A skeleton structure of self-replicating dynamics

Physica D 130 (1999) 73–104

A skeleton structure of self-replicating dynamics

Yasumasa Nishiuraa,∗, Daishin Ueyamaba Laboratory of Nonlinear Studies and Computations, Research Institute for Electronic Science, Hokkaido University,

Kita-ku, Sapporo, 060-0812, Japanb Department of Mathematics, Hiroshima University, Higashi-Hiroshima, 739-0046, Japan

Received 27 October 1997; received in revised form 18 December 1998; accepted 13 January 1999Communicated by C.K.R.T. Jones

Abstract

Self-replicating patterns (SRP) have been observed in several chemical reaction models, such as the Gray–Scott (GS)model, as well as in physical experiments. Watching these experiments (computational and physical) is like watching themore familiar coarsening processes but in reverse: the number of unit localized patterns increases until they fill the domaincompletely. Self-replicating dynamics, then, can be regarded as a transient process from a localized trigger to a stable stationeryor oscillating Turing pattern. Since it is a transient process, it is very difficult to give a suitable definition to characterize SRP. Itcannot be described in terms of well-studied structures such as the attractor or a singular saddle orbit for a dynamical system.In this paper, we present a new point of view to describe the transient dynamics of SRP over a finite interval of time. Wefocus our attention on the basic mechanism causing SRP from a global bifurcational point of view and take our clues fromtwo model systems including the GS model. A careful analysis of the anatomy of the global bifurcation diagram suggeststhat the dynamics of SRP is related to a hierarchical structure of limit points of folding bifurcation branches in parameterregions where the branches have ceased to exist. Thus, the skeleton structure mentioned in the title refers to the remains ofbifurcation branches, the aftereffects of which are manifest in the dynamics of SRP. One of the natural and important problemsis about the existence of an organizing center from which the whole hierarchical structure of limit points emerges. In oursetting, the numerics suggests a strong candidate for that, i.e., Bogdanov–Takens–Turing (BTT) singularity together with thepresence of a stable critical point, and so this indicates a universality of the above structure in the class of equations sharingthis characteristic. ©1999 Elsevier Science B.V. All rights reserved.

PACS:68.35.Fx; 82.20.Wt; 82.40.Bj; 82.20.Mj1991 MSC:34A47; 35B32; 35K57; 35R35; 58F14

Keywords:Self-replicating pattern; Reaction diffusion system; Pulse solution; Turing pattern; Wave splitting

∗ Corresponding author.

0167-2789/99/$ – see front matter ©1999 Elsevier Science B.V. All rights reserved.PII: S0167-2789(99)00010-X

74 Y. Nishiura, D. Ueyama / Physica D 130 (1999) 73–104

1. Introduction

Self-replicating patterns (SRP) have recently been observed numerically in a reaction diffusion system [1–4]as well as in laboratories [5–7]. A prototypical model is the irreversible Gray–Scott (GS) model that exhibits avariety of new patterns including spots that self replicate and develop into a variety of asymptotic states in twodimensions [1] as well as pulses that self replicate in one dimension [2]. Several interesting analytical works havealso appeared: for instance, construction of single-spot solution to the GS model and its stability has been doneby [8] with the aid of formal matched asymptotic analysis, which is closely related to the splitting phenomenon; arigorous analysis concerning the existence and stability of steady single pulse as well as nonexistence of travelingpulses has been done quite recently by [9,10]. These works are very suggestive, nevertheless, very little is knownabout the mechanism that drives the replication dynamics itself. The aim of this paper is to present a key mechanismfor the self-replicating dynamics for a class of model systems including the GS model. Two different types of SRPare depicted in Fig. 1, one is a SRP of static type for the GS model and the other is a SRP of propagating type forour model system. In both simulations SRP is observed as a transient process from a localized trigger to a periodicstructure.

It looks like a reverse process of the usual coarsening phenomena, i.e., the number of unit localized patternincreases until the domain is filled by them completely. More precisely, there are three different phases of dynamicsof SRP:1. Steady or traveling phase: an isolated pulse-like pattern (i.e., away from other pulses and boundary) stays or

travels almost without changing its shape.2. Splitting phase: some of the pulses split into two parts.3. Convergence to a final state: after several circulations of the above two phases, and the number of localized

patterns exceeds a critical level, then the orbit approaches an asymptotic state.

Fig. 1. Self-replication for PDE models: (a) The GS model (see Eq. (2)) whereDu = 2 × 10−5, Dv = 10−5, F = 0.04,k = 0.06075 and theL (=width of interval)= 1.6. (b) The P-model (see Eq. (4)) whereDu = 0.001,Dv = 0.004,α = 0.27169,m = 0.95, L = 4. The boundarycondition is of Neumann type and onlyv-profiles are drawn for both models.N (=the number of grid points) is 1500 (respectively 1000) for (a)(respectively (b)).

Y. Nishiura, D. Ueyama / Physica D 130 (1999) 73–104 75

Apparently the third phase appears due to the finiteness of the interval, in fact the splitting may last for everfor the extended system. Our main concern is the first two stages, especially we clarify what kind of dynamicalmechanism drives SRP. The two model systems presented in Section 2 look different and the resulting SRP’s arealso different as in Fig. 1, however, it turns out that the underlying driving mechanism of SRP is common for bothmodels. It is, therefore, natural to study these two different models simultaneously, and treat them in a unified way.The difficulty lies in the fact that SRP cannot be captured in the phase space neither as an invariant set like a steadystate or a time-periodic solution, nor an itinerary orbit traveling from a steady state of higher instability to that oflower instability along stable (or unstable) manifolds of them like a coarsening process of Cahn and Hilliard orbistable scalar reaction diffusion equations. It turns out that it is definitely necessary to work in theextendedsolutionspace to find such a driving mechanism, namelyX = S × P is appropriate for our purpose, whereS is an usualphase space andP is a parameter space. The goal is to reveal ahiddenglobal bifurcational structure inX, which isresponsible for SRP. It may be good to start off from a very simple ODE example to explain the main idea:

du

dt= −(u + 1)(u2 − α)(u2 − 2u + 1 − α), |α| <

1

4. (1)

The associated bifurcation diagram with respect toα is given by Fig. 2(a) . Note that there are two limiting pointslocated exactly at the same pointα = 0 and the one-dimensional unstable manifolds of each unstable critical pointare connected to the upper and lower stable critical points, respectively (here thelimiting point represents the pointwhere fold bifurcation occurs). Forα > 0, the orbit approaches one of the stable critical points depending on itsinitial value, however, if we chooseα slightly to the left of 0, sayα = −0.001, and take the initial valueU0 = 0.5,then the orbit behaves like Fig. 2(b). The height of each plateau region is almost equal to the value of limiting point,and the time spending there becomes longer whenα is closer to 0. It is obvious that the orbit eventually settlesdown to−1, however, it jumps from a neighbourhood of one limiting point to the next one and behaves like aquasi-equilibrium between jumps. It is easy to see that this behavior reflects theconnectionof unstable manifoldsmentioned above. In other words, the global connection has a strong influence on the dynamics nearby inX, andthe orbit starting fromU0 feels the influence of it. We call this characteristic transient phenomenon theaftereffectof hierarchy structure of limiting points.Note that the aftereffect becomes weaker whenα leaves 0, however, itsinfluence survives at least for certain finite amount of range in the parameter space. This is one of the reasons

Fig. 2. Hierarchy structure of limiting points for Eq. (1) and its aftereffect. (a) shows the bifurcation diagram of critical points of Eq. (1) and (b)shows the time-course of the orbit starting fromU0. The plateau regionsAE1 andAE2 indicate the aftereffect of limiting points.


Fig. 3. Intermittency of one-dimensional mapping

Fig. 4. Edge of hierarchy structure of limiting points (EHL). Each lower part of folding branch is unstable and the associated unstable manifoldsare connected to the stable parts of the folding branches.

why we observe SRP in a wider region. It might be instructive to compare the above hierarchy structure with itsone-dimensional counterpart, i.e., a series of intermittency as in Fig. 3.

Suppose that the upper (lower) parabolic branch in Fig. 2(a) were standing pulse of one-hump (two-hump), thenit is easy to imagine that the self-replication corresponds to the jump from the upper limiting point to the lower one,and the duration in plateau region in Fig. 1(b) corresponds to the quasi-static behavior in Fig. 1(a).

We shall show that this is indeed the case for the PDE models, and the difference of nature of SRP in Fig. 1simply comes from that of the bifurcating branches, namely if it is a standing pulse (respectively, time-periodictraveling pulse), the orbit behaves like Fig. 1(a)(respectively (b)). This suggests that SRP is rather universal since itonly depends on the global structure, but not on the nature of each branch. Here, it may be appropriate to introducea notionedge of hierarchy structure of limiting points(EHL) depicted schematically as in Fig. 4, which is definedby the following two conditions:


1. Line-up property: the stable solutions disappear at fold (i.e., saddle-node) bifurcations, all of which occur almostat the same parameter value.

2. Connectivity: prior to this disappearance, there are heteroclinic orbits from the saddle to two different stablesolutions as shown in Fig. 4.

This is a loose definition and a more precise description will be given in the subsequent sections. The connectionamong different branches of PDE models is not so simple like a linear array as in Fig. 2, in fact it is quite a subtletask to clarify the details of interweaving manner of stable and unstable manifolds of them, and at present, numericsis the only way to know the details. Nevertheless, we are led to an interesting conjecture on the connectivity amongbranches (see Section 4), which might be a good challenge for rigorous people. Also, the depth of the hierarchystructure depends on the system sizeL, i.e., the larger the system size, the deeper the structure.

Two natural questions arise: Firstly, why they line up? Coincidence of location of limiting points looks accidentaland seems structurally unstable. Secondly, what is the origin of each branch? There must be some organizing centerfrom which all these branches emanate. Although, we do not know yet complete answers to these questions, it ispossible to give a strong numerical evidence and intuitive arguments which shed light on the essential part of theseissues. Folding up principle is one of the keys for the line-up property, and the BTT singularity together with theexistence of stable equilibrium point seems an important ingredient for the quest of the origin. More details aregiven in Section 4 and the Appendix. It should be emphasized that the key concept characterizing self-replicatingphenomenon has a potential to be applicable to many model systems. Finally, we briefly mention the organizationof this paper. In Section 2 we show two representative models having SRP. In Section 3 we present two hierarchystructures responsible for two types of SRP in Fig. 1. It is clarified that how such a structure drives SRP dynamics. InSection 4 we discuss two questions in the previous paragraph by using numerics as well as semi-rigorous arguments.There still remain many open problems from mathematical point of view. Precise numerical tracing of the bifurcatingbranch with respect to the diffusivity is presented in the Appendix, which explains how such a global hierarchystructure is formed when the diffusivity decreases. Part of the results for the P-model was announced in [11].

2. The model

2.1. The Gray–Scott model (GS model)

The chemical reaction U+ 2V → 3V and V→ P in a gel reactor can be described by the following GS model[12]:

{∂u/∂t = Du∇2u − uv2 + F(1 − u),

∂v/∂t = Dv∇2v + uv2 − (F + k)v.(2)

whereu andv are concentrations of the chemical materials U and V, respectively,Du andDv the diffusion coef-ficients,F the in-flow rate of U from outside,F + k the removal rate of V from reaction field, andP is an inertproduct. The corresponding kinetics of the GS model is given by

{du/dt = −uv2 + F(1 − u),

dv/dt = uv2 − (F + k)v.(3)

The nullclines and the typical flows are drawn in Fig. 5. The kinetics (3) has a Bogdanov–Takens (BT) point togetherwith a stable critical point (1,0). BT-point is a singularity of codim 2 where fold and Hopf bifurcations merge therein two-dimensional parameter space(k, F ) (see Fig. 5).


Fig. 5. Unfolding of BT-point and flows of the kinetics for the GS model. BT-point is a codim 2 singularity where fold and Hopf bifurcationsmerge there in(k, F )-space. Solid (dotted) line represents fold (Hopf) bifurcation. Unfolding BT-point, there appears a homoclinic line asindicated.

In subsequent sections, we study Eq. (2) on a finite interval under Neumann (zero flux) boundary conditions withfixed parameters asDu = 2 × 10−5, Dv = 10−5 andF = 0.04. Then, there remain two control parametersk andL whereL is the system size, which is equivalent to the strength of diffusivity.

2.2. A model system of propagating type (P-model)

We propose another model system which shows SRP of propagating type as in Fig. 1(b).

{∂u/∂t = Du∇2u + u(u − v2 − α),

∂v/∂t = Dv∇2v + mu − v,(4)

whereDu andDv are diffusion coefficients,α andm are non-negative parameters. This is one of the simplest modelswhich have BT-point together with a stable critical point. We shall consider Eq. (4) on a finite interval subject toNeumann (zero flux) boundary conditions. In most cases we fix the parameters asDu = 0.001,Dv = 0.004, andm = 0.95. The length of intervalL andα are control parameters. As we will see, this model system and the GSmodel share a remarkably similar qualitative behavior. The corresponding ODE kinetics is

{du/dt = u(u − v2 − α),

dv/dt = mu − v.(5)

The two parametersα and m control the ODE dynamics. The nullclines and the typical flows are drawn inFig. 6. Whenα is small, Eq. (5) has two nontrivial constant states; the upper one is a stable focus and the lowerone is a saddle as in Fig. 6(2). Asα becomes larger, through fold bifurcation, Eq. (5) becomes monostable as inFig. 6(1).


Fig. 6. Unfolding of BT-point and flows of the kinetics for the P-model. Hopf and fold bifurcation lines emanate from the BT-point and theresulting periodic orbit is stable as indicated. The amplitude of this limit cycle is increased asα is decreased and the branch of the limit cycleeventually approaches the homoclinic line.

3. Edge of hierarchy structure of limiting points drives SRP

We show in this section that EHL is built in two model systems in Section 2, and that SRP is driven by EHL.First, we examine the after effect of limiting point with the associated unstable manifold for each model system inits simplest form. Second, by numerical tracing, we reveal that fold branches form a global structure in a hierarchyway, i.e., EHL, which is the hidden driving mechanism of self-replicating dynamics. We shall discuss in Section 4why such a structure appears naturally for the model systems. In what followsmode numberof the static branchmeans the Fourier-mode number of the associated eigenfunction at the bifurcation point.

3.1. Aftereffect of limiting points and the connections among them

We observed a very simple ODE example (1) in Section 1 which displays the aftereffect of hierarchy structureof limiting points. The aftereffect in fact consists of two parts; one comes from the limiting points themselves andthe other from the connection of those limiting points via unstable manifolds. The manifestation of the formeraftereffect is determined by the nature of the limiting point, namely, if it is an equilibrium point like Eq. (1), thenthe orbit stays there for a long time as in Fig. 2(b) (quasi-static behavior) followed by the sharp jump to the nextplateau region. This sharp transition is apparently guided by unstable manifolds. Less nontrivial case is when it isa periodic solution. The schematic picture Fig. 7 shows the three different dynamics whenα varies. Whenα = α2,there are stable and unstable time-periodic solutions which coalesce atα = αc and disappear after that, however,even atα = α1, there still remains a temporary oscillatory motion as anaftereffectof the limiting point. This issimply because the eventfold bifurcationoccurs abruptly, but the vector field varies continuously with respect tothe parameter.

Since quasi-behavior does not last for ever, the orbit escapes from the influence of the limiting point and movesto the next destination. For example (1) it is clear that we can predict the next destination, because the dimension ofthe unstable manifold of the limiting point isone. The orbit moves along this manifold until it arrives at the placewhich is close to the next limiting point in the extended phase space. This is the aftereffect due to the connection to


Fig. 7. Dynamics near the limiting point: black (white) disk stands for the stable (unstable) periodic orbit inR3. Even after the fold bifurcation,there still remains an oscillatory motion atα = α1. The temporary oscillatory motion lasts longer, whenα1 approachesαc. Each phase planeindicates the associated Poincare mapping.

the next limiting point via unstable manifold. Suppose the first limiting point was a steady state of one-hump, andthe second one of two-hump, respectively, then one can easily imagine that the connection apparently correspondsto splitting. In what follows we present two different types of aftereffects for the PDE models (2) and (4) in itssimplest form, which are the basic to understand the full hierarchy structure.

3.1.1. GS modelFig. 8 shows the bifurcation diagram of steady states for the GS model atF = 0.04 where several fold bifurcation

curves coexist. We check the aftereffect of the limiting point of 2-mode branch in the diagram by choosing anappropriate initial data of one-hump, which is close to the profile of the function corresponding to this limitingpoint. Each stationary branch in Fig. 8 has a labelTuring, since each branch was born from the constant state P (seeFig. 8) due to the Turing instability (i.e., diffusion-driven instability) for smallk (more precise explanation is givenfor the P-model in Appendix). It is clearly shown in both Fig. 8(a,b) that there is an aftereffect of the limiting pointand it lasts longer when the parameterk becomes closer to the location of limiting pointkc.

The unstable 2-mode solution atUs is given by Fig. 9(a). There is only one unstable eigenvalue (Fig. 9(c)), andthe profile of the associated eigenfunction looks like Fig. 9(b). Note that there is a dent in the middle of it, whichindicates the beginning of splitting. To know the destination of the associated one-dimensional unstable manifold,we add a perturbation toUs, which is a small constant multiple of the unstable eigenfunction. If the constant ispositive (negative), the orbit goes to the stable Turing pattern of 4-mode (2-mode). This apparently indicates thatthe unstable 2-mode branch is connected to stable 2-mode and 4-mode ones by the unstable manifold. Since there


Fig. 8. Aftereffect for the GS model: the limiting point of 1-mode is located at 0.06080211 and 2-mode is 0.06079793, which are very close toeach other. The right diagram is a magnification of the left one near the limiting points. The vertical axis representsL2-norm of the solutions.The parameter values are (a)F = 0.04, k = 0.06079,L = 0.3 (b) F = 0.04,k = 0.06075,L = 0.3, respectively. The aftereffect in (a) lastslonger than (b).

is only one unstable eigenvalue near the limiting point, this connection persists upto the limiting point, which drivesthe SRP of static type as in Fig. 9(β).

3.1.2. P-modelExactly the same thing happens also for the P-model, but in the form of periodic motion. Fig. 10(c) shows the

bifurcation diagram of P-model with respect to the parameterα atL = 0.65. There appears a time-periodic branchemanating subcritically from 2-mode steady branch. A magnified picture Fig. 10(d) clearly shows the existence oflimiting point atα = αc. Fig. 10(a,b) show the evolution whereα is fixed to be slightly smaller thanαc and the


Fig. 9. (a) Unstable steady stateUs atk = 0.6009 (see the magnified diagram in Fig. 8). Solid (dotted) line stands forv (u) profile. (b) The profileof the unstable eigenfunction atUs. There is a negative dent in the middle. (c) Spectral distribution atUs. Adding a small positive (negative)multiple of the eigenfunction toUs as a perturbation, the orbit behaves like(β) ((α)).

initial function is taken from a snapshot of the stable oscillatory solution at someα slightly larger thanαc. Note thatwe can make this temporary oscillatory motion keep running as long as we want, if we choose the parameterα andinitial data appropriately. In fact,α is taken to be much closer toαc in Fig. 10(a) than Fig. 10(b), then it behaves likean oscillatory pulse ten times longer than before. In other words, although nothing remains after fold bifurcationin the phase space, there remains a strong secondary effect due to the coalescence of large-amplitude oscillatorymotions whenα(< αc) is close toαc. It is numerically confirmed (see Fig. 11) that the unstable oscillatory pulse(white disk in Fig. 10(d)) atUp has only one unstable Floquet exponent, and that the associated unstable manifoldis connected to the 3-mode Turing patternT3 in Fig. 10 and the stable oscillatory pulse (black disk in Fig. 10(d)),respectively. Hence, by continuation with respect toα, we see that it is not a coincidence, but anecessitythat theorbit goes toT3 after splitting. Note that the profile of the unstable eigenfunction (see Fig. 11(b)) has a deep hollowin the middle indicating a break-up of the oscillatory pulse. There are two stages of the dynamics, firstly temporaloscillation of pulse type due to the aftereffect of the limiting point, secondly jumping to the Turing branch of highermode guided by the connection of the unstable manifold toT3.


Fig. 10. Aftereffect for the P-model: The diagram (d) is a magnification of the left one (c) near the limiting point. The fold branch of periodicsolutions is clearly shown in (d) with limiting point atαc. The aftereffct in (a) lasts longer than (b). (Du = 0.001,Dv = 0.004 (a)α = 0.2688(b) α = 0.2685)

3.2. Hierarchy structure of limiting points of static type

We observed 1-step replication of the GS model as in Fig. 8 as well as the aftereffect of the limiting point. Weshall show in this section that such a structure is a basic element for constituting a hierarchy when the system sizebecomes large, which is a key mechanism to drive SRP. More precisely we shall discuss:1. The limiting points of the stationary Turing branches line up at almost the same parameter valuekc.2. The self-replicating pattern of static type occurs near the edge of the limiting points where stable and unstable

Turing branches coalesce through fold bifurcation.3. The self-replicating pattern persists, to some extent, even if the parameter leaves the critical pointkc.4. Turing pattern ofn-mode loses its stability at the limiting point and the resulting unstable Turing pattern of

n-mode is connected to the stable Turing pattern of 2n-mode by the associated unstable manifold forn ≤ N

where the positive integerN depends on the system size.We call the above structure‘EHL of static type’(see Figs. 13,19 and 23). The fourth one is quite important in the

sense that such a connection between a lower mode Turing branch and a higher mode one constitutes a backboneof the self-replicating dynamics. For a typical parameter settingF = 0.04, k = 0.06075, L = 0.5, the GS modelshows SRP as in Fig. 12. The associated bifurcation diagram of stationary solutions is given by Fig. 13, whichcontains the information of their stabilities. The vertical axis denotes theL2-norm of stationary patterns and the


Fig. 11. (a) Unstable periodic solutionUp atα = 0.2694141 (see the magnified diagram in Fig. 10(d)). Solid (dotted) line stands forv (u) profile.(b) The profile of the unstable eigenfunction atUp. There is a negative dent in the middle. (c) Floquet exponents atUp. Adding a small positive(negative) multiple of the eigenfunction toUp as a perturbation, the orbit behaves likeβ(α). (Du = 0.001,Dv = 0.004.)

horizontal one is the bifurcation parameterk. The dark lines are stable stationary solutions and the light gray linesare unstable ones. Apparently the existence of the hierarchy structure of limiting points of stationary branches isobserved, which line up at the parameter valuekc ≈ 0.0608. The value ofk = 0.06075 adopted in Fig. 12 isvery close tokc where the limiting points of 1,2,3 and 4-mode Turing branches line up, although they coalesce inpairs and already disappear atk = 0.06075. A schematic bifurcation diagram Fig. 14 is convenient to explain thefollowing splitting process. Suppose one takesk = 0.06075 and starts with an initial data of 2-mode Turing patternwhich is taken from the stable Turing branch atk = 0.0609. Then it behaves like a 2-mode stationary pattern for awhile, and then jumps (splits) to a 4-mode pattern. The 4-mode pattern also behaves like a stationary pattern for awhile, and then jumps (splits) to a 8-mode pattern. The 8-mode Turing branch lies at the bottom of this hierarchystructure and it is stable atk = 0.06075. The time-plot ofL2-norm of this orbit gives us a clear evidence of theaftereffect (see Fig. 15). The norm of the first plateau region denoted by AE2 is about 8.8, which coincides withthe value of the limiting point of the 2-mode Turing branch in Fig. 13 (see also Fig. 14). Similarly the norm of thesecond plateau region AE4 is about 7.6, which is also close to the value of the limiting point of the 4-mode Turing


Fig. 12. A self-replicating pattern in the one-dimensional GS model (F = 0.04,k = 0.06075,L = 0.5, N = 500). The graph is a space–timeplot of v.

Fig. 13. A bifurcation diagram forF = 0.04,L = 0.5, N = 100. Dark (gray) lines indicate the stable (unstable) part of the branches. SteadyTuring branches up to 4-mode have limiting points neark = 0.0608.

branch. The norm att = 7500 is 6.05, which perfectly fits the value of the intersecting point of the stable 8-modeTuring branch with the linek = 0.06075 as in Fig. 14.

The backbone structure for each splitting process consists of the unstable manifolds emanating from the unstablepart of each folding branch. The unstable Turing pattern atU2 in Fig. 14 has the form as in Fig. 16(a). It has onlyone unstable eigenvalue and the associated eigenform as in Fig. 16(b). Note that the unstable eigenfunction has adent in the middle, which indicates the beginning of splitting.


Fig. 14. A schematic bifurcation diagram atL = 0.5. SRP of Fig. 12 is observed atk = 0.06075 and the initial data is taken from a stablesolution of 2-mode as indicated. The time-plot of itsL2-norm is shown in Fig. 15. The unstable solutionU2 has a one-dimensional unstablemanifold and the destinations are stable 2-mode and 4-mode solutions, respectively (see also Fig. 17).

Fig. 15. Time-plot of theL2-norm of the self-replicating orbit in Fig. 12. The norms at AE2 and AE4 are close to those of limiting points of2-mode and 4-mode branches, respectively, in Fig. 13.

To confirm the destination of the unstable manifold, we made a simulation as in Fig. 17. As was expected from theprofile of the eigenfunction, it is connected to the 2-mode (one-hump) and 4-mode (two-hump) patterns, respectively.The connection persists up to the limiting point, and ifk is close tokc(k < kc), the orbit is well-approximated by thisconnection provided the initial condition is close to the profile of the limiting point. Note that if we choosek to be alittle bit far away from the limiting point, the aftereffect of the limiting points becomes weak and the pattern splitsfaster than before (see Fig. 18) , however, its qualitative behavior is still controlled by EHL. Also, note that the stableTuring patterns of 6, 7 and 8 modes coexist (see Fig. 13) on a wide range of parameter values, but they are not regardedas members of the EHL, since they are affected too much by the boundaries and hence the locations of limitingpoints of them are shifted to the left. Nevertheless, they join to the members when the system size becomes larger.


Fig. 16. (a) The profile of the unstable Turing patternU2 in Fig. 14 (F = 0.04,k = 0.0609,L = 0.5, N = 100). The dotted line representsu

and the solid linev, respectively. (b) The unstable patternU2 has only one real positive eigenvalue and the associated eigenfunction has a dentas depicted. (c) The distribution of the eigenvalues. There is only one real unstable eigenvalue represented by small black disk. Note that theessential structure of Fig. 9 is kept even if the system size becomes larger, which indicates a structural stability of EHL.

Fig. 17. Final destinations of the unstable manifold of the Turing patternU2 of Fig. 16(a). (F = 0.04, k = 0.0609,L = 0.5, N = 100).(α) an addition of small negative multiple of unstable eigenform (Fig. 16(b)) toU2 leads to the stable 2-mode Turing pattern.(β) the positiveperturbation leads to the stable 4-mode Turing pattern (see also Fig. 14).

In fact, one sees a tendency that the larger the system size, the deeper the depth of the hierarchy structure asin Fig. 19. Note that the location of limiting points decreases when the associated mode-number increases. Theorbit starting from the 2-mode pattern experiences a similar splitting process as before, and finally, it touches downthe stable 8-mode Turing branch (Figs. 19 and 20). Although the initial pattern is slightly shifted to the right, theaftereffect is strong enough to control its orbital behavior.


Fig. 18. A self-replicating pattern when the parameter is slightly away from the location of limiting points (F = 0.04, k = 0.0605,L = 0.5, N = 500). The splitting occurs faster than that in Fig. 12, however, the aftereffect still persists and guides an orbital behavior.

Fig. 19. A deep hierarchy structure forF = 0.04,L = 0.8, N = 150.

3.3. Hierarchy structure of limiting points of propagating type

It is very plausible that similar type of hierarchy structure as in the previous section exists for SRP of propagatingtype for the P-model (see Fig. 1(b)), however, it becomes much harder to trace the time-periodic branches of Eq.(4) with respect toα, especially when the system sizeL becomes large. On the other hand, we can trace the steady


Fig. 20. A self-replicating pattern of the GS model forL = 0.8(F = 0.04, k = 0.06075, N = 750).

Fig. 21. Hierarchy structure of oscillatory pulses for the P-model (Du = 0.001,Dv = 0.004,L = 4). The associated evolutional picture is givenby Fig. 1(b).

branches and detect the Hopf points without difficulty, and then find astabletime-periodic solution ofn-pulsetype by choosing an initial data appropriately. Once such a stable propagating pulse were found, we changeα

gradually, and check the persistency of it by solving the evolutional equations. We continue this process until thestable propagating patterns disappear. The resulting existing intervals for such stable time-periodic solutions aredepicted in Fig. 21 forL = 4. The disappearing points almost coincide each other, which strongly indicates theexistence of hierarchy structure as in Fig. 21. A self-replicating process was observed like Fig. 1(b) at the edge ofthis structure. In Appendix we introduce a discrete P-model (called the compartment model), which has much less


grid points, and hence allows us to trace at least some of the periodic branches globally. It turns out that we are ableto find a whole time-periodic branch of 1-pulse type that has a fold bifurcation near the expected value computedas above (see Fig. 36). The compartment model also has a similar hierarchy structure to Fig. 21 (see Fig. 37).

A slightly loose, but intuitive explanation is possible for the fact that the disappearing points are almost the sameas in Fig. 21. First of all, both front and tail parts of such a 1-pulse decay quickly to zero, and hence there are veryfew interaction among pulses when we superpose several pulses on the interval without intersection. Secondly, sincethe boundary condition is of Neumann type, the pulses must be robust on symmetric collision. Hence, it is plausiblethat the solution remains as a stable oscillatory multi-pulse even after collision, when the evolution starts from sucha superposed initial data. This implies that all the oscillatory multi-pulses up to some number remain stable almoston the sameα-interval. Moreover, when 1-pulse becomes unstable or ceases to exist, all the other oscillatory pulsesare also expected to behave in a similar way, which implies the coincidence of location of limiting points.

4. Line-up property with saddle connections and an origin of the EHL

We observed two types of SRP in previous sections, which were driven by the EHL. Now, we return to the twonatural questions mentioned in Section 1, namely, first why the limiting points line up with connections?; secondwhat is the origin of such a hierarchy structure? We answer the first question in a semi-rigorous way under twonatural hypotheses, and try to reveal a universal characteristic necessary to build up EHL. The folding up principle,which is a key idea to solve the first one, allows us to simplify the second question substantially, in fact it is notnecessary to treat the whole hierarchy structure, but only deal with the basic (i.e., 1-pulse) branch. A detailednumerical tracing of branches strongly suggests that the origin of such a basic branch can be traced back to anunfolding of BTT singularity under the presence of stable equilibrium point.

4.1. Line-up property and folding up principle

At first sight the line-up property of limiting points may be regarded as an accidental coincidence, however, thereis a structural reason for that, and hence it is a part of intrinsic properties of the system. In what follows exceptSection 4.3, we mainly focus on the case of the GS model on a large interval, however, similar arguments are alsopossible for the P-model. The following simple idea called thefolding up principleis useful in the sequel: supposethe system is autonomous and the boundary condition is of Neumann type on(0, L), then any solution can beextended to a solution on(0, 2L) by even extension atx = L. Next, by flipping the solution on(L, 2L) to (2L, 3L),we obtain a solution on(0, 3L). Applying this proceduren-times, we immediately have a solution on(0, nL). Next,reducing the size of the interval from(0, nL) to (0, L) by multiplying 1/n2 to the diffusion matrix, we obtain amulti-bump solution on the interval(0, L) with reduced diffusion matrix. It is clear from this observation that if thebasic 1-pulse pattern exists for any small diffusivity (or for any large interval), we have a coexistence of differenttypes of pulses. Obviously, this principle can be applied to the orbits as well. The line-up property of limiting pointsis a direct consequence of this principle under the following hypothesis.

Hypothesis 1. There exists aL0 such that the steady 1-pulse (=2-mode) folding branchP1 of Eq. (2) with limitingpointLP1 exists forL ≥ L0, and the location of limiting point converges tok∗

c in monotonically increasing way asL tends to∞.

In other words, this is equivalent to assume that the periodic wave train converges to the homoclinic orbit to(1, 0) when the period tends to∞. Note that this is a plausible assumption in view of existence and stability resultsof 1-pulse solution by [9,10]. Monotone property is also confirmed numerically as in Figs. 8,13 and 19. There is


Fig. 22. Folding-up

an implicit indication about the disappearance of pulse branches in [8], although it has been unknown how theydisappear as parameters vary. Now, for any given positive integern, take a largeL such that the steady 1-pulsebranch with limiting point exists on the interval with lengthL/n. Applying the folding up principle to this 1-pulsebranch, we have a solution ofn-pulse branchPn on (0, L). Simultaneously 1-pulse branch also exists onL/m forany integerm with 1 < m ≤ n, which implies the existence ofm-pulse branchPm on (0, L) via folding up (seeFig. 22).

From the latter part of the above assumption, the location of limiting point of each branch constructed above isclose tok∗

c whenL is large. More precisely it holds that

limL→∞

kmc (L) = k∗

c (6)

wherekmc (L) is the location of limiting point ofPm(1 ≤ m ≤ n). Note thatkm

c (L) decreases whenm increasesfor a fixedL. Thus, we have shown the line-up property of limiting points up ton-pulse branch on a fixed interval(0, L). To show that this is really a hierarchy structure, we shall examine the stability properties and connectionsamong them in the next section.

4.2. Stability and connection among limiting points

In order to show that the branches constructed in the previous section form a hierarchy structure, we have tostudy the stabilities and connections among them. Here again we need to assume the following for the basic 1-pulsebranch.

Hypothesis 2. The stable steady 1-pulse branchP s1 disappear at the fold bifurcation, and prior to this disappearance,

there are heteroclinic orbits from the saddleP u1 to two different stable pulse solutionsP s

1 andP s2 . HereP s

m (P um)

denotes the stable (unstable) part of the branchPm (see Figs. 14 and 23). Moreover along the folding branch of thesteady 1-pulse branchP1, a simple real eigenvalue crosses the origin from left to right at the limiting pointLP1.

This is confirmed numerically in Section 3. The goal is to show that the exactly the same thing happens toPm,namely it loses the stability atLPm primarily and the associated one-dimensional unstable manifolds ofP u

m areconnected toP s

m andP s2m, respectively. It is obvious from the construction thatP s

m is stable to spatially periodicperturbations, but may be unstable to non-periodic ones. However, whenL tends to∞, both head and tail of eachhump decay quickly to (1,0), and hence each steady pulse is almost independent of others and close to the homoclinicorbit to (1,0) (see [9]). Therefore, for largeL, P s

m becomes stable also to non-periodic perturbations owing to the


Fig. 23. Connection among folding branches

stability of (1, 0) in PDE sense. As for the connectivity fromP um to P s

2m, first note that the unstable eigenfunction(denoted byUm(L)) atP u

m should coincide withm-folding up ofU1(L/m) atP u1 due to the uniqueness of unstable

eigenvalue. Here, we used a generic assumption that only one simple real eigenvalue crosses the origin at LPm.Applying the folding up principle to the connection orbits fromP u

1 to P s2 on (0, L/m), we immediately obtain the

connection orbits fromP um to P s

2m. Thus, we have proved formally that the one-dimensional unstable manifold ofP u

m is connected toP s2m near the limiting pointLPm. The connection fromP u

m to P sm can be proved in a similar way.

Applying this procedure successively and extending the connection up to the limiting pointLPm, we obtain a seriesof connections of limiting points via unstable manifolds in the extended phase space likeLP1 → P s

2 , LP2 →P s

22, LP22 → P s23 · · · → P s

2n (see Fig. 23). We call this sequence of orbits the 2n-sequence of connecting orbits.Obviously, this is not a real orbit, since the locations of limiting points are different. However, for fixedn, whenL

tends to∞, the locations of all limiting points converge tok∗c , therefore, the above connection up ton is expected

to approach direct connection among the limiting points in this limit. This leads to the following conjecture on achain reaction of splitting.

Conjecture. For a givenn, there exists a large numberL and an appropriatek close tok∗c such that the orbit starting

from asuitablychosen initial data of 1-pulse type splitsn-times on the interval(0, L), which is an aftereffect of theabove 2n-sequence of connecting orbits.

It is highly possible that there are many other unstable eigenmodes near the limiting pointLPm besidesU2m, sincenot only the simultaneous splitting, but also various types of asymmetric splitting might be possible depending oninitial data on a wider interval. This indicates that the aftereffect near EHL is not so simple like the ODE example inSection 1, and the condition ‘suitably chosen initial data’ in the above Conjecture is necessary even if all the abovescenario is correct. Nevertheless, EHL is apparently a key concept to understand the self-replication phenomena,and more precise analysis might reveal the complex dynamical structure near EHL.


Fig. 24. (a) Time-periodic traveling pattern for the GS model (F = 0.025,k = 0.0544,L = 0.5, N = 500). (b) SRP of propagating type forthe GS model (F = 0.025,k = 0.0542,L = 0.5, N = 500).Du = 2 × 10−5 andDv = 10−5 for (a) and (b).

4.3. An origin of hierarchy structure

It looks a tough problem to clarify the origin of such a hierarchy structure, since it contains many branches aswell as connections among them, however, in view of the discussions in the above two subsections, we have agreat reduction of the problem, namely, it suffices to check the Hypotheses 1 and 2. Existence and connections ofhigher-mode branches follow from it by using the folding-up principle and so on. In this subsection, we especiallyfocus on the origin ofbasicpattern (i.e., 1-pulse branch). Even so it is not a priori clear how such a folding branchof steady 1-pulse was born. In fact, SRP is observed in most cases when the kinetics is of mono-stable type wherethere is only one stable equilibrium from which no bifurcations occur, therefore the branch, if exists, looks floatingin the function space and has nothing to do with the equilibrium. However, as we saw in Section 3, those branchesare not isolated if one traces them globally in parameter space. Our strategy is that we first start from a simplebifurcation diagram for large diffusivity (i.e., the system size is small), which is equal to the ODE kinetics, then weseek the onset of stationary 1-pulse and trace the deformation of it when the diffusivity decreases (i.e., the systemsize becomes large). If there occurs secondary or tertiary bifurcations, we trace all the bifurcating branches as well.This program was accomplished for the P-model (i.e., SRP of propagating type) and the details are delegated toAppendix. In fact, the main idea EHL was first hit upon through the numerics in Appendix. Here, we only summarizethe key features obtained by this computation. The bifurcation parameters for the P-model areα and the ratio ofdiffusivitiesd, and we draw a bifurcation diagram inα versus solution space for each fixedd and then observe thedeformation of the diagram whend decreases. The resulting metamorphosis is as follows (the details are given inAppendix):1. The unstable 1-pulse (2-mode) branch was born from the unstable equilibrium point (node) via Turing instability

(Fig. 32).2. It recovers stability by the interaction with 1-mode branch and then experiences a supercritical Hopf bifurcation

(Fig. 34).3. The Hopf branch changes from supercritical to subcritical and forms a folding structure with limiting point

(Fig. 36).


Fig. 25. Self-replicating spots of the P-model (Du = 0.001,Dv = 0.004,α = 0.275, k = 1, system size= 2 × 2).

Apparently the Hopf bifurcation plays a crucial role for the Turing branch to become a time-periodic one. Thefold bifurcation of equilibrium points seems relevant to the subcriticality of the Hopf branch, and the existence ofstable critical point (0,0) is indispensable to have a stable pulse (i.e., homoclinic orbit to (0,0)). In fact, supposethere does not exist such a critical point, the oscillatory Turing pattern does not look like pulse shape when itsamplitude grows, and hence, we do not have a coexistence of stable oscillating multi-pulses, since quick decay tostable critical point reduces the stability problem of multi-pulse solution to that of 1-pulse. Overall, the numerics inAppendix strongly suggests that BTT singularlity with the stable equilibrium point (0,0) is a kind of an organizingcenter to form a hierarchy structure of limiting points, however, more precise unfolding analysis is necessary tojustify the role of BTT singularity.

5. Concluding remarks

1. What is the fundamental process?A natural question is that once we understand thefundamental process, i.e., 1-spot splits into 2-spots, can we

explain the whole sequence of splitting phenomena? The answer is affirmative as was discussed in Section 4 forone-dimensional case, for instance, the existence of a folding branch of 1-pulse type generates multiple-pulse


Fig. 26. Unfolding of BT-point atm = 0.95.

branches by folding-up principle. The connections of unstable manifolds among those branches have a stronginfluence on the dynamics off the edge of hierarchy structure of limiting points. The mechanism proposed inSection 4 seems rather universal and has a potential to become a general framework to explain SRP.

2. SRP from general initial dataWe discussed in this work about the basic process of SRP, typically starting from a 1-pulse shape. The

mechanism proposed here seems relevant to a wider class of initial data provided that the pulses are well-separated initially, since each pulse starts to replicate according to our mechanism at least for some finite time,then enters into a more complicated stage where the pulses from different parents meet and interact.

3. SRP of propagating type for the GS modelWe confirmed that the GS model has time-periodic traveling patterns in a different parameter regime ofF

as in Fig. 24(a). This type of patterns only exist in a restricted area in a parameter space, and we found thatthe GS model also has SRP of propagating type as in Fig. 24(b) near the boundary of this existence region. Inview of our previous discussions, it is highly possible that the GS model also has an edge of hierarchy structureof limiting points of propagatingtype, which is currently being investigated. In fact, Reynolds et al. [8] didan asymptotic analysis to seek a mechanism of self-replication and found that SRP is closely related to theinstability of spot patterns. Their observation is apparently consistent with our result and we believe that thedestruction (or disappearance) of spot patterns is due to the folding of the solution branch.

4. Two-dimensional SRPIn two-dimensional space we found the self-replicating spots as in Fig. 25 for the P-model, which is quite

similar to those in [1,5,6] for the GS model. Although we do not know the detailed structure of bifurcatinglocalized spots, it might be possible to know the basic mechanism if the system size is rather small. Namely,for instance, take a domain so that only two localized spots can survive in it, then, if we start from a localized1-spot, it may split and settle down to the two spots state, which is quite similar to the fundamental processdiscussed in Section 3.1.

Acknowledgements

The authors thank the referees for useful comments and suggestions. This research was supported in part byGrant-in-Aid for Scientific Research 09440071 and 09874039, Ministry of Education, Science and Culture, Japan.


Fig. 27. Turing bifurcation curves in(Du, α)-space(n = 1, 2, . . . , 4) for PDE model (4) on(0, π) with m = 0.95; (a) (respectively (b)) showsthe location of bifurcation points on the upper (respectively lower) part of the folding branch of equilibria in Fig. 26. Note that the bifurcationpoints on the upper branch are accumulated near the folding pointαc = 0.277, and the smallerDu, the larger mode-number primarily emergesasα decreases.


Fig. 28. Overhang of Turing branch(d = 1.6, n = 8).

Fig. 29. Hopf barrel (d = 0.555, n = 8); the right diagram is a magnification of the left one. The Hopf branch emanates atH1 and comes backto the 1-mode branch.

Fig. 30. Behavior of eigenvalues near the Hopf barrel(d = 0.555, n = 8): a new pair of complex eigenvalues appears and crosses the imaginaryaxis along the 1-mode branch. The complex eigenvalues associated with the ODE Hopf bifurcationH0 remain in the left-half plane. Each spectraldistribution is computed at the steady state indicated by an arrow.


Fig. 31. Stable oscillatory motion of 1-mode (d = 0.555,α = 0.277, n = 8).

Fig. 32. Emergence of Turing branch of 2-mode (d = 0.53, n = 8).

Fig. 33. Interaction of Turing branches (d = 0.52, n = 8).

Appendix A

We shall discuss an origin of folding branch of 1-pulse type which is a member of the hierarchy structure forthe P-model (see Fig. 21). The meaning of theorigin is to find a kind of organizing center in the following sense.When the system size is small (or the diffusivity is quite large), the dynamical behavior of the system is essentiallyequal to that of ODE kinetics as in Section 2, however, when the diffusivity decreases, various types of spatially


inhomogeneous patterns appear through bifurcations and the hierarchy structure of folding branches is a part ofthem. It is, therefore, natural to ask from where such a structure was born, in other words, what is theparentsingularity, if exists, from which the hierarchy structure comes out? As was discussed in Section 4, it suffices totrace 1-pulse branch and not necessary to trace all the members of the hierarchy structure. In this appendix we try tofind such a singularity by computing the emergence and deformation of the branches by using AUTO [13] where weemploy two bifurcation parametersα and diffusivity. It turns out that BTT point (see the next section for definition)with a stable critical point seems to play a role of such an organizing center. This speculation relies on the fact thatboth kinetics (3) and (5) have BT singularities and each folding branch of EHL can be traced back to the onsetof Turing instability of an equilibrium point obtained by unfolding the BT-point. Moreover, as we shall see later,a good combination of these three bifurcations is indispensable to form EHL. In what follows, we only deal withthe propagating model (4), since it contains periodic branches which are much harder to trace them numericallycompared with the static case (for the static SRP of GS model, see [14]). Recalling the folding up principle and thearguments in Section 4, it suffices to know the origin of the basic 1-pulse branch. Even so, the codimension of BTTsingularity is at least four if we take into account the first two Fourier modes. If we discretize the PDE models withfine mesh, it becomes quite tough to trace all the periodic branches globally with respect toα and diffusivity. We,therefore, employ much coarse discrete approximation like{

dui/dt = ui(ui − v2i − α) + d(ui−1 − 2ui + ui+1),

dvi/dt = mui − vi + dβ(vi−1 − 2vi + vi+1), i = 1, . . . , n,(A.1)

whereβ = Dv/Du, d = Du(n + 1)2/L2 holds whereDu is the diffusion coefficient ofu andL is the length ofinterval for the original PDE. In what follows most of the computations was done forn = 8 or 32. One may thinkn = 8 is too coarse, however, we will see at the end of this appendix that the AUTO diagram of Eq. (A.1) forn = 8is qualitatively very close to the PDE case as far asd is not too small (see Fig. 36). Hence it is plausible that thePDE model also experiences a similar metamorphosis of branches as Eq. (A.1) does.

A.1. Bogdanov–Takens–Turing point

There are three parametersk, α, and diffusivityd. In order to have a BTT singularity at an equilibrium point,it is necessary to tune all these parameters. In fact, as we saw in Section 2, the ODE kinetics employed in Eq. (4)has a BT singularity atUBT = (uBT, vBT) = (1, 1/

√2) whenm = 1/

√2 andα = 1/2, namely fold and Hopf

singularities coalesce at this point in the parameter space (see Fig. 6). When the diffusive effect is taken into account,we also have the diffusion-driven (Turing) instability at an equilibrium point of Eq. (5) when the parameters lie inthe region-2 in Fig. 6. Combining BT-point and Turing instability, we have a highly degenerate singularity, what wecall, the BTT point by tuning three parameters, which is at least of codimension 3. By computing the eigenvaluesof the linearized system of Eq. (4) atUBT, it is easy to see that Turing instability of any mode could occur atUBT

in the limit of Du, Dv → 0. Formally speaking,UBT has a BTT singularity of codim∞ at m = 1/√

2, α = 1/2andDu = Dv = 0. Apparently the codimension becomes finite when the parameters deviates slightly, however,at present, it seems too complicated to treat this singularity in its full generality, so we focus on a special sectionof this singularity by fixingm = 0.95 andβ = Dv/Du = 4. We draw the bifurcation diagrams inα versussolution space for a givend, and trace the deformation of diagrams when the diffusivityd decreases. It shouldbe noted that the trivial state (0,0) always exists and is locally asymptotically stable. Therefore, the kinetics is ofbistable type whenα < αc, and mono-stable whenα > αc, whereαc is theα-value of the fold bifurcation point(see Fig. 26).

For larged (i.e., the system size is small), sayd = 2.5, the bifurcation diagram is the same as that of the ODEkinetics (see Fig. 26), i.e., there appears fold bifurcation of equilibria atαc and the Hopf bifurcation on the upper


Fig. 34. Emergence of Hopf barrel on Turing branch of 2-mode (d = 0.2, n = 8).

Fig. 35. Stable oscillatory 1-pulse (d = 0.2, α = 0.275, n = 8).

branch which ends up with a homoclinic orbit (see Fig. 6). Whend decreases, the Turing bifurcations occur on theupper and lower branches of equilibria. When we plot the location of Turing bifurcation on each branch in(Du, α)-plane for the PDE model, we have Fig. 27. For the compartment model, we have a similar bifurcation curves to thePDE model when the number of compartments is not too small. The mode-number stands for the correspondingFourier mode-number at the onset. Note that, for small diffusivity, higher-mode-number of branches are able tobifurcate primarily from the upper branch asα is decreased, therefore, we have intersections of different mode-number of bifurcation curves on the upper branch (see Fig. 27(a)) where codimension 2 (Turing type) bifurcationoccurs. These codim 2 points are very important, since the branch of higher-mode-number successively recoversits stability through the interactions with other branches (a quite similar thing was already observed in [15]). TheTuring bifurcations on upper and lower branches are not indifferent, in fact, whend = 1.6, the bifurcating branchesform a Turing loop of 1-mode type connecting the upper branch to the lower branch of equilibria as in Fig. 28. TheTuring branch in Fig. 28 does not look like a loop, since the norm of the ordinate isL2. If we switch to a differentnorm which distinguishes a solution from its reflection at the center of the interval, the loop shape becomes visiblelike Fig. 36. This type of Turing bifurcation of higher-mode successively occurs asd decreases and interactionamong those branches causes many secondary and tertiary bifurcations, and after interaction, EHL is eventuallyformed.


Fig. 36. Subcriticality and the limiting point of Hopf branch of 1-pulse for the PDE model (Eq. (4)) ((a-1) and (a-2)) and the compartmentmodel (Eq. (A.1)) ((b-1) and (b-2)):(b-1) and its magnification (b-2) show the existence of limiting point of 1-pulse for the compartment modelat d = 0.175 withn = 8, which are qualitatively very close to the PDE diagrams (a-1) and (a-2) atDu = 0.001,Dv = 0.004 withL = 0.65.The parameter values of these two models almost coincide with each other through the relationd = Du(n + 1)2/L2. The vertical axis of thediagram stands for the modulus of the value at the left boundary grid.

Fig. 37. Hierarchy structure of oscillatory pulses for compartment model (d = 0.3, n = 32): each arrow indicates the interval where theassociated pulse is stable.

A.2. Overhang of Turing loops and emergence of Hopf barrels

The Turing loop always overhangs the fold branch of equilibria as in Fig. 28 and, therefore, Turing patternspersist over the parameter region where there are no nontrivial constant states (i.e., only zero solution survivesthere, namelymono-stable region). This explains partially why EHL is formed in mono-stable region. When the


Fig. 38. Oscillatory 2-pulse (d = 0.3, α = 0.272,n = 32).

Turing bifurcation point on the upper branch come close to the limiting point of equilibriaαc (see Fig. 27a) asd isdecreased, there appears a ‘Hopf barrel’ as a secondary branch near the Turing bifurcation point as in Fig. 29.

This Hopf instability is different from the one appeared atH0 in Fig. 26, in fact, a new pair of complex eigenvaluecrosses the imaginary axis at the bifurcation pointH1 of Hopf barrel (see Fig. 29) and the associated eigenfunctionresembles the spatial derivative of the steady Turing pattern there. This implies that the resulting oscillatory patternrolls from side to side, not up and down (see Figs. 31).

A.3. Interaction of Turing branches and deformation of Hopf barrels

For an appropriate smalld, the Turing branch of 2-mode appears as in Fig. 32. Note that the bifurcation points of2-mode are shifted to the upper branch compared with those in Fig. 27, since in the latter the computation was forthe continuous PDE model. The 2-mode branch swells and starts interacting with 1-mode branch asd is decreasedthrough a double-zero singularity (codim 2-bifurcation), then 1-mode branch is split into two parts as in Fig. 33.Nevertheless, the Hopf barrel of 1-mode persists as a tertiary bifurcating branch. More precise local analysis neardouble-zero singularity was done, for instance, in [15]. Stilld is decreased, the Turing branch of 3-mode appearson the upper side of 2-mode branch, and then Hopf barrel appears on the 2-mode branch as in Fig. 34, which is ofsupercritical type. The resulting oscillatory pattern is of pulse type rolling from side to side as in Fig. 35. As we willsee later that this oscillatory solution deserves to be called a ‘pulse’ as the system size becomes larger, namely bothfront and back sides decay quickly to zero. For simplicity, we call time-periodic traveling solution of pulse typeon finite interval theoscillatorypulse. The size of this barrel becomes larger asd decreases, and at the same time,the barrel changes its form from the supercritical type to thesubcriticalone. In fact, whend = 0.2, the whole partof the Hopf branch is represented by black disk (i.e., it consists of only stable solutions), however, asd becomessmall, the Hopf branch starts to bend and take the form as in Fig. 36(b-2) atd = 0.175. Although this deformationseems innocent from bifurcational point of view, it has a strong influence on the dynamics, namely there appears anaftereffectnear the limiting point discussed in Section 3. It should be noted that the bifurcation diagram Fig. 36(b-2)for the compartment model is qualitatively very close to Fig. 36(a-2), hence it is plausible that the subcritical branchof Fig. 36(a-2) for the PDE model was born through the similar successive bifurcations and deformations as above.Recalling the discussion in Section 4.1, we easily see that, once we have such a subcritical branch of 1-pulse typeon a large interval, there exists a hierarchy structure of limiting points by the folding-up principle.


Fig. 39. Hierarchy structure of limiting points of the oscillatory branches.

Fig. 40. From 1-pulse to Turing pattern of 8-mode (d = 0.3, α = 0.27028,n = 32).

A.4. EHL of time-periodic branches for the P-model

When the system size becomes large, the compartment model (A.1) is expected to form an EHL of time-periodicbranches. However, it is still a hard task, at present, to trace all the periodic branches even for the compartmentmodel when the size is large. Therefore, we instead trace thestableperiodic branches of multi-pulse type withrespect toα by solving the evolutional system. In fact, forn = 32, it has already 3-level hierarchy structure as inFig. 37.

The profile of oscillatory 2-pulse looks like Fig. 38 atα = 0.272, but it ceases to exist aroundα = 0.27082.Similar thing also happens to 3-pulse type. The ceasing points of these three branches are almost equal, whichstrongly suggests that the schematic picture Fig. 39 is correct. In fact, we made two numerical experiments startingfrom almost identical initial shapes atα = 0.27028 to check whether the orbit starting from 1-pulse shape behavesaccording to Fig. 39 or not. Fig. 40 shows a symmetric splitting, i.e., 1-pulse→ 2-pulse→ Turing pattern of 8-mode.


Fig. 41. From 1-pulse to oscillatory Turing pattern of 6-mode (d = 0.3, α = 0.27028,n = 32).

On the other hand, for a slightly different initial data, Fig. 41 shows asymmetric splitting, i.e., only one-hump of2-pulse pattern splits at the second splitting and the orbit is trapped in the basin of 3-oscillatory pulse. This indicatesa subtle dependence on the initial data as well as the influence of boundary, however, our view point is useful tounderstand those transient behaviors.

In conclusion, the above numerical tracing strongly indicates that BTT singularity is responsible for formingthe subcritical branch of 1-pulse type, and hence the associated hierarchy structure EHL when the system sizebecomes large. This observation motivates us to try to prove the existence of such a subcritical branch analyticallyby unfolding BTT singularity, which is left for future work.

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A skeleton structure of self-replicating dynamics