Spiral waves are a type of self-organization observed in alarge variety of spatially extended, thermodynamically non-equilibrium systems of physical, chemical and biological na-ture �1–19�, where wave propagation is supported by asource of energy stored in the medium. If the system can beconsidered spatially uniform and isotropic and its propertiesdo not depend on time, the corresponding mathematicalmodels possess corresponding symmetries. For many practi-cal applications, considerable interest is in nonstationary dy-namics of spiral waves, which is usually defined separatelyeither as drift, which is displacement of the average positionof the core of the spiral with time due to external symmetry-breaking perturbations, or meandering, which is spontaneoussymmetry breaking due to internal instability rather than ex-ternal forces and which is manifested by complicated move-ment of the spiral with the average position of the core typi-cally unmoved.
The numerical simulation of drift and meander of spiralwaves, particularly when models are complicated and highaccuracy is required, can be challenging. There are sometheoretical considerations which suggest some way of deal-ing with this challenge. So it has been observed that as far asdrift is concerned, spiral waves behave like particle-like ob-jects, which results from effective localization of the criticaleigenfunctions of the adjoint linearized operator �20–24�, soit should be sufficient to do the computations only around thecore of the spiral to predict its drift. On the other hand, in theabsence of external symmetry-breaking perturbations, mean-dering of spirals can be understood by explicitly referring tothe Euclidean symmetry of the unperturbed problem�25–30�. Specifically, an idea of dynamics in the space ofsymmetry group orbits �31�, when applied to a reaction-diffusion system of equations and the Euclidean symmetrygroup, leads to a description which is formally equivalent toconsidering the solution in a moving frame of reference�FoR� such that the spiral wave maintains a certain positionand orientation in this frame �29�. We shall call it comovingFoR for short.
The purpose of this paper is to present a computationalapproach based on these considerations. We calculate the dy-namics of the spiral wave in a comoving FoR; as a result, thecore of the spiral never approaches the boundaries of thecomputation box, which allows computations of drift and
meandering of large spatial extent using small numericalgrids. A simple software implementation of this approach,which is based on the popular spiral wave simulator “EZ-
SPIRAL” �32,33�, and which we called “EZRide,” is providedon the authors’ website �34�.
Our approach can be compared to the approach proposedby Beyn and Thummler �35� and further developed by Her-mann and Gottwald �36�. Their approach also exploits sym-metry group orbits but is different in some essential details.We shall discuss the similarities and differences when wewill have introduced our method.
The structure of the paper is as follows. In Sec. II we layout mathematical basics of the approach and briefly compareit with �35�. In Sec. III we describe the numerical methoditself. In Sec. IV we illustrate the work of the method bysimple and quick examples. The potential for numerical ac-curacy is demonstrated in Sec. V. The subsequent three sec-tion are dedicated to examples of applications of the methodsto problems where the conventional methods would be strug-gling: Sec. VI for the degenerate case of meandering whichresults in “spontaneous drift” of spirals; Sec. VII for thedynamics near to, and beyond, the parametric boundary atwhich the core radius of the spiral becomes infinite; and Sec.VIII for drift caused by a symmetry-breaking perturbation.We conclude with a brief discussion of the results in Sec. IX.
II. SYMMETRY GROUP REDUCTION
Following �29�, we start from a perturbed reaction-diffusion system of equations in a plane,
�u
�t= D�2u + f�u� + h�u,�u,r�,t� , �1�
where u= �u�1� , . . . ,u�n���=u�r� , t��Rn is a column vector ofreagent concentrations varying in space and time, f= f�u� is acolumn vector of reaction rates, D�Rn�n is the matrix ofdiffusion coefficients, h�Rn represents symmetry-breakingperturbations, �h��1, n�2, and r�= �x ,y��R2. If h=0, thenEq. �1� is equivariant with respect to Euclidean transforma-tions of the spatial coordinates r�.
The following technical discussion is necessary to placeour method in the context of other works in the field. Read-ers not interested in technical details may skip down to sys-tem �16�.
The idea of the symmetry group reduction is convenientto describe if we view Eq. �1� as an ordinary differentialequation �ODE� in a suitably chosen functional space B,
dU
dt= F�U� + H�U,t� �2�
where U :R→B represents the dynamic field u, F :B→Brepresents the unperturbed right-hand side D�2u+ f, andH :B�R→B represents the perturbation h.
Let us suppose that Eq. �2� at h=0 is equivariant withrespect to a representation T of a Lie group G in B. Thismeans that for any g�G and any U�B, we have
F„T�g�U… = T�g�F�U� . �3�
In our case, G=SE�2�, the special Euclidean transformationsof the plane R2→R2 �including translations and rotations�,and T is its representation in the space of functions u�r��defined on this plane, acting as
T�g�u�r�� = u�g−1r�� . �4�
We consider a subset B0�B such that G acts freely on B0;i.e., for a U�B0, any nontrivial transformation g�Gchanges U; and in other words, T�g�U=U⇒g=id. In theterminology of �31�, B0 is the principal stratum of B, corre-sponding to the trivial isotropy subgroups. In our case, thismeans that the graph of the function u�r���B0 is devoid ofany rotational or translational symmetry, which is of coursetrue for functions describing single-armed spiral waves.
It is straightforward that at H=0, the set B0 is an invariantset of Eq. �2�. Moreover, we shall restrict our considerationto such perturbations H�t� that resulting solutions U�t� re-main in B0 for all t. This means, that the perturbations aresupposed to be so small that they cannot impose incidentalsymmetry on the otherwise unsymmetric spiral wave solu-tions.
A group orbit of a given U is defined as the set T�G�U= �T�g�U �g�G�. That is, it is a set of all such functionsu�r�� that can be obtained from one another by applying anappropriate Euclidean transformation to r�. A group orbit isa manifold in B0, of a dimensionality equal to d=dim Gless the dimensionality of the isotropy group. In our case,dim SE�2�=3, the isotropy group is trivial and the orbits aresmooth three-dimensional manifolds.
From the definition of the set B0 it follows that this set isfoliated by group orbits. The principal assumption for thefollowing analysis is that there exists an open subset S�B0,also invariant with respect to G, in which the foliation has aglobal transversal section, i.e., we can select one representa-tive from each orbit in S, such that all such representativesform a smooth manifold M�S, which is everywhere trans-versal to the group orbits. We call this manifold a represen-tative manifold �RM�. That would mean that any orbit in Scrosses M transversally and exactly once. Hence
∀U � S, ∃!�g,V� � G � M: U = T�g�V . �5�
The RM has codimensionality equal to the dimensionality ofthe group orbits, i.e., in our case codim M=d=3. It is as-sumed to be smooth and we expect that it can locally be
described by equations ���V�=0, �=1, . . .d, where functions�� :B→R, i.e., are functionals when interpreted in terms oforiginal reaction-diffusion Eq. �1�.
A convenient pictorial interpretation for our case is interms of spiral wave solutions and their tips. Suppose theconditions �1�V�=0, �2�V�=0 determine that the tip of thespiral wave is located at the origin, and condition �3�V�=0fixes its orientation, so M consists of such functions V thatlook like spiral waves which have the tip exactly at the originand in a standard orientation. Then Eq. �5� states that anyspiral wave solution u�r��, considered at a fixed moment oftime, can be transformed by a Euclidean transformation, in aunique way, to a solution v�r�� which has its tip at the originand in the standard orientation. This is equivalent to sayingthat v�r�� is the same as u�r�� only considered in a differentsystem of coordinates, with the origin at the tip of u�r�� andoriented accordingly to the orientation of that tip. We shallsay this is the system of coordinates attached to the tip. Anexample of ��, as used, e.g., in �29�, is
�1�v�r��� = v�l1��0�� − u�, �6a�
�2�v�r��� = v�l2��0�� − v�, �6b�
�3�v�r��� = �xv�l3��0�� , �6c�
where �l1 , l2 , l3�� �1, . . . ,n� are suitably chosen components,and l1� l2. This means that the tip of u�r�� is defined as thepoint of intersection of isolines of the components l1 and l2of the field u at appropriately chosen levels u� and v� respec-tively, and the orientation of the attached coordinate systemis such that gradient of component l3 �which may or may notcoincide with l1 or l2� is along the y-axis in that system. Thischoice of �� is of course not prescriptive, and later in thispaper we shall consider some variations.
Regardless of the exact definition of the tip, i.e., choice offunctionals ��, an essential assumption that we have to makeis that our spiral waves have one tip only, otherwise therewould be more than one way to transform them to the stan-dard position or to chose the attached system of coordinates.Hence the reason for a further constraint to the subsetS�B0, which we now can define as consisting of such one-tip spiral wave solutions, or functions that look like it: with-out such constraint, the whole set B0 includes solutions withno tips or more than one tip, for which decomposition �5�would not hold. As before, we assume that set S is invariantwith respect to dynamic Eq. �2� for not too big �H�; that is, ifU�0��S, then U�t��S for all t�0 and �H��Hmax.
A further restriction is on the manifold M. It is easy tosee that equations such as Eq. �6� may not be sufficient todefine the manifold with the required property that any orbitcrosses it only once. For instance, if v�r�� satisfies Eq. �6�,then v�−r�� also satisfies it, so a rotation by 180° around theorigin transfers a point on S to another point on S. So tomake representation �5� unique, rather than just requiringthat the gradient of the l3 component of v�r�� is along the yaxis, one would need to specify in which direction it is, sayadd to the definition of M by the equations �1,2,3�v�=0 afurther inequality
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�4�v� � 0, where �4�v�r��� = �yv�l3�. �6d�
This comment extends to the variations of Eq. �6� which weconsider later.
By performing decomposition �5� for every t�0, we de-compose motion in S to motion along the RM and motionalong group orbits which are transversal to the RM �see il-lustration in Fig. 1�.
So for all t�0, we have
U�t� = T�g�V�t� �7�
Substituting Eq. �7� into Eq. �2� and applying T�g−1� to bothsides, we get
T�g−1�dT�g�
dtV +
�V
�t= F�V� + H�V,g,t� �8�
where
H�V,g,t� = T�g−1�H�T�g�V,t� . �9�
We note that if H=0, the right-hand side of Eq. �8� is inde-pendent of g.
By the assumptions made, intersection of the group orbitT�G�V with the manifold M at the point V is transversal.
This means that the vectors F�V� and H�V ,g , t� can beuniquely decomposed into the sums of the components alongthe group and along the manifold,
F�V� = FG�V� + FM�V� , �10a�
H�V,g,t� = HG�V,t� + HM�V,t� . �10b�
Hence Eq. �8� splits into two components, along the RMand along the group orbit �GO�,
�RM��V
�t= FM�V� + HM�V,t� , �11a�
�GO� T�g−1�dT�g�
dtV = FG�V� + HG�V,t� . �11b�
Note that Eq. �11a� is the equation on the infinite-dimensional manifold M, i.e., corresponds to a partial dif-ferential equation �PDE�, whereas the left- and right-handsides of Eq. �11b� are in the tangent space to the finite-dimensional group orbits, and the dynamic variable g is anelement of the finite-dimensional manifold G, so Eq. �11b� isin fact a system of ordinary differential equations of orderd=dim G.
At this point we comment on what we see as a significantdifference between our approach and that proposed by Beynand Thummler �BT� �35�. Using our notation, in place ofour “pinning” conditions ���V�=0, �=1, . . . ,d, they defined“phase conditions” of the form ���V ,g�=0 �see Eq. �2.22� in�35��, subsequently further generalized to ���V ,g ,dg /dt�=0 �Eq. �2.33� in �35��. This means that their decompositionU=T�g�V is not uniquely determined by the current state Ubut depends on history. Such generalization may have itsadvantages and, apparently, works well for relative equilib-ria, i.e., steadily rotating spirals �35,36�. However, the situa-tion is different if the solution is a meandering spiral, i.e., isperiodic with period P in the orbit space �as illustrated inFig. 1�. This means that U�t+ P� is equivalent to U�t�up to some Euclidean transformation. In our approach,it is then guaranteed, that V�t+ P�=V�t�, as by Eq. �5�,���T�g−1�U�=0 has a unique solution for g at a given U.However, in the BT approach, typically V�t+ P��V�t�, since���T�g−1�U ,g ,dg /dt�=0 does not uniquely define g, asdg /dt is not fixed. So in our approach, study of meanderingspirals reduces to study of periodic solutions for V�t�, but itdoes not do so in the BT approach.
A practical approach to the problem of decomposing thevector fields as in Eq. �10� is as follows. Equation �11� to-gether with the definition of the RM via functionals �� canbe rewritten in an equivalent form
�V
�t= F + H + A , �12a�
���V�t�� = 0, � = 1, . . . ,d , �12b�
T�g−1�dT�g�
dtV = − A , �12c�
where A=A�V , t�=−FG�V�−HG�V , t� is a vector belongingto the three-dimensional tangent space of the group orbitT�G��V� at V. In this formulation, at any given moment oftime, Eqs. �12a� and �12b� together define the evolution of Vand the current value of the vector A, whereas Eq. �12c�defines the evolution of g.
By definition, vector A is a result of action of a linearcombination of the generators of the Lie group T�G� as linearoperators on V. To write the explicit expression for the gen-eral form A for our case, let us introduce coordinates �R� ,��
BG
M
gg′
g′′
UU′
U′′
V = V′′
V′
FIG. 1. �Color online� Sketch of skew-product decomposition ofan equivariant flow using a representative manifold M, which hasexactly one transversal intersection with every group orbit g�Gwithin the relevant stratum of the phase space B and is diffeomor-phic to the orbit manifold. Trajectory �U ,U� ,U�� of an equivariantflow in B is a relative periodic orbit, since it projects onto thetrajectory �V ,V� ,V�=V� on M which is periodic. The flow on Mis devoid of symmetry G.
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on G=SE�2�, where R� = �X ,Y� is the translation vector, � isthe rotation angle and a group element acts as
g = �R� ,��:r� � R� + e��r� , �13�
where �= � 0 −11 0 �, so exp���� is the matrix of rotation by
angle �.Using this representation, differentiating the definition of
T�g�v given by Eq. �4�, and substituting the result into Eq.�12c�, we get
A = �v + �c� · ��v , �14�
where
= �, c� = e−��R� , �15�
and is the polar angle in the �x ,y� plane, so �=x�y −y�x.With this result, system �12� in the original PDE notation
states that
�v
�t= D�2v + f�v� + h�v,e�� � v,R� + e��r�,t�
+ �c� · ��v + �v
�, �16a�
v�l1��0� ,t� = u�, v�l2��0� ,t� = v�, �16b�
�v�l3��0� ,t��x
= 0,�v�l3��0� ,t�
�y� 0, �16c�
d�
dt= ,
dR�
dt= e��c� , �16d�
where the dynamic variables are v�r� , t�, c��t�, �t�, R� �t�, and��t�.
In terms of the tip of the wave, Eq. �16a� is the originalreaction-diffusion Eq. �1� written in the comoving FoR, Eqs.�16b� and �16c� define the attachment �pinning� of the tip tothis FoR, and Eqs. �16d� describe the movement of the FoRand, therefore, of the tip.
Equations �16b� and �16c� imply that the position�xtip ,ytip� and orientation � of the tip during calculations inthe laboratory FoR are defined as
and the comoving FoR is chosen so that in it, �xtip ,ytip�= �0,0� and �=� /2 at all times. Unlike other equations ofsystem �Eq. �16��, these are not prescriptive and is essentiallyan arbitrary choice, dictated by properties of particular sys-tems. We shall refer to pinning conditions �16b� and �16c� as“choice 1,” as below we shall consider a variation in these,which we call “choice 2.”
When h=0, system �16� decouples, as its upper part in-cluding Eqs. �16a�–�16c� becomes independent of the lowerpart �Eq. �16d��. This is the “skew-product” decomposition,the upper part describing the dynamics in the space of grouporbits, so-called “quotient system,” and the lower part the“symmetry group extension,” i.e., dynamics along the group,which depends on but does not affect the quotient dynamic.The connection between the quotient system and the groupextension is via the dynamic variables �c� ,�; in the follow-ing, we refer to these three quantities as “quotient data” forbrevity.
The skew-product representation has been useful forthe analysis of various types of meander of spiral waves�29,30,37,38�. Note that the approach used in �30,38� �alsosee references therein� is based on the assumption that themeandering pattern in question is considered in the vicinityof a bifurcation from the rigidly rotating spiral wave solutionso that the quotient dynamics can be reduced to the centermanifold, hence instead of Eqs. �16a�–�16c�, these studiesconsidered normal forms on the corresponding center mani-folds. However, as noted in �39�, the centre manfold theoremis not applicable for spiral waves, so this approach seems tobe fundamentally flawed. This technical difficulty of coursedoes not in any way affect the validity of system �16�, which,as we have just demonstrated, is derived by elementarymeans without recourse to any bifurcations.
In the rest of the paper, we consider system �16� as acomputational tool rather than an instrument of theoreticalanalysis. The disadvantage of original system �1� as a com-putational tool is that it requires a big computational grid tosimulate dynamics of a spiral in an infinite medium, particu-larly when the tip of the spiral performs excursions to largedistances. This is actually not necessary, as the dynamics ofthe spiral is mostly determined by the events in some finitevicinity of its tip �24�. System �16� takes advantage of thisproperty so that the PDE calculations are done always insome fixed vicinity of the spiral wave, whereas the move-ment of the tip is described by the ODE part.
III. NUMERICAL IMPLEMENTATION
A. Discretization
We use time discretization with constant step t andsquare spatial grid with step x, covering spatial domain�x ,y�� �−L /2,L /2�2, so that
v„�i − i0� x,�j − j0� x,k t… vi,jk = �vi,j
�l�,k�l = 1, . . . ,n�,
i = 0, . . . ,Nx, j = 0, . . . ,Ny ,
Nx = Ny = L/ x,
and the grid coordinates of the origin are
i0 = �Nx + 1�/2, j0 = �Ny + 1�/2
�we only use odd values of Nx=Ny�. We designate
the kth time layer of the numerical solution as Vk
= �vi,jk � i=1, . . . ,Nx , j=1, . . . ,Ny�. We discretize the ODE dy-
namic variables on the same time grid, i.e., R� �k t�R� k, etc.
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B. Operator splitting
We rewrite Eq. �16a� in the form
�v
�t= F�v� + H�v;R� ,�� + A�v;c�,� �18�
where differential operators F, H, and A are defined as
F�v� = D�2v + f�v� , �19a�
H�v;R� ,�� = h�v,e�� · �v,R� + e��r�,t� , �19b�
A�v;c�,� = �c� · ��v + �v
�= �cx − y�
�v
�x+ �cy + x�
�v
�y.
�19c�
Let F, H, and A be discretizations of F, H, and A. Ourcomputations proceed as follows:
Vk+1/3 = Vk + tF�Vk� , �20a�
Vk+2/3 = Vk+1/3 + tH�Vk+1/3,R� k,�k� , �20b�
Vk+1 = Vk+2/3 + tA�Vk+2/3,c�k+1,k+1� , �20c�
�1,2,3�Vk+1� = 0, �4�Vk+1� � 0, �20d�
�k+1 = �k + tk+1, �20e�
R� k+1 = R� k + te��k+1
c�k+1. �20f�
C. Kinetics
As specific examples, we consider two models, theFitzHugh-Nagumo �FHN� model �40,41�,
f:u
v� � �−1�u − u3/3 − v�
��u + � − �v� � �21�
and Barkley’s model �32,42�,
f:u
v� � �c−1u�1 − u��u − �v + b�/a�
u − v , �22�
both with D= � 1 00 0 �.
D. Reaction-diffusion step
The computational scheme is designed as an extension tothe standard approach to simulation of spiral waves. Specifi-cally, we chose Barkley’s EZ-SPIRAL �32,33,42� as the start-ing point and extended it to add the other computationalsteps. So reaction-diffusion step �20a� is as implemented inEZ-SPIRAL, with central five-point difference approximationof the Laplacian, without any features specific to the Barkleymodel, such as implicit treatment of the kinetic terms, and
with appropriate modifications when FitzHugh-Nagumomodel is used.
E. Perturbations
We consider one particular type of nonzero perturbation,the electrophoresis,
h = E�xu ,
h = E�cos����xv�r� − sin����yv�r�� , �23�
where E is a diagonal matrix, E= �E1 00 E2
�, �E��1. For areaction-diffusion system this perturbation can describemovement of the reagents in response to electric field withvelocities −E1 and −E2 along the x axis. For E=�D, thisperturbation can also approximately describe the movementof an axially symmetric scroll ring. For a cylindrical systemof coordinates �r , ,z�: x=r cos , y=r sin , z=z, the diffu-sion term has the form D�2u=D��r
2+r−1�r+r−2�2+�z
2�u,which for �=0 and large r is equivalent to an unperturbeddiffusion term with a two-dimensional Laplacian in �r ,z�plane plus a small perturbation 1
r D�ru. If the filament of thescroll is located at large values of r�1 /� and as the dynam-ics of the scroll is mostly determined by the events near itsfilament, then 1 /r can be approximately replaced with �.
Perturbation �23� violates only rotational symmetry of theproblem, preserving symmetry with respect to translations
in space and time. Hence h explicitly depends only on �.This limitation is not principal and translation symmetry-breaking perturbations can be considered similarly, in which
case h would also explicitly depend on X ,Y and/or t. Wediscretize the first spatial derivatives in the perturbation termusing upwind second-order accurate differences, and use ex-plicit Euler time stepping. In the absence of perturbations,h=0, perturbation step �20b� is of course omitted and
Vk+2/3= Vk+1/3.F. Tip definition and pinning conditions
Discretization of pinning conditions �16b� and �16c�, us-ing l1= l3, and the right-side first-order discretization of thex-derivative, gives
vi0,j0
�l1�,k = u�, �24a�
vi0,j0
�l2�,k = v�, �24b�
vi0+1,j0
�l1�,k = u�, �24c�
where �i0 , j0� are grid coordinates of the origin. This worksin principle but gives rather inaccurate and noisy approxima-tions for , which get worse for finer discretizations. This istypical for numerical differentiation. We overcome this byenhancing the spatial discretization step, by replacing condi-tion �24c� with
vi1,j1
�l1�,k = u�, �25�
where the grid point �i1 , j1� was chosen some way away fromthe center point, �i1 , j1�= �l0 , j0�+ �iinc , jinc�. This means re-placing third pinning condition �16c� with
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v�l1��r�inc,t� = u�, �26�
where r�inc= � xiinc , xjinc�. Empirically, we have found thatthe length of the displacement �r�inc� should be of the order of,but not exceeding, one full wavelength of the spiral.
This revised orientation-pinning condition still does notdefine the position uniquely, as illustrated by Fig. 2. An extrainequality is required to distinguish between different solu-tions satisfying conditions �24a�, �24b�, and �25�. We use
vi1,j1
�l1�,k � v�. �27�
corresponding to
v�l1�,k�r�inc,t� � v�. �28�
Specifically, we chose l1= l3=1 and l2=2. Conditions �27�and �28� then mean that the third pinning condition �Eqs.�25� and �26�� ensures that the front, rather than the back, ofthe excitation wave passes though the grid point �i1 , j1�. SoEqs. �16b�, �26�, and �28�, with discretizations �24� and �27�are our “choice 2” pinning conditions.
The choice 1 and choice 2 pinning conditions define dif-ferent RMs and different quotient data c��t� ,�t�, for thesame solution u�r� , t�. However, the two FoRs they definehave a common origin and differ only by the orientationangle. So if �c� ,� are quotient data for choice 1 pinningconditions, and �c�� ,�� are quotient data for choice 2 pin-ning conditions, then we have
c� = e���−�/2�c��, = � + d�/dt , �29�
where � the tip orientation angle in choice 2 comoving FoR,so �−� /2 is angle of one FoR against the other.
G. Advection
We use an upwind second-order accurate approximation
of the spatial derivatives in A. Steps �20c� and �20d� aredone in conjunction with each other. The discretization of
Vk+1 at the tip pinning points, resulting from Eq. �20c�, isused in the three equations �Eq. �20d�� to find the three un-knowns c�x
k+1, c�yk+1, and k+1 so that the pinning conditions
�Eq. �20d�� are always satisfied exactly �to the processor pre-cision� after every step �52�.
H. Boundary conditions
Since the boundaries in the comoving FoR do not repre-sent any physical reality but are only a necessity of numeri-
cal approximation, the results can only be considered to bereliable if they do not depend on the boundary conditions. Sowe use both Dirichlet and Neumann boundary conditions andcompare the results. For Dirichlet conditions, we use bound-ary values of the resting state vr, such that f�vr�=0.
I. Tip trajectory reconstruction
Steps �20e� and �20f� are simple first-order implemen-
tations of the corresponding ODEs. The resulting � is usedin calculations of the H step when the perturbation is on.
Otherwise, � and R� are calculated only for the record.
J. Some details of software implementation
For stability purposes, we ensure that the following in-equalities are observed during computations:
�cx� � x
2
2 t,
�cy� � x
2
2 t,
�� �1
NX t.
This is an empirical choice motivated by von Neumann sta-bility analysis.
When the absolute values of cx and cy found in Eqs. �20c�and �20d� are beyond these limits then they are restricted tothe intervals stated above. Also, we eliminated the need torestrict the values of cx and cy to their stability limits bymoving the spiral wave solution so that the tip of the spiralwave is in the center of the box, using the standard EZ-
SPIRAL’s “mover” function, which performs translation of thesolution by an integer number of grid steps, suitably extrapo-lating the solution where necessary near the boundaries.
For , we implemented the restriction that if �� exceededits maximum stability value, then we set =0. Effectivelythis means that unless the orientation of the spiral wave isalready very near the standard orientation satisfying Eq. �25�and inequality �27�, the code computes a solution of theproblem
instead of Eq. �16�. That is, it performs reduction by thesubgroup of translations of the Euclidean group.
A typical run of the program in the interactive mode startsfrom obtaining a spiral wave solution in the standard “ride-off” mode, by solving initial-value problem �1�. When the
A
B
FIG. 2. �Color online� Nonuniqueness of the revised tip pinningcondition.
A. J. FOULKES AND V. N. BIKTASHEV PHYSICAL REVIEW E 81, 046702 �2010�
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spiral wave is initiated so there is one tip in the solution, theuser switches the program to the “ride-on” mode, with cal-culations according to the above scheme. On the switch, theprogram first of all moves the tip of the spiral to the center ofthe box via EZ-SPIRAL’s “mover” function, i.e., by paralleltranslations of the solution, supplementing the missingpieces near boundaries by duplicating the existing boundaryvalues. From then on, the spiral continues to rotate with itstip fixed at the center of the box, thus solving problem �30�.In this regime, only the first two pinning conditions are sat-isfied, and only cx and cy are calculated and used, where as is calculated but replaced with zero, until it falls within thestability limit and the fourth inequality-type pinning condi-tion is satisfied. From that point, the program proceeds in thefully engaged mode, calculating problem �16�.
IV. PRIMARY EXAMPLES: RIGIDLY ROTATIONAND MEANDER
First we illustrate how our approach works using two ex-amples. One example uses Barkley model with rigidly rotat-ing spiral waves, and the other is FitzHugh-Nagumo modelwith meandering spiral waves.
Figure 3 illustrates the work of EZRide in the case of arigidly rotating spiral wave. The panels represent three con-secutive runs, in different regimes: the “direct numericalsimulations” �DNSs� of system �1�, then the “skew-product”calculation in the comoving FoR, and then again the DNS inthe laboratory FoR. The skew-product calculation in turnconsists of two parts. The first part is described by Eq. �30�
where only the two translation pinning conditions are en-gaged, so that the position of the tip of the spiral is fixed, butnot its orientation, so the FoR is cotranslating but not coro-tating. The second part is where all four pinning conditionsare engaged, and the FoR is cotranslating and corotating. It isseen from Fig. 3, that after a transient period, the solution inthe fully comoving FoR becomes stationary. This corre-sponds to the definition of a rigidly rotating spiral wave as arelative equilibrium.
Figure 4 shows a similar set of runs for a different case,where the spiral wave is not stationary but is meandering. Inthis case, the solution in the comoving FoR is not stationary,but periodic in time. This corresponds to the definition of ameandering spiral wave as a relative periodic orbit.
Figures 5�a� and 5�b� show selected pieces of tip trajecto-ries obtained as a result of the runs shown in Figs. 3 and 4.The discretization steps there are deliberately chosen crudeto allow very fast running simulations and also to illustratethe difference introduced by the change in method of com-putation. The tip trajectories obtained by reconstruction fromthe quotient data are qualitatively similar to the tip trajecto-ries obtained in DNS. However, the quantitative difference isalso quite evident. In the case of rigid rotation, the recon-structed trajectory radius is noticeably bigger than that fromDNS, and the centers of the meandering patterns in differentruns are offset against each other. As panels �c� and �d� in thesame figure show, these discrepancies decrease when the dis-cretization steps are refined.
Figure 6 shows the tip and quotient system trajectories,obtained in laboratory and comoving FoR calculations, for ameandering spiral. This is drawn for the finer discretization
0 1 2 3 4 5
6 7 8 9 10 11
11 12 13 14 15 16
17 18 19 20 21 22
22 23 24 25 26 27
28 29 30 31 32 33
FIG. 3. �Color online� Three consecutive runs of Barkley model,a=0.52, b=0.05, c=0.02, L=20, x=1 /5, t=1 /2000, and r�inc
= �2,0�. The runs t� �0,11� and t� �22,33� are direct simulations.The run t� �11,22� is a quotient system simulation, the pinningpoints are indicated by small white crosses. The third pinning con-dition is engaged at t�16.5.
0 2 4 6 8 10
12 14 16 18 20 22
22 24 26 28 30 32
34 36 38 40 42 44
44 46 48 50 52 54
56 58 60 62 64 66
FIG. 4. �Color online� Three consecutive runs of FHN model,�=0.2, �=0.7, �=0.5, L=30, x=1 /3, t=1 /720, and r�inc
= �20 /3,0�. The runs t� �0,22� and t� �44,66� are direct simula-tions. The run t� �22,44� is a quotient system simulation, the pin-ning points are indicated by small white crosses. The third pinningcondition is engaged at t�27.5.
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steps, as in Fig. 5�d�. For comparison, quotient data for boththe laboratory and comoving FoR calculations were recalcu-lated for choice 1 pinning conditions using Eq. �29�. There isgood agreement between the two methods of calculations,within the expected accuracy. More detailed analysis of thenumerical accuracy of our method is given in the next ses-sion.
V. NUMERICAL CONVERGENCE
Figure 7 illustrates the convergence of the results of cal-culations of rigidly rotating spiral, using EZRide with Neu-mann and Dirichlet boundary conditions, and DNS usingNeumann boundary conditions. In these calculations, the boxsize is fixed at L=60 and the time step is changed with thespace step so that t= x
2 /40. For � x2� dependence, we also
show the angular velocity measured in direct numericalsimulations. We do not show �c�� x
2�� found in DNS, sinceobtaining it involves numerical differentiation which givesaccuracy insufficient for the convergence study.
Our discretizations are second order accurate in x andfirst order accurate in t both in DNS and in the riding mode,which corresponds to linear dependence of any results on x
2
for x→0. We see in Fig. 7 that this is indeed the case.Linear extrapolation of the � x
2� gives the values of �0�
for laboratory and comoving calculations coinciding towithin 10−3.
One of the advantages of EZRide is the fact that the simu-lations can be done in a smaller box compared to DNS. So,the last test is convergence in box size. We have calculatedthe rigidly rotating spiral by EZRide at fixed x=1 /15, t=1 /9000 and L varying through �15,60� and found that both�c�� and �� vary by less than 10−3.
VI. APPLICATION I: THE 1:1 RESONANCEIN MEANDERING SPIRAL WAVES
One of the cases where the DNS would meet with diffi-culties is the study of the meandering of spiral waves forparameters near the “1:1 resonance” between the Euclideanand the Hopf frequencies. This case is marginal between me-andering patterns with inward petals and outward petals.
-2
0
2
4
6
-4 -2 0 2 4
1234
X
Y
-3
-2
-1
0
-2 -1 0 1
1234
X
Y
(a) (b)
-2
0
2
4
6
-4 -2 0 2 4
1234
X
Y
-3
-2
-1
0
-2 -1 0 1
1234
X
Y
(c) (d)
FIG. 5. �Color online� �a� and �b� Reconstructed tip trajectoriesfrom �a� simulation shown in Fig. 3 and �b� simulation shown inFig. 4. The pieces labeled 1 are trajectories obtained in direct simu-lations in the laboratory FoR. The pieces labeled 2 are trajectoriesobtained via cotranslating simulations, with first two pinning con-ditions engaged. The pieces labeled 3 correspond to comoving�cotranslating and corotating� simulations with all three pinningconditions engaged. The final pieces labeled 4 correspond to directsimulations in a nonmoving FoR, which has been displaced withrespect to the laboratory FoR during the quotient system simula-tions. �c� Same as �a�, with x=1 /10, t=1 /4000. �d� Same as �b�,with x=1 /10, t=1 /4000.
-2
-1
0
1
2
3
-2 -1 0 1 2 3
12
3
4
5
6
7
EZRideDNS
X
Y
0.1
0.2
0.3
0.4
0.5
0.6
0.2 0.3 0.4 0.5 0.6
EZRideDNS
cx
c y
(a) (b)
-0.7
-0.6
-0.5
-0.4
-0.3
0.2 0.3 0.4 0.5 0.6
EZRideDNS
cx
ω
-0.7
-0.6
-0.5
-0.4
-0.3
0.2 0.4 0.6
EZRideDNS
cy
ω
(c) (d)
FIG. 6. �Color online� Meander in the FHN model, calculated inthe laboratory frame of reference �DNS�, and from quotient system�EZRide�. In �a�, the meandering pattern is shown, which for theEZRide curve is obtained by numerical integration of quotient datausing Eq. �15�. In �b�–�d�, the projections of the quotient data areshown, which for the DNS curves are obtained by numerical differ-entiation of the tip trajectory, using Eq. �15�.
2.05
2.06
2.07
2.08
2.09
2.10
0 0.02 0.04 0.06 0.08
NeumannDirichlet
∆2x
|�c|
0.75
0.80
0.85
0.90
0.95
1.00
0 0.02 0.04 0.06
NeumannDirichlet
DNS
∆2x
|ω|
(a) (b)
FIG. 7. �Color online� Convergence of the rigidly rotating spiralwave solution in the Barkley’s model.
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Near the resonance, the spatial extent of the meandering tra-jectory becomes large, and for the case of exact resonance,infinite, and the spiral appears to be spontaneously drifting�25,43�. Hence, following the dynamics of the spiral wave inthe comoving FoR presents an advantage.
We illustrate this using the FHN model. We fix the dis-cretization parameters at x=1 /8, t=1 /2560, and L=20.The choice of model parameter is influenced by Winfree’s
“Flower Garden” �44�, which gives a rough estimate forthe location of the 1:1 resonance line in the �� ,�� plane at�=0.5. Using this information, we have selected two values�=0.2 and �=0.25, and scanned values of � across the re-sonance value, which we determined as �0�0.935 35 for�=0.2, and �0�0.813 62 for �=0.25 at our discretizationparameters.
The results are presented on Figs. 8–11. The shape oftrajectories is well known from the theory, and is outward
9
15
21
27
33
9 15 21 27 33
X
Y
0
10
20
30
40
-20 -10 0 10 20
XY
(a) β = 0.91535 (b) β = 0.92535
-350
-250
-150
-50
50
-250 -50 150
X
Y
-55
-45
-35
-25
-15
-20 -10 0 10 20
X
Y
(c) β = 0.93535 (d) β = 0.93535
0
10
20
30
40
5 15 25 35 45
X
Y
9
13
17
21
25
1 5 9 13 17
X
Y
(e) β = 0.94535 (f) β = 0.95535
FIG. 8. �Color online� The reconstructed tip trajectories inFitzHugh-Nagumo system with �=0.2, �=0.5 and varying �.
cxcy
ββ
0.955350.945350.935350.925350.91535
-0.2
-0.2-0.4-0.6-0.8-1
00.20.4 cx ω
ββ
0.955350.945350.935350.925350.91535
-0.2-0.2-0.4
-0.60
0.20.4 cyω
ββ
0.955350.945350.935350.925350.91535
-0.2-0.4
-0.4
-0.6-0.6
-0.8-1
0
(a) (b) (c)
FIG. 9. �Color online� Various projection of the limit cycles in the quotient system corresponding to the trajectories shown in Fig. 8.
-8
-4
0
4
8
-4 0 4 8 12
X
Y
-15
-5
5
15
-25 -15 -5 5
X
Y
(a) β = 0.80362 (b) β = 0.80862
-200
-155
-110
-65
-20
-90 -45 0 45 90
X
Y
-90
-70
-50
-30
-20 0 20 40
X
Y
(c) β = 0.81362 (d) β = 0.81362
-13
-9
-5
-1
3
7
-5 -1 3 7 11 15
X
Y
-3
0
3
6
-7 -4 -1 2
XY
(e) β = 0.81862 (f) β = 0.82362
FIG. 10. �Color online� Same as Fig. 8, for �=0.25.
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petals for ���0 and inward petals for ���0, degeneratinginto spontaneous straightforward drift at �=�0. The trajec-tory at �=�0 in Fig. 8 is shown twice: once for the wholeduration as it was calculated, Fig. 8�c�, and then a close-up ofsmall part of it, Fig. 8�d�. Calculation of this particular tra-jectory using DNS would require, by our estimate, about fiveweeks, as opposed to 2.5 h used by EZRide.
The change in the quotient dynamics with changing � isillustrated in Fig. 9. As opposed to the tip trajectories, thereis no evident qualitative changes in the shape of the limitcycle across �=�0. Note the very elongated shape of thelimit cycles in all three projections. We do not know whetherthis has some theoretical explanation or is merely incidental.
The parametric line �=0.25 exhibits similar behavior, asshown in Figs. 10 and 11. This is closer to the Hopf bifur-cation line in the quotient system, called �M line in �44�.Correspondingly, the size of the limit cycles in the quotientsystem is smaller and they become more oval shaped. Notethat the scale of cy axis is disproportionately stretched inFig. 11; i.e., the Hopf central manifold appears to be nearlyorthogonal to that axis. Again, there is no qualitativechange in the quotient system dynamics when crossing the1:1 resonance.
VII. APPLICATION II: LARGE CORE SPIRALS
Another example where the spatial extent of the spiralwave dynamics is large is the vicinity of Winfree’s “rotorsboundary” �R in the parametric space �44�. In the vicinity ofthis boundary, the period of rotation and the radius of thecore of the spiral wave grow infinitely.
There are at least two different asymptotic theories, basedon different choice of small parameters, which aim to de-scribe the vicinity of �R. Hakim and Karma �45,46� devel-oped a “free-boundary” asymptotic theory applicable toFitzHugh-Nagumo type models in the limit c→0 or �→0 interms of our chosen kinetics, where angular velocity typi-cally decreases as
�� � �p − p��3/2, p → p�, �31�
where p is a parameter of the model such that p= p� corre-sponds to the �R boundary.
Elkin et al. �47� obtained an alternative asymptotic basedon assumptions which were not restricted to kinetics of anyparticular kind, but which were not directly validated. Theirprediction was
�� � �p − p��, p → p�. �32�
Further analysis has suggested that these two alternatives arenot actually antagonistic and may be even observed in thesame system in different parametric regions �48�. Reliablydistinguishing between the two asymptotics is challengingfor DNS as it requires a rather close approach to the criticalpoint p= p�, which is not known a priori, implying large tiptrajectory radii and correspondingly significant computa-tional resources.
In here we present an example of studying this depen-dence using calculations in the comoving FoR, which is freefrom the above complication, as it can be performed withinthe box of fixed size for all p.
For this study, we use Barkley’s model with varying pa-rameter p chosen to be a, varying from a=0.48 downwardswith step 0.001 until 0.43, with other parameters fixed at b=0.05 and c=0.02. The discretization parameters are L=30, x=1 /8, t=1 /2560, and r�inc= �0,7 /4�.
Selected stationary solution obtained in this way are illus-trated in Fig. 12, and the graphs of �a� and cy�a� are shownin Fig. 13. We compare the features of the observed solutionswith those that are given by the two asymptotic theories �48�,and observe that:
cxcy
ββ
0.823620.818620.813620.808620.80362
-0.5-0.55
-0.6 0.10.20.30.40.50.6 cx ω
ββ
0.823620.818620.813620.808620.80362
0.1 0.2 0.3 0.4 0.5 0.6
-0.3-0.4
-0.5-0.6 cyω
ββ
0.823620.818620.813620.808620.80362
-0.3-0.55
-0.4-0.5
-0.5-0.6 -0.6
(a) (b) (c)
FIG. 11. �Color online� Same as Fig. 9, for �=0.25.
(a) a = 0.430 (b) a = 0.450 (c) a = 0.453
(d) a = 0.456 (e) a = 0.460 (f) a = 0.480
FIG. 12. �Color online� Snapshots of relative equilibria inBarkley model obtained at different values of parameter a. Thearrows indicate the direction of the vector c�.
A. J. FOULKES AND V. N. BIKTASHEV PHYSICAL REVIEW E 81, 046702 �2010�
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�1� there is a critical value of the parameter a��0.456, atwhich the behavior of the solution changes qualitatively. Ata=a�, we observe a nearly straight broken excitation wave.
�2� For a�a�, the solutions are spiral waves, that is, bro-ken excitation wavelets, which become less and less convexas a→a�, and have macroscopic angular velocity whichhowever diminishes in the same limit;
�3� for a�a� the solutions are retracting nearly straightbut slightly concave wavelets, with very small angular veloc-ity;
�4� for a=a�, the direction of movement of the tip seemsapproximately orthogonal to the overall orientation of thewave itself;
�5� For a�a�, the vertical component of vector c� dependson a in a way which is consistent with the asymptotic �cy�� �a−a��1/2 �see Figs. 13�e� and 13�f��. Since the overall ori-entation of the wavelets, as seen in Figs. 12�a�–12�c�, isnearly vertical we can take cy as a crude estimate of the“global tip growth rate” as defined in �48�.
�6� For a�a�, the angular velocity of solutions dependson a in a way which is consistent with the asymptotic��� �a−a��, see Fig. 13�b� but not ��� �a−a��3/2 �seeFig. 13�c��.
All these observations are in agreement with the theory in�48� and can be used to empirically distinguish between the
Elkin et al. asymptotics �corresponding to the “I/V” paramet-ric boundary in �48�� and Hakim-Karma asymptotics �respec-tively, “J/C” boundary in �48��.
Feature 1 is inconclusive: existence of a critical solution,called “critical finger” by Hakim and Karma, is common forboth J/C and I/V boundaries, but the shape of this solution isdifferent. It is asymptotically linear for I/V boundary, andasymptotically logarithmic for J/C boundary. Looking at Fig.12�d� and considering the effect of the boundary conditions,it is not clear which case is nearer to the observed reality.
Feature 2 is common for I/V and J/C boundaries. Thephenomenological difference is that spirals close to I/Vboundary can be “growing” or “shrinking,” while spiralsclose to J/C boundary can only be “growing.” The movementof the tip in Figs. 12�d�–12�f� seems approximately orthogo-nal to the orientation of the wavelet near the tip, which isconsistent with both cases.
Feature 3 tips the balance in favor the I/V boundary sincethe broken wavelets are concave. According to �48�, thetranslating waves near an I/V boundary should be concave,and those near an J/C boundary should be convex.
Feature 4 is common for I/V and J/C, as in both cases thecritical fingers should have zero “global growth rate.”
Feature 5 is common for I/V and J/C boundaries.Feature 6 is, in our opinion, a convincing evidence in
favor of an I/V boundary since according to �48�, near I/Vboundary the dependence ��� is linear, whereas near J/Cboundary it is ������ ���3/2.
An unequivocal interpretation of all theoretical predic-tions in the view of our present numerical results would re-quire further investigation, as the asymptotics of �47,48� op-erate with a “crest line” of an excitation wave. There is noobvious operational definition of this line which would bevalid up to the tip, and some of the predictions concern themutual orientation of this line and the tip velocity. Howeverthe predictions that can be unambiguously interpreted, seemto indicate that for the model considered here, we have thecase of I/V boundary, i.e., Elkin et al. asymptotics, ratherthan J/C boundary corresponding to Hakim-Karma asymp-totics.
The last observation here is that of the small angular ve-locity calculated for the “retracting waves” at a�a� �seenin Figs. 12�a�–12�c��. As we already noted, the smallness ofthese values is consistent with the theoretical prediction oftranslating but not rotating waves. However when these val-ues are magnified, we observe that they demonstrate a pecu-liar power law ��a��� �a−a��p where p�1 /4.3 �see Fig.13�d��. A theoretical explanation of this requires furtherstudy; it is clear, however, that in this area is stronglyaffected by the boundaries, as the curves for L=30 andL=35 differ quite significantly.
VIII. APPLICATION III: ELECTROPHORESISOF MEANDERING SPIRAL
Finally, we illustrate calculation of the movement of spiralwaves in a perturbed reaction-diffusion system. We considerFitzHugh-Nagumo kinetics at the same parameters as inFig. 4, and add to it “electrophoresis” perturbation �23� in theright-hand side, with E=�D.
0
0.05
0.1
0.15
0.2
0.25
0.43 0.44 0.45 0.46 0.47 0.48
L=35selected
ω
a
0
0.1
0.2
0.46 0.47 0.48
L=30L=35
L=35 fit
ω
a(a) (b)
0
0.1
0.2
0.3
0.4
0.46 0.47 0.48
L=30L=35
L=35 fit
ω2/3
a
02468
1012141618
0.43 0.44 0.45
L=30L=35
109ω4.3
a(c) (d)
-2
-1
0
1
2
3
0.43 0.44 0.45 0.46 0.47 0.48
L=30L=35
c y
a
0
2
4
6
8
0.43 0.44 0.45
L=30L=35
c2 y
a(e) (f)
FIG. 13. �Color online� Dependencies �a� and cy�a� of therelative equilibria, for different L as indicated. On panel �a�, thesymbols correspond to the selected values of a used in Fig. 12.
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Results of the simulations are presented in Fig. 14. Theunperturbed spiral waves for these parameters are meander-ing, so with the perturbation present, we observe meanderingwith drift. The drift proceeds with a constant average veloc-ity, which is consistent with the fact that the perturbationviolates only the rotational but not the translational symme-try of the problem. The average drift is to the left, whichcorresponds to a collapsing scroll wave. So at these param-eter values, the scroll waves have positive tension, inasmuchas this concept can be applied to meandering scrolls.
In the calculations in the laboratory FoR, the time duringwhich the drift can be observed is limited, as when the spiralreaches the left boundary, it terminates. In the comovingFoR, this drift can be observed indefinitely. Comparing thetraces in Fig. 14 we see that although, as we know from Figs.5 and 6, the discretization is too crude to give quantitativeagreement between laboratory and comoving calculations indetail, the drift velocities obtained in these two ways are verysimilar.
We illustrate the relative advantages of the two methodsof calculation by comparing the computation costs. The labo-ratory FoR simulation, for L=30 and t� �0,300� has taken325 s �the spiral has annihilated at the left at t�237�. Thetime taken by the comoving FoR simulation for the same boxsize L and the same t interval is 462 s, i.e., is naturallysomewhat longer due to the extra effort required for the ad-vection term calculations. However, the comoving FoR cal-culation proceeded unabated where the laboratory FoR cal-culation failed due to annihilation with the border. Tocontinue the laboratory FoR calculation to the same extentwe would have to increase the box size L with a correspond-ing increase in computation cost. Moreover, virtually thesame result, as far as drift velocity is concerned, can be ob-tained by comoving FoR calculation with L=20, and it takesonly 202 s. Of course the drift in the laboratory FoR withL=20 would terminate even earlier.
IX. DISCUSSION
We have described a numerical method of solving areaction-diffusion system of equations describing a spiralwave in a frame of reference which is moving with the tip ofthat wave.
We have shown the method can provide accurate solu-tions, and that there are applications where the computationalcost of our method can be considerably lower than that of theconventional approach, or the conventional approach is justinapplicable. As always, the computational advantages areparticularly essential in case of parametric studies, for whichthe method is well suited. Although the applications werechosen just to provide some meaningful examples of use ofthe method, the results obtained there can be of scientificvalue themselves.
So, we have investigated the vicinity of the “1:1 reso-nance” manifold in the parametric space, which correspondsto spontaneous drift of spirals, and which separates meander-ing patterns with outward petals and inward petals. Henry�49� proposed a theory which implies that this manifold co-incides or is an analytical continuation of the manifold wherethe filament tension of scroll waves vanishes. There are re-ports in literature confirming that change of sign of filamenttension is associated with change from outward to inwardpetals in meandering patterns, but also examples where thereare no such correlation, e.g., �50� and references therein. Oursimulations indicate that as far as orbit manifold dynamics ofthe spiral is concerned, the 1:1 resonance is not characterizedby any special features. Hence any special features of thisresonance ought to be due to the Euclidean extension of theorbifold dynamics. Since scroll filament tension can also bedefined via properties of the spiral wave solutions within thecomoving FoR, any genetic and generic relationship betweenthe two manifolds seems unlikely �but, of course, one cannotexclude the possibility of such relationship in some specialcases�.
-15
-10
-5
0
5
10
15
-20 -15 -10 -5 0 5 10 15
Lab L=30Com L=30Com L=20
X
Y
(a)
-20
-15
-10
-5
0
0 50 100 150 200 250 300
Lab L=30Com L=30Com L=20
t
X
(b)
FIG. 14. �Color online� Trajectories of tips of drifting meandering spirals calculated in the laboratory FoR �for L=30� and in thecomoving FoR �for L=30 and L=20�. The thin black dotted lines designate the boundaries of the calculation box in the laboratory FoR wherethe initial position of the tip is in the center. The parameters are the same as in Fig. 4 and the perturbation is h=�D�xu, where �=0.1.
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We have also investigated the vicinity of the “�R” mani-fold in the parametric space, which has provided a strongevidence toward one of the two theoretical possible asymp-totics, namely, Elkin et al. �47� asymptotics as opposed toHakim-Karma �45� asymptotics. It should be noted here thatwhile Hakim-Karma asymptotic theory was based on as-sumptions which have been well established, the Elkin et al.asymptotic theory was using assumptions, validity of whichcould not be asserted at that moment. Here we have pre-sented firm evidence that Elkin et al. asymptotics is not amere theoretical possibility but is indeed observed in reality�see also �36� and a discussion below�. A direct confirmationwould be via calculation of the “response functions,” i.e.,critical eigenfunctions of the adjoint linearized operator ofthe critical finger solution. This would require obtaining firsta good quality critical finger solution, so the method de-scribed here can be a significant step toward this goal, too.
Finally, we have demonstrated that calculations in the co-moving FoR can be efficiently used to study perturbation-caused drift of spirals, including meandering spirals. Al-though the asymptotic theory of drift of meandering spirals isyet to be developed �see, however, a preliminary draft ofsuch theory in �51��, we can expect, for instance, that scrollwaves in the FitzHugh-Nagumo model with the parametersas in Figs. 4 and 14 will have “positive tension,” i.e., tend tocollapse, rather than develop a scroll wave turbulence. Theadvantage of calculating drift in the comoving FoR, apartfrom computation cost, is absence of “pinning” effects ofspatial discretization, both in terms of discrete space stepsand discrete spatial directions, on the drift.
Our approach can be compared to the approach proposedby Beyn and Thummler �BT� �35�. BT use a similar math-ematical idea of decomposing the evolution of the nonlinearwave into the motion of the wave and evolution of its shape,which in the functional space appears as decomposition intomotion along and across the Euclidean symmetry group or-bits. But there are also differences. There are technical de-tails of implementations which are probably of lesser impor-tance, such as choice of polar vs Cartesian grid, central vsupwind discretization of spatial derivatives and explicit vssemi-implicit discretization in time. More significant differ-ences are in the “phase conditions” they use, which play thesame role as, but are qualitatively different in nature from,
our “pinning conditions.” One aspect is that the phase con-ditions involve integral functionals. We show here that this isnot necessary, and local conditions such as Eq. �24� are sim-pler. The other aspect is the one we discussed in Sec. II: theBT phase conditions appear to be well suited for calculationof relative equilibria �rigidly rotating spirals� but not neces-sarily for relative periodic solutions �meandering spirals�.Further, the phase conditions proposed by BT were not in-tended for use with symmetry-breaking perturbations thatproduce drift of spirals. And indeed, BT comment in theirpaper that “it seems quite a challenging task to freeze driftingspirals or recognize meandering spirals as periodic orbits.”As we have demonstrated, our approach works both for me-andering spirals and for drifting spirals.
After completing this study we became aware of a workby Hermann and Gottwald �HG� �36� who also investigatedthe large core limit, using a further development of the BTmethod. HG paid a great deal of attention on refining theboundary conditions so as to minimize the effect of bound-aries onto the quotient dynamics. This has allowed them, inparticular, to verify linear scaling law �32� for seven decadesof variation in ��, compared to mere one decade as shown inFig. 13. Notice that as shown in the same figure, our progresstoward smaller values of �� is limited precisely by the in-fluence of boundaries. HG also have explicitly addressed theissue of the numerical stability of the computations, whichwe treat in this study purely empirically.
We believe that combining the advantageous features ofthe approach developed by BT and HG, and the one pro-posed here, is an interesting topic for future work, whichmay yield further results about spiral wave dynamics, thatare not possible, or very difficult, to obtain by direct numeri-cal simulations.
ACKNOWLEDGMENTS
This study has been supported in part by EPSRC-GBGrant No. EP/D074789/1. The preliminary stage for thisstudy was done in collaboration with B. N. Vasiev, who pro-posed the idea of solving for quotient data simultaneouslywith the advection substep. V.N.B. is also grateful to A. M.Pertsov and G. A. Gottwald for inspiring discussions, and toG. A. Gottwald also for informing us about the results of �36�prior to publication.
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