaces. This approach is similar to the one in [JS16], where we treated the one dimensional case and showed existence of target patterns in a large one dimensional array of oscillators with nonlocal coupling. For the applications we have in mind, e.g. neural field models, we need to extend these results to two dimensional arrays.
The two dimensional case is technically more interesting because a regular perturbation expansion in does not always provide the correct ansatz. Indeed, in the case with local coupling, the equation
which results from inserting the ansatz into the perturbed viscous eikonal equation, is conjugate to a Schrödinger eigenvalue problem via the Hopf-Cole transform, :
In two dimensions, it is well known that the Schrödinger eigenvalue problem has bound states if [Sim76]. Notice that the ground state eigenfunction can be chosen to be everywhere positive, so that does define a phase function solving the viscous eikonal equation with inhomogeneity. At the same time, the eigenvalue corresponding to the ground state is small beyond all orders of (see [Sim76] and Section 2 below), and is therefore not accessible to a regular perturbation expansion. This is the other analytical challenge that we have to overcome, and our approach is to develop a superasymptotic perturbation expansion for , i.e. an approximation whose error is and captures behaviors that are small beyond all orders in , [Boy99].
To show the existence of traveling waves for equation (2) we make the following assumptions on the convolution kernels and . First, we assume the kernel is a diffusive and exponentially localized kernel that commutes with rotations. Consequently, its Fourier symbol depends only on . We also impose additional properties that we specify in the Hypotheses 1 and 2. A representative example to keep in mind throughout the paper is the convolution kernel that would result in the formal operator .
We reiterate that the model (2) is derived under the assumption that the phase varies slowly in time, with no assumptions on its spatial variation. If we assume that the solutions also vary slowly in space, then hypothesis 2 implies that the nonlocal operator can be (formally) replaced by , and (2) reduces to the “local” viscous eikonal equation. Indeed, this is the setting for a substantial body of work on weakly coupled nonlinear oscillators [Kur84, SL12]. Our additional contribution is that we rigorously show the existence of target solutions of (2) that vary slowly (on a scale ) in space and time, if the model satisfies:
The multiplication operator is a function of . Its domain can be extended to a strip in the complex plane, for some sufficiently small and positive , and on this domain the operator is uniformly bounded and analytic. Moreover, there is a constant such that the operator is invertible with uniform bounds for .
Our main result, Theorem 1 requires in the following hypothesis. We state the hypothesis in more generality because some of the intermediate results also hold more generally with .
The multiplication operator has a zero, , of multiplicity which we assume is at the origin. Therefore, the symbol admits the following Taylor expansion near the origin.
The kernel is radially symmetric, exponentially localized, twice continuously differentiable, and
Our strategy to show the existence of traveling waves will be to first establish the Fredholm properties of the convolution operator following the ideas described in [JSW17]. This will allow us to precondition our equation by an operator , resulting in an equation which has as its linear part the Laplace operator. We then proceed to show the existence of target patterns in the nonlocal problem. More precisely, we prove Theorem 1 where we use the following notation:
Here denotes the space with weight .
Similarly, denotes the Hilbert space with weight .
Lastly, the symbol describes the completion of functions under the norm
We describe these last spaces with more detail in Section 3.
where and , that allows us to construct an -dependent family of target pattern solutions to (2). Moreover, these solutions have the form
In particular, as ,
, for and ; and
, where represents a constant that depends on , and is the Euler constant.
In addition, these target pattern solutions have the following asymptotic expansion for their wavenumber
If with , it follows that
Consequently, there is such that and , so that the Schrödinger operator satisfies the hypothesis in [Sim76].
Here and henceforth in the paper the function is a cut off function, whose precise form is immaterial, and satisfies for and for .
The rest of this paper is organized as follows: In Section 2 we analyze the case with local coupling and show the existence of traveling waves for the viscous eikonal equation using matched asymptotics. In Section 3 we review properties of Kondratiev spaces and state Fredholm properties of the Laplacian and related operators, leaving the proofs of these results for the appendices. Finally, in Section 4 we derive Fredholm properties for the convolution operator and then, guided by the results from Section 2, we proceed to prove Theorem 1. We present a formal derivation of the nonlocal eikonal equation (2) in Appendix A. In Appendix B, we prove various subsidiary results that are needed for the proof of Theorem 1.
2. Matched asymptotics for 2D Target patterns
As we discussed in the introduction, the viscous eikonal equation
is an abstract model for the evolution of the phase of an array of oscillators with nearest neighbor coupling [DSSS05]. The perturbation , a localized function, represents a small patch of oscillators with a different frequency than the rest of the network. It is well known that this system can produce target patterns that bifurcate from the steady state, , when the parameter is of the appropriate sign. Our aim in this section is to determine an accurate approximation to these target wave solutions using a formal approach based on matched asymptotics.
We therefore consider solutions to equation (3) of the form , where solves
We also assume in this section that is a radial and algebraically localized function that satisfies
This simplifies our analysis since we can restrict ourselves to finding radially symmetric solutions. In addition, because the viscous eikonal equation (4) only depends on derivatives of , we can recast it as a first order ODE for the wavenumber :
Here, is an eigenparameter, i.e. it is not specified, rather it is determined in such a way as to ensure that the solutions satisfy the required boundary conditions.
For radial solutions that are regular at the origin, is as , and we can rewrite the ODE in an equivalent integral form
Moreover, because we are bifurcating from the trivial state , we can assume that is small if is small, and posit the following regular expansions
Substituting these expressions in (7), we obtain that at ,
Now, because we are interested in target patterns, solutions should satisfy as , where is the asymptotic wavenumber. This requires us to consider functions that have a finite limit as , and forces us to pick . The result is that , in the limit of going to infinity.
At the same time, from assumption (5) on the algebraic localization of , we have the quantitative estimate
so that at order we find
with in . Again the boundary conditions force , and as a result we obtain for large values of . Here the constant depends on and is given by the following limit
which we know exists from the estimate for .
Note that these expressions for and imply that the asymptotic wavenumber is . In fact, continuing this procedure it is easy to check that we get for all , so that the frequency is for all orders in . In addition, the expressions for and yield the expansion
whose terms are not uniformly ordered. For instance,
This suggests that the above inner expansion is not uniformly valid. We therefore need to introduce an outer expansion and match both solution in an intermediate region given by .
In this intermediate region the inhomogeneity, , and the frequency, , are small compared to the other terms in the equation, so that the radial eikonal equation (6) reduces to
We can solve this explicitly to find that
where is a, possibly dependent, constant of integration. Comparing this result with the inner expansion (8) leads to , to leading order. We can then write , and for fixed and as , obtain
Comparing again with the inner expansion (8), we see that .
In the outer region, where we retain the frequency, , and neglect the inhomogeneity, g, solutions are described by the equation
If we define the ’outer’ variable and scaling function so that , then satisfies the (and hence also ) independent equation . Using the (differentiated) Hopf-Cole transformation , and then solving for gives
where is the modified Bessel’s function of the first kind [WW96]. Consequently, for fixed and , a solution of the outer equation is given by
where is the Euler constant [AS92]. This approximation is also valid in the intermediate region, allowing us to match it to the inner expansion,
and obtain the following approximation for the frequency
Hence, if we assume , this does indeed show that is small beyond all orders in .
The viscous eikonal equation (3) is conjugate to a Schrödinger eigenvalue problem via the Hopf-Cole transform. The frequency in the viscous eikonal equation corresponds to the ground state energy for the Schrödinger operator . Indeed, the expression above is a refinement of the results from [Sim76] and [KS07] for the ground state eigenvalue/frequency respectively, in that we have an expression for the numerical prefactor.
The matching procedure above relied strongly on the coupling kernels and being local, and having a radial inhomogeneity , so the results do not immediately carry over to the case of nonlocal coupling and/or nonradial and algebraically localized inhomogeneities. Figure 1 depicts numerical results for particular cases of the nonlocal eikonal equation (2) given by
with and in 1(a) and (b) (respectively in 1(c) and (d)). This evolution equation was integrated using a spectral discretization for the spatial operator and exponential time differencing (ETD) for the time stepping [CM02, KT05]. We will present a full discussion of our numerical methods and results in future work; here we only note that the far-field behavior of the target waves are (nearly) radially symmetric (see Figure 1) even in the general nonlocal problem. Indeed, setting with , and rescaling to the “outer variables” , we get . Further, from the algebraic localization of , it follows that
Consequently, the (formal) limit equation in the outer variables is the viscous eikonal equation
This argument, together with the numerical results depicted in Figure 1, suggests that, even for general inhomogeneities, we may approximate the dynamics of target waves solutions in the outer region by a radial viscous eikonal equation
This intuition will guide our analysis of the nonlocal equation (2) for general coupling kernels and nonradially symmetric, algebraically localized perturbations . We will show that the frequency indeed scales as , with , and that the target waves that bifurcate from the steady state are radially symmetric far away from the inhomogeneity. Our strategy consists of first finding, in Section 4.2, solutions to equation (2), with , which give the appropriate intermediate approximation. Then in Section 4.3 we find the “outer” solutions to equation (2) by treating the frequency as an extra parameter. Finally, in Section 4.4 we derive a relation between the frequency and the parameter using asymptotic matching, and then proceed to prove the results of Theorem 1.
3. Weighted Spaces
We define the Kondratiev space [McO79], , with , as the space of locally summable, times weakly differentiable functions endowed with the norm
From the definition, it is clear that these spaces admit functions with algebraic decay or growth, depending on the weight , and that these functions gain localization with each derivative. Moreover, given real numbers , such that , the embedding holds, and additionally if and are integers such that then . As in the case of Sobolev spaces, we may identify the dual with the space , where and are conjugate exponents, and in the case when we also have that Kondratiev spaces are Hilbert spaces. In particular, given the pairing
satisfies all the properties of an inner product. This is not hard to see once we notice that for every the function is in the familiar Hilbert space .
We will use this last property to decompose the space into a direct sum of its polar modes. Here we restrict ourselves to the case which is relevant for our analysis, but mention that a similar decomposition is possible in higher dimensions. More precisely, using the notation to denote the subspace of radially symmetric functions in , we show that
Given and , the space can be written as a direct sum decomposition
The proof of this result follows the analysis of Stein and Weiss in [SW16].
We need to show that each element can be well approximated by an element in the direct sum . To obtain a candidate function in the latter space we first identify with the complex plane by letting for . Then, using the notation we write for each function . By Fubini’s Theorem the function is in for a.e. , so we may express this function as a Fourier series in .
Notice that the functions are in the space , so that this sum is the desired candidate function. Because the series is monotonically increasing and because by Parseval’s identity it converges to , letting a straight forward calculation shows that
Then, by the monotone convergence theorem we may conclude
as desired. Moreover, since for a.e. the function , a similar argument can be carried out to show that as the expressions for all integers . This completes the proof of the lemma. ∎
We also have the following result describing how elements in decay at infinity (see Appendix B.1 for a proof).
Given , then as .
In addition, the next lemma characterizes the multiplication property for Kondratiev spaces. The lemma is more general than we need in the sense that it holds for complete Riemannian manifolds that are euclidean at infinity, . A proof of this result can be found in [CBC81]. We have adapted the notation so that it is consistent with our definition of Kondratiev spaces.
If is a complete Riemannian manifold euclidean at infinity of dimension , we have the continuous multiplication property
provided and .
3.1. Fredholm properties of the Laplacian and related operators
The main appeal of Kondratiev spaces for us is that the the Laplace operator is a Fredholm operator in these spaces. This is summarized in the following theorem, whose proof can be found in [McO79]. This result is the basis for deriving Fredholm properties for other linear operators that will be encountered in Section 4.
Let , , and or , for some . Then
is a Fredholm operator and
for the map is an isomorphism;
for , , the map is injective with closed range equal to
for , , the map is surjective with kernel equal to
Here, denote the harmonic homogeneous polynomials of degree .
On the other hand, if or for some , then does not have closed range.
Notice that we can use the result from Lemma 1 to diagonalize the Laplacian. That is, given ,
One can now combine this decomposition together with Theorem 2 to arrive at the following lemma.
Let , and . Then, the operator given by
is a Fredholm operator and,
for , the map is invertible;
for , the map is injective with cokernel spanned by ;
for , the map is surjective with kernel spanned by .
On the other hand, the operator is not Fredholm for integer values of .
The next lemma requires that we specify some notation. In this paper we use the symbol to denote the space of locally summable, times weakly differentiable functions endowed with the norm
Then, we write to denote the subspace of radially symmetric functions in . With this notation we can summarize the Fredholm properties of the operator , defined in the next lemma.
Given , , and , the operator defined by
and with domain , is Fredholm for . Moreover,
it is invertible for , and
it is surjective with for .
Lastly, we include the following proposition whose proof can be found in Appendix B.2.
Let , , , and . Then, the operator,
is a Fredholm operator for with kernel and cokernel given by
and not Fredholm for values of .
4. Nonlocal Eikonal Equations
At the outset, we recall our model nonlocal eikonal equation (2) and the hypotheses on the coupling kernels. Our goal for this section is to show the existence of target wave solutions for (2) which bifurcate from the spatially homogeneous solution. We concentrate on the equation
where again we assume is a diffusive kernel that commutes with rotations. As a consequence its Fourier symbol, , is real analytic and a radial function. We also recall the following assumptions.
1 The multiplication operator is a function of . Its domain can be extended to a strip in the complex plane, for some sufficiently small and positive , and on this domain the operator is uniformly bounded and analytic. Moreover, there is a constant such that the operator is invertible with uniform bounds for .
Note that, because is analytic, its zeros are isolated.
2 The multiplication operator has a zero, , of multiplicity which we assume is at the origin. Therefore, the symbol admits the following Taylor expansion near the origin.
In particular, we pick .
3 The kernel is radially symmetric, exponentially localized, twice continuously differentiable, and
The idea behind our proof for the existence of target wave solutions is to first show that the convolution operator behaves much like the Laplacian when viewed in the setting of Kondratiev spaces. In other words, both operators have the same Fredholm properties. Consequently it is possible to precondition equation (2) by an appropriate operator, , with average one, and obtain an expression which has the Laplacian as its linear part,
We then proceed in a similar manner as in Section 2 and look for solutions of the form, . Dropping the tildes from our notation we arrive at
We will look at the following two problems.
Finding a solution to the intermediate approximation, described by equation (10) with the value of the frequency, , equal to zero.
Finding a solution valid on the whole domain described again by equation (10), but where we let be a non-negative parameter.
These two solutions are matched, and then the results of Theorem 1 are shown.
This section is organized as follows. In the next subsection we will derive Fredholm properties for the convolution operator , as well as mapping properties for a number of related convolution operators. Then in Sections 4.2 and 4.3 we prove the existence of solutions to the intermediate approximation and to the full problem, respectively. Finally, in Section 4.4 we prove Theorem 1.
4.1. Nonlocal Operators
The following proposition is the 2-dimensional version of the results form [JSW17], but for convolution kernels with radial symmetry. The results below follow very closely the proofs outlined in Ref. [JSW17], and we include them for the sake of completeness. The proof shows that, with the Hypothesis 1 and the more general version of Hypothesis 2 in which we assume , the convolution operator has the same Fredholm properties as the operator .