Towards nonlinear selection of reaction-diffusion patterns in presence ...

PAPER

www.rsc.org/pccp | Physical Chemistry Chemical Physics

Towards nonlinear selection of reaction-diffusion patterns in presence of advection: a spatial dynamics approach Arik Yochelis* and Moshe Sheintuch* Received 16th February 2009, Accepted 25th June 2009 First published as an Advance Article on the web 17th August 2009 DOI: 10.1039/b903266e We present a theoretical study of nonlinear pattern selection mechanisms in a model of bounded reaction–diffusion–advection system. The model describes the activator–inhibitor type dynamics of a membrane reactor characterized by a differential advection and a single diffusion; the latter excludes any finite wave number instability in the absence of advection. The focus is on three types of different behaviors, and the respective sensitivity to boundary and initial conditions: traveling waves, stationary periodic states, and excitable pulses. The theoretical methodology centers on the spatial dynamics approach, i.e. bifurcation theory of nonuniform solutions. These solutions coexist in overlapping parameter regimes, and multiple solutions of each type may be simultaneously stable. The results provide an efficient understanding of the pattern selection mechanisms that operate under realistic boundary conditions, such as Danckwerts type. The applicability of the results to broader reaction–diffusion–advection contexts is also discussed.

I.

Introduction

Open spatially extended systems are known to exhibit universal behaviors under nonequilibrium conditions, such as spiral and solitary waves, standing-wave labyrinths, and oscillating spots.1,2 Most of these phenomena are found to persist in chemical media3,4 which in turn support the use of autocatalytic (nonlinear) reaction–diffusion (RD) systems, as case models to study the general mechanisms of self-organization phenomena. As such, the seminal work of Turing5 (and its extensions1,2,4,6) became central in the theory of pattern formation and led to many fundamental insights of pattern selection mechanisms in many fields of applied science. Yet, the selection mechanisms in situations when several distinct spatially nonuniform states coexist under the same conditions, are still intriguing open problems bounded to high sensitivity to initial and boundary conditions (BC).7 In such contexts, the terminology nonlinear wavenumber selection is often used to distinguish from the linear (pattern forming) instabilities that are dominant in vicinities of instability points,1 such as the Turing instability. The central obstacle to predict such patterns is in the absence of analytical tools, like the Amplitude equation reductions that rigorously capture the weakly nonlinear spatiotemporal behaviors near the onset of an instability.6 Recently, spatial dynamics and numerical continuation methods8–14 have been shown to be useful to scrutinize the nonlinear pattern selection problem in RD type systems. This theoretical approach allows one to uncover a number of complex nonlinear mechanisms, such as pattern formation in presence of homoclinic snaking15 and propagating nonlinear waves,16–20 in systems with (non-variational) activator–inhibitor dynamics.

Department of Chemical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel. E-mail: [email protected], [email protected]

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Most often RD systems should account for reactant supply and product removal by advection. Advective transport, which may represent a flow of certain components21 or motion of ions in the presence of an externally imposed electric field,22 modifies the resulting patterns significantly. Such a class, is often referred to as a reaction–diffusion–advection (RDA) medium. These processes may arise in a broad class of applied sciences, including tubular reactors,23–26 axial segmentation in vertebrates,27 biochemical oscillations in the amoeboid organism Physarum,28 autocatalytic reactions on a rotating disk,29 and vegetation patterns.30 Although patterns have been studied in several RDA models,26–29,31–49 the theoretical results are limited to the vicinity of the instability onset. Thus, many questions remain open, for example the impact of nonlinear instabilities and boundary conditions on spatiotemporal dynamics.32,45,47,50–56 Here we theoretically study pattern formation under one of several distinct BC in a one-spatial dimension RDA system. We focus on three types of asymptotic behaviors: traveling waves (TW), stationary periodic (SP) patterns and excitable pulses. The paper extends and provides details to pattern selection mechanisms that have been briefly introduced in ref. 57. In section II, we present a two component activator– inhibitor type model that contains a single diffusive field: such a situation arises in cross-flow (membrane) chemical reactors.25,35 Notably, the absence of two diffusive fields excludes, in the absence of advection, the presence of any finite wavenumber instabilities.1 Next, in section III we focus on a linear analysis of uniform states in an extended system and identify two finite wavenumber Hopf bifurcations but then demonstrate that in most cases the spatiotemporal dynamics cannot be deduced from it. Consequently to uncover the mechanisms of pattern selection, we employ in section IV, the spatial dynamics approach: looking at bifurcations of nonuniform periodic solutions in the comoving frame. In these investigations we exploit numerical branch-following methods This journal is

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coupled to temporal stability computations. In section V, we use the above insights and predictions to explain the effect of BC on selection mechanisms in finite domains (x A [0,L]), i.e. with realistic BC of Danckwerts type. Finally in section VI, we conclude the methodology and discuss the relevance of the results to broader RDA contexts.

III.

II. Reaction–diffusion–advection model: a chemical reactor

and show the system dynamics for certain Da values for the set of parameters

To demonstrate our methodology we use an RDA model that describes a pseudohomogeneous one-dimensional catalytic membrane (or cross-flow) reactor in which a single first order exothermic reaction occurs, or a simple flow reactor with two consecutive reactions A - B - C, where the first reaction proceeds at a constant rate; in both cases we can find a homogeneous (space-independent) solution. The reactants in a cross-flow reactor are supplied along the systems to avoid temperature runaway or poor selectivity that may be associated with feed at one point. The mass and energy balances can be written in dimensionless form that reads25,35

Le

@u @u þ ¼ f ðu; vÞ  u; @t @x

ð2:1aÞ

@v @v 1 @2v þ ¼ Bf ðu; vÞ  av þ ; @t @x Pe @x2

ð2:1bÞ

where  f ðu; vÞ  Dað1  uÞ exp

 gv ; gþv

ð2:1cÞ

is the rate of a first order activated reaction which is controlled by a kinetic parameter (Damko¨hler number, Da). In (2.1) u(x,t) stands for conversion (u = 1 implies zero reactant concentration) and can be viewed as a fast inhibitor while v(x,t) is the temperature or a slow activator in the context of RD systems. Le (Lewis number), the ratio of solid- to fluid-phase heat capacities, is a large number and also Pe (Pe´clet number), the ratio of convective to conductive enthalpy fluxes, is large resulting in steep gradients. Consequently the lumped version of the system (i.e., a mixed reactor or CSTR) does not predict temporal uniform oscillations. Cross-flow systems are bounded and the BC are expected to impact the spatiotemporal dynamics.50,51,55 For chemical reactors, the common BC are Danckwerts type,25,35 which imply mixed at the inlet u ¼ 0;

@v ¼ Pe v @x

at

x ¼ 0;

ð2:2aÞ

and no-flux (Neumann) @v ¼0 @x

at

x ¼ L;

ð2:2bÞ

at the outlet, where L is the physical domain size. Similar equations describe high-switching asymptote of a loop reactor, when the feed is periodically switched between several units,58 but the BC are periodic in that case. This journal is

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Stability of uniform states

In this section we first assume an unbounded case and apply the linear analysis around the uniform steady states, u = u0 and v = v0  Bu0/a, where u0 is a solution to   u0 gu0 ¼ 0; exp  Da  1  u0 ga=B þ u0

B = 16.2, a = 4, g = 10 000.

(3.3)

While g value is unrealistically large, we use this value as there exists an asymptotic solution as g - N. For (3.3), variation in Da yields a narrow bistable region for  57,59 as depicted in Fig. 1. Da+ SN r Da r DaSN, A

Linear stability of extended domain

Our analysis starts by looking at the growth rate of infinitesimal periodic perturbations about the uniform state1       u u u0 þ e k estþikx þ c:c: þ Oðe2 Þ: ð3:4Þ ¼ v0 vk v Here s is the (complex) perturbation growth rate, k 4 0 is the wavenumber, e { 1 is an auxiliary parameter, and c.c. denotes a complex conjugate. As we had already showed in ref. 57, for (3.3) and Le = 100, Pe C 14.976, numerical calculations admit two dispersion relations, to which we refer as s(k); among the two only s+(k) is relevant, since Re[s(k)] o 0, 8k. The inspection of s+(k) yields (see Fig. 1):  The ‘top’ branch of uniform states goes through a finite wavenumber Hopf bifurcation at Da = Da+ W = 0.2, + Re[s+(k+ )] = 0 while Re[s (k )] o 0 otherwise, where W + W + + l+  2p/k C 0.94 and c C 0.0009 at the onset. W W W  The ‘bottom’ branch goes through a finite wavenumber Hopf bifurcation at Da = Da W C 0.1264, with a higher speed   c W C 0.007 and a longer period lW  2p/kW C 3.06 at the onset. Here, the imaginary part of the dispersion relation admits Im[s+(k)] o 0, 8k 4 0 while Im[s+(0)] = 0 persists. In RD systems, a finite wavenumber Hopf bifurcation is encountered in a three component system with at least two diffusing fields,20 giving rise to both traveling and standing waves.60,61 Here, the advective terms in (2.1), break the spatial reflection symmetry of right-left propagating waves and for the same reason also the presence of standing wave oscillations that respect the reflection symmetry, is excluded. In addition, Danckwerts BC [see eqn (2.2)], which are of interest here, readily exclude the existence of spatially uniform solutions of (2.1) and the validity of the above analysis should be questioned. Nevertheless, we will show that on relatively large domains, the system properties can be still analyzed by means of spatial instabilities of the uniform states.57 B Motivation: direct numerical integrations and the pattern selection problem To test the results of the linear analysis and the impact of BC we integrate eqn (2.1) on a relatively large domain (L = 20) subjected to either periodic or Danckwerts BC [see eqn (2.2)]. Phys. Chem. Chem. Phys., 2009, 11, 9210–9223 | 9211

Fig. 1 Bifurcation diagram for spatially (u0,v0) steady states as a function of Da, underlying the hysteretic behavior for ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffihomogeneous + Da u2 þ v2 . The right- and the left-most insets indicate the onset of finite wavenumber Hopf bifurcations SN r Da r DaSN, where N ¼   (Re[s+(k W)] = 0 and Re[s+(k)] o 0 otherwise) while the other dispersion relations mark the condition at which Re[s+(kH)] = Im[s+ (kH)] = 0 (see text for details); solid and dashed lines mark Re[s+(k)] and Im[s+(k)], respectively. Consequently, in the bifurcation diagram, dark solid lines mark the temporal stability of uniform states in unbounded domains. Other parameters are given in (3.3) with Le = 100 and Pe C 14.976.57 + Within the domain Da W r Da r DaW, where the uniform states are linearly unstable, we test the spatiotemporal evolutions [Fig. 2(a)] in response to small amplitude random fluctuations at t = 0 and near x = 0, in a background of a uniform state (u0,v0). Below Da W, we use large amplitude perturbations but over a finite spatial extent.

1. Periodic domains. Above Da = Da+ W the uniform steady state is stable and all perturbations decay.57 However, when Da is decreased below Da+ W, we find a transition from smallamplitude [Da t Da+ W, e.g. Da = 0.19 in Fig. 2(a)] downstream-moving TW with l C 0.94, to upstream-moving TW with l C 1.18 [Da = 0.15 in Fig. 2(a)]. At intermediate values (Da = 0.16), the downstream propagation of the initial

Fig. 2 Space–time plots for different values of Da, where the dark color indicates larger v field values (normalized for all images); eqn (2.1) was integrated with (a) periodic or (b) Danckwerts [see eqn (2.2)], boundary conditions. In (a), for Da = 0.19,0.16,0.15,0.131 a small-amplitude random perturbation around the uniform state (u0,v0) and a small spatial region near x = 0 were admitted at t = 0 while for Da = 0.12 both the amplitude and the spatial region were increased. In (b), the initial state is (u,v) = (u0,v0) for all, whereas the perturbation is constantly present at x = 0, due to the Danckwerts boundary condition. Other parameters as in Fig. 1.

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Fig. 3 Schematic spatiotemporal evolution of waves under (a) convective and (b) absolute instabilities.52,62 Direction of the advection is from left to right.

perturbation gives rise to upstream propagating waves with l C 1.11; at later times the TW change their propagation direction. As Da is decreased towards Da W, the period of the upstream propagating TW increases significantly (l C 3.3 at Da = 0.131); as we show later, with different initial conditions we may find distinct patterns. Further decrease in Da, to the regime where (u0,v0) is stable (Da = 0.12 o Da W), leads to large amplitude upstream moving TW, after large-amplitude initial perturbations are used; finite amplitude perturbations are required here since the uniform state is linearly stable. The presence of an advective flow introduces additional complexity to the linear analysis, distinguishing between convective and absolute instabilities of a wave packet, as shown in Fig. 3. However, these two instabilities can be also nonlinear,31,52,62 and we wish here to better understand the nature of these nonlinear instabilities and the related wave evolutions. The spatiotemporal behavior near the onset Da+ W describes typical dynamics in the presence of a linear supercritical convective instability: both boundaries (left and right fronts) of a region where the perturbation is developed, move in the advective direction. Thus, on a finite domain with no-flux BC also at the inlet, the waves are expected to be swept out while leaving the system at the rest state, see Fig. 2(a) at early times and Fig. 3(a). On the other hand, if the instability is of the absolute type, the left front of the perturbation expands against the advective direction and the perturbation evolves throughout the whole domain and is not swept out of the system, see Fig. 3(b). For Da C 0.16 the instability resembles an absolute type due to the evolution of the perturbation to left. However, this manifestation is different from the standard transition from convective to absolute instability:52,55,62 normally the propagation direction of the resulting waves should persist (see Fig. 3) while here the absolute instability is associated with the change in propagation direction of the TW. In addition, in the vicinity of Da W, the period of the resulting waves is of large amplitude and is different from the period of counter propagating waves at Da = 0.15, indicating a nonlinear instability of TW at Da = Da W. These observations enhance the need in development of a nonlinear theory even for periodic domains. 2. Finite domains with Danckwerts boundary conditions. Danckwerts BC [see (2.2)], break the translational symmetry of (2.1) and due to the spatial locking of the phase at the inlet,21 may stabilize stationary nonuniform patterns [Fig. 2(b)]. Here, for all Da values the initial state is (u0,v0), This journal is

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and the perturbations are naturally exerted at x = 0. The transient spatiotemporal nucleation of TW at Da = 0.19,0.16,0.15 are similar to TW dynamics on periodic domains, but then asymptotically SP are being formed with periods larger (Da = 0.19), similar (Da = 0.16) or smaller (Da = 0.15) than the periods of respective nucleating TW. On the other hand, the dynamics for Da \ Da W are different: The upstream moving TW form SP states but the period of SP states is much smaller than the corresponding TW. For Da = 0.131, the process repeats itself via spontaneous excitations in a moderate distance from the rightmost stationary stripe so that the SP domain continuously increases. Surprisingly, as Da is decreased to Da = 0.127 \ Da W, the perturbation generates only one sequence of upstream propagating waves which consequently generate a bounded SP state about the uniform state (u0,v0), i.e. no secondary excitations at x = L. Notably, the nonuniform stationary states are consequences of nonuniform propagating states, i.e. there is no instability to perturbations (small or large) as in the case of nucleation of TW. Nevertheless, as we will show in the following section, the existence of such states can be efficiently analyzed by means of spatial instabilities.

IV.

Coexistence of nonuniform states

To explain the pattern selection of nonuniform states, some described in Fig. 2, we pursue here both the existence and stability of such solutions. We consider (2.1) in a comoving frame and on periodic domains; after the transformation x = x  ct. Since the reflection symmetry of the TW is not present, the sign of c was chosen to be negative according to the dispersion   relation at the onset: c W = Im[s(kW)]/kW o 0. Thus, we replace qt - qt  cqx, qx - qx and look at time independent version, qt = 0: du f ðu; vÞ  u ¼ ; dx 1c

ð4:5aÞ

dv ¼ w; dx

ð4:5bÞ

dw ¼ Pe½ð1  cLeÞw  Bf ðu; vÞ þ av: dx

ð4:5cÞ

Similarly to the original model, the first step constitutes a linear analysis of the uniform states or alternatively looking at the asymptotic behavior in space of nonuniform solutions: 0 1 0 1 u u0 @ v A  @ v0 A / emx ; ð4:6Þ 0 w where the spatial eigenvalues m satisfy a third order algebraic equation. In the next sections we show that the knowledge of spatial eigenvalues reveals a significant information on the propagating or stationary nonuniform states, in the context of (2.1). The investigation of distinct solutions that emerge from the spatial bifurcation points is then performed by a numerical continuation AUTO package,63 where c is obtained by a nonlinear eigenvalue problem on periodic domains, Phys. Chem. Chem. Phys., 2009, 11, 9210–9223 | 9213

i.e. L = l  2p/k. Branches of the resulting solutions are plotted in terms of the norm sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 l 2 ðu þ v2 þ w2 þ u2x þ v2x þ w2x Þ dx: ð4:7Þ N¼ l 0 At the end, temporal stability of such solutions is computed via a standard numerical eigenvalue method using the timedependent version in the comoving frame.20 A

Spatially periodic orbits

The two finite wavenumber Hopf instabilities identified earlier  at Da = Da W with c = cW, correspond in (4.5) to Hopf bifurcations (for details see Appendix A). In this section, we discuss the primary periodic orbits bifurcating from these two onsets (Fig. 4), while keeping the periods l W fixed. However, according to the analysis of an unbounded system (3.4), beyond the bifurcation onsets additional wavenumbers become unstable, to which we refer as secondary solutions. In some parameter regions the latter can be simultaneously stable, thus, to gain understanding of the selection mechanisms we also uncover the multiplicity and properties of solutions with l a l W. For simplicity, we portray only two such secondary branches [Fig. 5(a)] but there are infinitely many solutions pertaining to various periods. The insights are used to both explain and predict distinct numerical observations in the nonlinear regime. 1. Propagation of downstream traveling waves (TW+). The TW+ solutions, emerge supercritically from Da = Da+ W as small amplitude periodic orbits which are right propagating waves in the context of (2.1) (see Fig. 4). Continuation of these oscillatory states to lower Da values, shows stable states of c 4 0 (with l = l+ W) up to a turning point (saddle-node) which connects to an unstable TW branch that emerges at the (codimension-two) saddle-node/Hopf bifurcation (see Appendix A), Da = Da SN, as shown in the inset and the respective profiles (a–c) in Fig. 4. (While this scenario is generic in a three component system,16,64 the mathematical

aspects of the situation are beyond the scope of this paper.) The temporal linear stability computations of TW+ were performed via solving a numerical eigenvalue problem in the time dependent system, over L = l+ W C 0.94. 2 Propagation of upstream traveling waves (TW). The onset of TW solutions also corresponds to a Hopf but at Da = Da W (see Appendix A). Here continuation of the orbits with fixed l = l W, resulting in a subcritical bifurcation which forms a branch of unstable orbits in direction Da o Da W (see Fig. 4). The branch turns around at Da C 0 and gains stability before extending to large Da values (Fig. 4). After additional saddle-node (rightmost) it terminates as a small amplitude periodic orbit at Da C 0.155 with c C 0.007 (Fig. 4). (Notably the latter is also a Hopf onset but there is no analogue bifurcation in the context of (2.1), since the dispersion relation at that point admits a continuous band of unstable wavenumbers.) Stable solutions along TW branch correspond mostly to c o 0 (counter propagating waves). The temporal stability was calculated for periodic solutions (TW) with L = l W C 3.06. The subcritical nature of the TW branch well predicts the direct numerical integration of eqn (2.1): for Da 4 Da W the emerging states are of large amplitude while for Da o Da W the uniform state is indeed stable to small perturbations [see Fig. 2(a)]. 3. Multiplicity of stable wave solutions on large domains. As we indicated before, we are primarily interested in large domains, where L is much larger than the critical periodic solutions. It is known that stability to long wavelength perturbations (L = nl, n 4 1 with L \ 10), admits narrowing of the portion of stable periodic solutions, cf. ref. 20, 65 and 66. Numerically we show that the new onsets accumulate at Da values that are different than that for L = l W [compare the TW branches in Fig. 4 and in Fig. 5(a)]. We demonstrate this instability in Fig. 6 (left panel), by starting with a slightly perturbed periodic sinusoidal state with period l+ W, in domain

Fig. 4 Middle panel: Bifurcation diagram for spatially homogeneous (u0,v0,0) steady states and traveling waves TW+ and TW, as a function of  Da, where N is given by (4.7) with respective periods l+ W C 0.94, lW C 3.06. Eqn (4.5) was integrated on periodic domains, where solid lines indicate temporal stability in domains L = l , in the context of eqn (2.1). The inset shows the respective oscillatory TW branches in terms of the W maximum value of v in vicinity of termination points: for TW+ (Da,c) C (0.1323,0.00878) and for TW (Da,c) C (0.155, 0.00772). Left panel (a–c) and right panel (d–f) correspond to profiles at Da values as marked in the middle panel; the arrows in (b) and (e) mark the propagation direction in the context of eqn (2.1). Parameters: (a) Da C 0.198, (b) Da C 0.147, (c) Da C 0.1295, (d) Da C 0.1225, (e) Da C 0.1351, (f) Da C 0.1551, while other parameters as in Fig. 1.

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Fig. 5 (a) Branches and stability (in large domains) of traveling wave solutions in (Da,c) parameter space while l along each branch is fixed, where for branch (A) l = lA C 1.18, and for branch (B) l = lB C 1.10. (b) Nonlinear dispersion relation, l vs. c, at Da = 0.126, which is slightly below Da = Da W. The right inset magnifies the turning point where (A) mark (c,l) C (0.0088,lA) and (B) mark (c,l) C (0.00879,lB). The left inset depicts profiles of spatially localized solutions of homoclinic type [connection to a fixed point (E) and to a periodic orbit (’)], on a periodic domain l = 20. Other parameters as in Fig. 1.

L = 10l+ W. The asymptotic period is l = lA \1.18, which implies the existence of additional (secondary) periodic states. An efficient way to track secondary periodic solutions, is by computing a nonlinear dispersion relation, l vs. c (at fixed Da), for which the locus of nonlinear TW solutions is computed. Here we employ the subcritical regime of TW, since secondary wavenumbers that bifurcate from Da \ Da W inherit the subcriticality of the primary periodic states.67 Thus, we set Da = 0.126 t Da W and integrate (4.5) by varying simultaneously c and l. The resulting dispersion relation admits for l B O(1), periodic orbits with positive and negative velocities [see Fig. 5(b)]. The rightmost turning point, corresponds to the fastest downstream propagating TW with periodicity slightly larger than TW+, l = lB C 1.10, see point (B) in the top inset in Fig. 5(b). As we will show, solutions in vicinity of this fold play a significant role in pattern selection mechanisms, determining both the wavelength, the speed and the propagation direction of the asymptotic pattern, in the context of (2.1). To obtain the properties of such a secondary solution, we vary Da and monitor c while keeping l = lB, as shown in

Fig. 5(a). Continuation indeed shows that the secondary periodic orbit emerges at Da 4 Da W and terminates supercritically as a small amplitude state at Da o Da+ W. As a comparison, we have followed another (already mentioned before) periodic orbit lA C 1.18 [marked as (A) in Fig. 5]; notably an infinite number of such orbits is possible. All periodic solutions change the speed sign but at distinct Da values, which decreases with the period length. Notably, the slope near the speed transition c = 0 decreases with the period length and implies a respective slow propagation over a larger interval of Da values. The large multiplicity of coexisting stable periodic solutions implies also the following selection mechanisms:  The small slope of branch (B) in vicinity of c = 0 [Fig. 5(a)], explains the ‘competition’ (a rapid alternating direction) between upstream and downstream TW at Da = 0.16 [see Fig. 2(a)].  The emergent upstream moving TW at Da = 0.15 [in Fig. 2(a)], with a period larger than l+ W, are associated with the coexisting secondary TW solutions which admit a respective negative speed. This is also confirmed by a direct numerical integration of (2.1) at Da = 0.12 for L = 10lA and L = 10lB, as shown in Fig. 6 (middle panel) and Fig. 6 (right panel), respectively. Beyond the above explanations, we will attempt to gain (in section V) more general principles of pattern selection mechanisms that operate under other initial conditions in the nonlinear regime. B

Fig. 6 Spatiotemporal dynamics of (2.1) under periodic boundary conditions and distinct sinusoidal initial conditions, presented in space–time plots; dark color indicates larger v(x) values. Left panel: Instability of TW+ in domain L = 10l+ W = 9.4. Middle panel: Propagation of traveling waves upstream in domain L = 10lA = 11.8. Right panel: Propagation of traveling waves downstream in domain L = 10lB = 11. Other parameters as in Fig. 1.

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Spatially homoclinic orbits

In addition to spatially–oscillatory solutions, eqn (2.1) admits propagating solitary waves (or excitable pulses) which correspond [in the context of eqn (4.5)] to homoclinic orbits in space (l - N).57 Thus, in the following analysis we consider only solutions with large period orbits. In Fig. 5(b), we show that since the two branches in the nonlinear dispersion relation can be extended to large periods, homoclinic orbits are likely to form.64,68,69 Indeed, solutions with c o 0, approach a homoclinic connection to a fixed point, see profile (E) in the left inset in Fig. 5(b). On the other hand, Phys. Chem. Chem. Phys., 2009, 11, 9210–9223 | 9215

solutions with c 4 0 add additional spatial peaks to the profile, as the period increases, and by that approach to a homoclinic connection to limit cycle (periodic orbit), see profile (’). We will show that these homoclinic connections are in fact important since they are related to stable bounded excitable states and spatial defects. 1. Single pulses. The homoclinic connection to a fixed point [left inset in Fig. 5(b)], is found in the subcritcal regime of TW, Da t Da W. Indeed linearization about the uniform steady state yields a pair of complex conjugated eigenvalues (m, with Re(m) o 0) as well as one real (mr 4 0) where |Re(m)| o mr, corresponding to a Shil’nikov type homoclinic orbit,64,68,69 i.e. a monotonic excitation at the front and an oscillatory decay at the rear. Importantly, the pulse shape in the profile does not change with an increased period so that we regard this solution as a homoclinic orbit, for which formally l - N can be expected. Continuation in (Da,c) with a large fixed period (l = 30), shows a section of stable single pulse states (c o 0) extending over a large interval (Fig. 7), while the amplitude of the pulse is decreased as Da is increased towards Da = Da W [profiles (a) and (b) in Fig.7]. Above Da = Da , the homogenous state W becomes unstable and the branch after a number of excursions (saddle-nodes) approaches a periodic orbit [see the respective inset and profiles (c) and (d) in Fig. 7].

The solutions along the branch with c 4 0, are all unstable. The amplitude of the peaks along this branch decreases when approaching Da = Da W [see profile (e)] and eventually it decreases into an ‘‘oscillatory tail’’,16 as depicted in profile (f). Then after the fold (saddle-node), the solutions take the form of small amplitude bounded state,19 i.e. solutions that are composed from two peaks [see profile (g)]. As Da decreases the distance between the two pulses increases however persists in a double pulse state [see profile (h)]. This situation suggests the existence of a Belyakov bifurcation (see Appendix A), at which a saddle-focus of the linearized fixed point becomes a saddle and nearby multi pulse states can form.19 2. Grouped pulses. Next, we turn to analysis of the second class of homoclinic connections, corresponding to the branch with c 4 0 in the nonlinear dispersion relation [see Fig. 5(b)]. Continuation of the latter results in a branch that also distinguished by the propagation speed, but with a much more complicated structures (Fig. 8): bounded states of two peaks with c o 0 and intertwined branches of periodic orbits with c 4 0. In the following, we treat each of the cases separately. The double pulse states (c o 0) are being formed through a number of saddle-node bifurcations in which the number of oscillations within the profile is decreased, as shown in the inset in Fig. 8 and the respective (a) and (b) profiles. The consequent saddle-nodes resemble a slanted homoclinic

Fig. 7 Top left panel: Branches of the propagating long period solutions in a parameter plane (Da,c) for a fixed l; solid lines mark temporal stability on a whole line. The inset shows the enlargement in vicinity of the onset to TW states, i.e. Da = Da W, in terms of N (see eqn (4.7)). Top right panel: shows magnification of the turning point at which a single pulse becomes a double pulse state (all unstable). Bottom panels: Profiles of the solutions at the respective positions as indicated in the top panels. Parameters: (a) Da C 0.0021, (b) Da C 0.126, (c) Da C 0.132, (d) Da C 0.1317, (e) Da C 0.1135, (f) Da C 0.1249, (g) Da C 0.114, (h) Da C 0.1067, while other parameters as in Fig. 1.

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Fig. 8 Top left panel: Branches of the propagating long period solutions in a parameter plane (Da,c) for a fixed l; solid lines mark temporal stability. The inset shows the enlargement in vicinity of the onset to TW states, i.e. Da = Da W, in terms of N (see eqn (4.7)). Top right panel: shows magnification of the complex intertwining structure of solutions with positive speed (all unstable) where the dashed line marks the branch of unstable (small amplitude) oscillatory TW states. The inset shows one period of the branch oscillations. Bottom panels: Profiles of the solutions at the respective positions as indicated in the top panels. Parameters: (a) Da C 0.1308, (b) Da C 0.1319, (c) Da C 0.1301, (d) Da C 0.0674, (e) Da C 0.0818, (f) Da C 0.0745, (g) Da C 0.1245, (h) Da C 0.0701, while other parameters as in Fig. 1.

snaking structure,70 with increasing (in Da) oscillations as c is decreased. As a result, the double pulse solutions gain stability only after the turning point (rightmost saddle-node) and about Da = Da W [see profile (c)]. Then as Da is decreased the distance between the two pulses increases and the solution approaches a periodic orbit due to the finite domain size, as depicted by profile (d). The second part of the branch, c 4 0, admits a complex structure of intertwined (in Da) oscillations; along this branch all solutions are unstable, see top-right panel in Fig. 8. Before reaching the complex intertwined structure additional oscillations are being added to the profile, via a similar oscillating saddle-node structure as in the case of c o 0 branch, but here the oscillations rather decay with the increase in c. Then the complex oscillations of the branch at large c values, approach towards the branch of TW by variation in the profile structure. To observe this approach we plotted in the inset in Fig. 8 (top-right panel), a single representative oscillation along with the profiles at the respective saddle-nodes, see profiles (e–h). The approach of the solutions towards the TW (again a finite domain size effect), is through the top saddle-nodes [see profile (e)] while other saddle-nodes respectively ‘‘tune’’ the solution’s period by adding additional oscillations to the profile. Consequently, the envelope of the This journal is

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intertwined structure is decreasing in Da and eventually expected (on finite domains) to coincide with the TW branch, as shown in Fig. 8 (top-right panel). C

Stationary nonuniform solutions in space

Motivated by the formation of time independent solutions [Fig. 2(b)], we use spatial dynamics to also study (in this section) the behavior of stationary spatially periodic orbits (bounded to c = 0); while the c = 0 is a subset of the spatial dynamics, it prescribes a wavelength (l) which may not match the system length (L). Therefore, we distinguish between domain filling SP states and stationary patterns with defect near the outlet, such as Da = 0.15, 0.127 with Danckwerts BC [Fig. 2(b)]. As such, Da is an independent parameter and l(Da) is a dependent one. Notably, due to the coexistence of stable uniform and periodic states for Da o Da W, wide defects are naturally anticipated, yet we surprisingly find that a similar effect may also be present Da 4 Da W, where the uniform states are linearly unstable. 1. Periodic orbits, and homoclinic orbits to a fixed point. Although stationary solutions cannot be associated with any instability in the original model (2.1), their effective onset can be obtained from (4.5) by setting c = 0 (for all computations Phys. Chem. Chem. Phys., 2009, 11, 9210–9223 | 9217

that follow) and looking for a respective Hopf bifurcation.57 The branch of spatially periodic orbits (SP) bifurcate from  the respective locations at Da = Da H C DaW [see Fig. 9] and inherit the respective properties of the TW branches. We mark these branches by dashed lines since in (2.1) with periodic BC, these states inherit the instability of the steady state (u0,v0). In the top inset in Fig. 9(a), we show that the period of the SP+ solutions is slowly increased as Da is decreased. As Da approaches Dahom C 0, a rapid increase in the period length occurs and the branch terminates onto a stationary homoclinic orbit, as demonstrated in the respective profiles in Fig. 9(b) and (c). In the context of (2.1), both onsets   Da = Da H satisfy the condition Re[s(kH)] = Im[s(kH)] = 0 56 (see Fig. 1), where k = kH is related to the pair of imaginary +  spatial eigenvalues, m = ik+ H , at Da = DaH and m = ikH,  at Da = DaH. Notably, the homoclinic orbit at Da = Dahom is not of Shil’nikov type since there are no oscillations in the rear tail. The reason is that at Da  DaB C 0.079, the complex eigenvalues m collide on the real axis and become real, in the so called Belyakov bifurcation (see Appendix A). It means that multi-pulse stationary states are also characteristics of

(2.1) but evidently they can be stable only on non-periodic domains and be positioned near the inlet, similar to what is demonstrated in Fig. 2(b) at Da = 0.127. 2. Homoclinic orbits to a limit cycle. As opposed to homoclinic connections to a fixed point that yield excitable behavior in the context of eqn (2.1), here we show that homoclinic orbits to limit cycles are linked to spatial defects that may emerge at the outlet (x = L), under non-periodic BC. The bottom inset in Fig. 9(a), shows a narrow region of the SP solutions that subcritically emerge from the Hopf bifurcation at Da = Da H. As was already presented before, the increase in l is involved with inclusion of additional oscillations to the profile, see Fig. 9(b) and (c). As the spatial domain (L = l) is increased, the inner oscillations within the profile approach l C 1.136 as Da - 0.1285 [Fig. 9(c)]. This period is the period of the solution at Da C 0.1285 along the SP+ solutions, which lies exactly above the extension of the SP branch. Consequently, this subcritical situation of SP branch terminates rather on a homoclinic orbit to a limit cycle. These type of solutions have been already observed and suggested in Fig. 8 at c = 0 or in Fig. 5(b) with c 4 0. Notably, the spatial oscillations in the profiles are rather around the upper branch of (u0,v0) although the latter is not present for Da o Da H, due to the homoclinic bifurcation. This suggests that under different BC the existence regime of such solutions can be broader (i.e. for Da 4 Da H), so that defects of distinct length are possible to form, such as defects near the boundary x = L in the cases Da = 0.15 and Da = 0.127 in Fig. 2(b).

V. Pattern selection on finite domains

Fig. 9 (a) Bifurcation diagram for spatially uniform and nonuniform steady states SP that emerge respectively from Da H, as a function of Da. Eqn (4.5) was integrated on periodic domains with c = 0. The SP+ solutions approach a homoclinic orbit to a fixed point, l-N as Da - Dahom C 0, while the SP solutions approach a homoclinic orbit to a limit cycle, as Da - 0.1285. The left inset magnifies the respective region of the SP branch. The right inset presents the SP branches in terms of a spatial period, l. (b–c) Profiles of SP states for l = 24 and l = 30, respectively. N is given by eqn (4.7) while other parameters as in Fig. 1.

9218 | Phys. Chem. Chem. Phys., 2009, 11, 9210–9223

The approach of spatial dynamics discussed above allows explanations of a number of pattern selection mechanisms that were observed in Fig. 2. Moreover, a number of predictions were also made, such as sensitivity of TW to initial conditions (due to the coexistence of periodic orbits) or the formation of grouped excitable pulses. In this section, we elaborate on the nonlinear pattern selection mechanisms by testing the obtained predictions via direct numerical integrations of eqn (2.1) on finite domains. The aim is to unfold the emergence of patterns by exploiting distinct types of BC:  On periodic domains, there is an infinite number of coexisting periodic solutions, varying in their wavelength (l) and the corresponding speed (c). We have shown in Fig. 5(a) the primary two branches TW and only two secondary branches [(A) and (B)]. Therefore, in the context of eqn (2.1) and for any domain of size L, we will find a TW that is an integer multiple of one of these periodic solutions. Thus, the selected oscillatory state will be one of several possible solutions whose periodicity and propagation direction depend on the initial state and on L; with various initial conditions we may find different patterns. If we start with an initial state that incorporates two wavelengths, as we do below, we show that the system converge to a single wavelength that is intermediate between the two.  In contrast, with Danckwerts BC, there is no sensitivity to initial conditions: only the nonuniform solutions with c = 0 This journal is

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can be stable, i.e. solutions that belong either to SP+ or SP states. This implies having a small number (one or two) of solutions in a system of certain size L with L/l not an integer; the discrepancy between these values is satisfied with the defects at the outlet, evident in Fig. 2(b) at Da = 0.15, 0.027. A

Traveling waves

We have demonstrated in Fig. 2(a), a relatively ‘simple’ wavenumber selection due to the linear instability to a small amplitude perturbation. In Fig. 10 we want to demonstrate, the principles underlying wavelength ‘‘minimization’’ after initially imposing a step-distribution of two distinct wavenumbers, i.e. two different finite amplitude sinusoidal perturbations with each period filling half of the domain (l = ll at x A [0,L/2] and l = lr at x A [L/2,L], with ll o lr). We show that the system always converges to a single intermediate wavelength, since a system with large L may admit several such wavelengths multiples (or combinations). At early times, the TW propagate respectively upstream (ll) and downstream (lr), as depicted by the c(l) relation (Fig. 5), but then the system goes through secondary phase instabilities which homogenizes the wavelength. For discussion purposes, we find it useful to (artificially) distinguish between two Da regimes: ‘high’ and ’low’. In the high regime, 0.156 t Da o Da+ W (where Da C 0.156 marks the right TW boundary), we used ll = 1.0 and lr = 1.1 (Da = 0.19, 0.16 in Fig. 10). Initially the two packets follow the velocity resulting from the respective c(l) relation [e.g., Fig. 5(a)], but then the system undergoes phase-diffusion67,71,72 before settling on a single intermediate wavelength l C 1.05, which is smaller that the period of branch (B) in Fig. 5(a). Since the speed reversal condition c = 0 for branch (B) is at Da C 0.1735, TW at Da = 0.16 with l t 1.1 essentially predicted to propagate downstream (c 4 0), as being confirmed in Fig. 10. In the low regime, 0.024 t Da t 0.156 (where Da C 0.024 marks the left TW+ boundary), both TW are present, thus we set lr = 3.0. Here lr waves go through phase slips (Da = 0.15, 0.131, 0.12 in Fig. 10) and asymptotically approach an intermediate period l C 1.2 (although a solution with a period l W is stable). These phase slips are manifestations of the Eckhaus–Benjamin–Feir instability,73,74 and the resulting

TW correspond to upstream propagating waves (c o 0), as shown in Fig. 10. B

Stationary patterns

Unlike the TW families, the stationary states admit low coexisting multiplicity, one or two wavelengths to choose from (those with c = 0), but the chosen l may not fill the entire system length, resulting near the outlet (x = L) in a defect-like domain which is locally close to a ‘homogeneous’ solution. Here we focus on selection of such stationary patterns effectively stabilized under non-periodic BC. In what follows, we repeat the steps from previous section, distinguishing between two types of non-periodic BC (Danckwerts and no-flux), under the same domain size and initial conditions. The emphasis here is rather on asymptotic stationary patterns since in most cases the transients are similar to those of the TW discussed above. 1. Danckwerts boundary conditions. In Fig. 11, we show that the simulated wavelength agree with those predicted from periodic domains, see the top inset in Fig. 9(a). In the high regime, 0.156 t Da o Da+ W, the asymptotic stationary state exhibits the same periodicity that results from a small amplitude perturbation (Da = 0.19 in Fig. 11). In the low regime, Da t 0.156 but Da c Dahom, the wavelength is increased as Da is decreased, with a minor change in l (l C 1.0, see Da = 0.15, 0.131, 0.12 in Fig. 2(b) and 11). In addition, Fig. 9(a) suggests that as Da - Dahom C 0 the period should rapidly increase since the solution approaches a homoclinic orbit. This prediction is also supported: short wavelength perturbations x A [0,10] decay to a rest state (u0,v0) while only long period perturbations (lr), develop to a stationary stripped state with a large period (Da = 0.005 in Fig. 11). Notably, for Da o Da W the supplied perturbation must be supra-threshold implying that the extension of the asymptotic state is sensitive to initial conditions. 2. Spatial defects. The other property of SP states is the formation of defects near the outlet (x = L), for Da W o Da t 0.156. (For Da o Da W this property is absent since the uniform state is linearly stable so that periodic states can coexist or be embedded in background of a uniform state

Fig. 10 Spatiotemporal dynamics of (2.1) under periodic boundary conditions presented in space–time plots for different values of Da, where dark color indicates larger v(x) values. For Da = 0.19,0.16 we combined at t = 0, two finite amplitude sinusoidal perturbations that vary around the respective uniform state (u0,v0): l = 1l, x A [0,10] and lr = 1.1, x A [10,20], while for Da = 0.15,0.131,0.12 we used lr = 3.0, x A [10,20]. For Da = 0.19,0.16 the pattern approach (through a phase diffusion) a uniform periodicity of l C 1.05 while for Da = 0.15,0.131,0.12 the approached uniform periodicity (via phase slips) is slightly larger l C 1.2. Other parameters as in Fig. 1.

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Phys. Chem. Chem. Phys., 2009, 11, 9210–9223 | 9219

Fig. 11 Spatiotemporal dynamics of (2.1) under Danckwerts boundary conditions presented in space–time plots for different values of Da, where dark color indicates larger v(x) values. For Da = 0.19,0.16 we combined at t = 0, two finite amplitude sinusoidal perturbations that vary around the respective uniform state (u0,v0): l = 1.0, x A [0,10] and lr = 1.1, x A [10,20], while for Da = 0.15,0.131,0.12 we used lr = 3.0, x A [10,20]. Other parameters as in Fig. 1.

(u0,v0).) This feature is best studied via imposing no-flux (Neumann) BC at both ends, although in a real system such a BC at the inlet is not realistic. For Da t Da+ W (where the selection favors the TW with c 4 0), the waves are swept out due to the convective instability so that the asymptotic state is a uniform state (Da = 0.19 in Fig. 12). In the intermediate Da regime, the convective instability competes with the upstream propagating TW (c o 0), and consequently upstream waves decay near the inlet (x = 0) while new TW nucleate at the outlet x = L (Da = 0.16 in Fig. 12). For Da t 0.156, the upstream waves reach the inlet and evolve into a stationary periodic state, see Da = 0.15, 0.131, 0.12 in Fig. 12. This manifests the transition from a nonlinear convective to a nonlinear absolute instability. The width of the defects is decreased, due to increasing SP state wavelength, as Da is decreased (compare between Da = 0.15, 0.131, 0.12 in Fig. 11). These defects are a remote consequence of the SP states which are present in a narrow range on periodic domains but apparently exist over a much wider range of Da values if non-periodic BC are imposed. C

Excitable waves

Solitary waves can be generated in the regime Da o Da W via a localized finite amplitude perturbation,57 and their stability range is quite extended (see section IVB). Grouped excitable

states of two-peaks can also form (Fig. 8). Importantly, for these solutions, the interspacing between the pulses is increased as Da is decreased, with a minor speed variationsc. As Da is decreased further Da { Da W, the distance between the pulses will significantly increase and the speed will significantly decrease due to emergence to a periodic orbit, as representatively predicted by the TW branch in Fig. 5(a) and partially supported by a direct numerical integration at Da = 0.005 in Fig. 11. An extension of the double pulse branch depends on the domain size since the solution approaches eventually a periodic state, see profiles (c) and (d) in Fig. 8. To confirm the above predictions, we impose initially two localized adjacent stimulations and follow their spatiotemporal dynamics (Fig. 13). For Da t Da W, we indeed confirm not only the existence of double pulse states but their speed and interspace align with those predicted above; notably, for Da = 0.005 { Da W, the large spacing and slow speed agree with computations performed in the comoving frame. Stability of double pulse states [Fig. 13 (left and middle panels)] should not come as a surprise since non-monotonic dispersion relations [see Fig. 5(b)] often admit such a property.75 Under non-periodic (Danckwerts or Neumann) BC, stationary periodic (with large wavelength) bounded states emerge (see Da = 0.005 in Fig. 11); the number of peaks within the bounded state depends on the initial large amplitude and localized in space, perturbations.

Fig. 12 Spatiotemporal dynamics of (2.1) under Neumann boundary conditions (at x = 0 and x = L) presented in space–time plots for different values of Da, where dark color indicates larger v(x) values. For Da = 0.19 we combined at t = 0, two finite amplitude sinusoidal perturbations that vary around the respective uniform state (u0,v0): ll = 1.0, x A [0,10] and lr = 1.1, x A [10,20], while for Da = 0.15,0.131,0.12,0.005 we used lr = 3, x A [10,20]. Other parameters as in Fig. 1.

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Fig. 13 Emergences of patterns in the excitable regime (Da o Da W), presented in space–time plots for different values of Da, where dark color indicates larger v(x) values; eqn (2.1) was integrated with periodic boundary conditions. Here at t = 0, we induced two confined but separated large amplitude stimulations in a background of the uniform state (u0,v0) (the bottom branch); the perturbations are uniform, i.e. of a top-hat form. Other parameters as in Fig. 1.

VI.

Conclusions

We want to address here the physical and the pattern selection implications of our results. We chose a model that represents an activator-inhibitor type dynamics in the context of a crossflow reactor with a single Arrhenius kinetics,25,35 but the results are much wider and apply, as we argue below, to variety RDA models. Moreover, the motivated BC of Danckwerts type, are realistic for the studied system, but the results apply to other BC. RDA systems exhibit a large plethora of nonuniform states,26–29,31,32,34–49 including TW, SP patterns and solitary waves, thus it is natural to inquire: what are the nonlinear selection mechanisms in presence of multiplicity of coexisting solutions and the effect of BC? Recently, in a companion paper,57 we have partially demonstrated these mechanisms, and here we pursue the subject in more details by developing a methodology for as many as possible general spatiotemporal properties of bounded (in physical space) RDA systems. In pursue of the pattern selection mechanisms we employed the technique of spatial dynamics which provides a powerful framework for analyzing problems of this type: bifurcation analysis of nonuniform states coupled with numerical continuation. The spatial dynamics approach allowed us to view TW and SP states as periodic orbits in a comoving frame and identify excitable pulses with homoclinic orbits in space. The results were complemented by numerical temporal eigenvalue computations to identify the stability of these nonuniform states. The gained insights both explain and predict (see section IV) almost all the phenomena observed via direct numerical simulations of the original model [eqn (2.1)]. Among the results from our analysis, we find the following as most fundamental: Coexistence and nonlinear wavenumber selection: Under periodic BC, the model admits a large multiplicity of coexisting stable downstream- and upstream-propagating TW (section IV). The TW emerge due to the instability of the uniform states to two finite wavenumber Hopf bifurcations (see Fig. 1) and accompanied by an infinite number of stable This journal is

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secondary spatially periodic states [Fig. 5(a)]. Consequently, the wavenumber selection and the propagation (direction) speed are intimately depend on these coexisting solutions, domain size, and the form of the initial perturbation (for details see section VA). All together on large periodic domains, give rise to a large number of distinct asymptotic patterns when different initial conditions are used (Fig. 10). Boundary conditions and stationary patterns: Danckwerts or no-flux BC destroy the translational symmetry of (2.1) and may stabilize stationary nonuniform patterns. However, the mechanism behind SP states is not of Turing type since a single diffusive field cannot generate a spatial symmetry breaking of a uniform state.1,4 Nevertheless, although the onset of SP patterns cannot be deduced from the linear analysis of (2.1), it is revealed as a spatial Hopf bifurcation in a comoving frame (for details see section IVC). The bifurcation is to periodic orbits identical to the onset of TW but readily with speed c = 0. In particular, the solutions along the SP branch allowed us to explain the origin of spatial defects near the outlet (see section VB2). Excitability: Subcritical finite wavenumber Hopf instability in RD systems can yield intriguing dynamics of excitable bounded wave trains.20 Bifurcation of such type occurs in our RDA model at Da = Da W and indeed upstream propagating pulses and groups of two pulses can be generated for Da o Da W (for details see section IVB). Implications to cross-flow reactors: Here we solely emphasize the potential application of stationary pulses that appear at low Da values (such as Da = 0.005), i.e., in systems of very low activity. Since the reaction rate (2.3) is highly nonlinear, the reactor performance is determined by the highest temperature (v) that the feed meets, and the pulse peak (even at low Da values) acquires very high values. Notably, our aim was to construct a framework of the spatiotemporal behavior in RDA systems. Thus, a large Lewis number limit Le c 1 was employed, to arrest the Hopf instability at k = 0 (in the absence of advection). Evidently, Le B 1 and inclusion of an additional diffusing field will contribute to the complexity of the pattern selection,39 but this is beyond the scope of this paper and will be discussed elsewhere. Most of the results obtained here appear as model independent, partially due to the presence of global bifurcations which are organizing centers of spatiotemporal behavior.64 Indeed some of the phenomena have been observed numerically in other RDA models, including FitzHugh–Nagumo,21,44 Gray–Scott,36,43 and Brusseletor31 models. Thus, we hope that the survey provided here will trigger further explorations of autocatalytic systems with diffusing and advective transport, particularly spatiotemporal behavior in apparently distinct but nevertheless related subcellular biological media76,77 and vegetation patterns.30

Appendix A: Spatial bifurcations In this Appendix we review the eigenvalues representations for the three spatial bifurcations that have been discussed in the text, and are shown in Fig. 14. In the context of homoclinic connections to a fixed point, the eigenvalues represent the asymptotic behavior at x - N, i.e. departure (decay) of the Phys. Chem. Chem. Phys., 2009, 11, 9210–9223 | 9221

Fig. 14 Schematic representation of the three eigenvalues, a complex conjugated pair () and a real (*), at distinct bifurcation onsets: (a) Hopf at  Da = Da W, (b) saddle-node/Hopf at Da = DaSN, and (c) Belyakov at Da = DaB. The arrows show the motion of the complex eigenvalue pair as Da is varied.

orbit from (to) a fixed point; for more details we refer the reader (for example) to ref. 64, 68 and 69. 1.

Hopf

 At the Hopf bifurcation onsets, Da = Da W and c = cW, the configuration of the three spatial eigenvalues is described in Fig. 14(a). The crossover (through the real axis) of the complex conjugated pair () is: for TW+, purely imaginary  pair becomes complex as Da o Da+ W, while for TW , purely  imaginary pair becomes complex as Da 4 DaW. In both cases, the real part of the pair is smaller than the third real eigenvalue  (*), so that in regions Da 4 Da+ W and Da o DaW the linearization about the fixed point corresponds to a saddlefocus. However, only in the subcritical Da o Da W case, this leads to homoclinic connections.20

2.

Saddle-node/Hopf

Here at the bifurcation onset, Da = Da SN and c C 0.008785, the real eigenvalue (*) is also zero, as shown in Fig. 14(b). This bifurcation corresponds to a (codimension-two) saddle-node/ Hopf type at which pure imaginary pair eigenvalues (of the Hopf) coexists with a zero eigenvalue [of a fold of the uniform state (u,v,w) = (u0,v0,0)]. 3.

Belyakov

This bifurcation represents the collision of the complex conjugated eigenvalue pair () on the real axis, at Da  DaB C 0.079 and certain c, where for Da 4 DaB there is a complex pair and for Da o DaB the splitting is on the real axis so that all eigenvalues are real, as depicted in Fig. 7(c). This bifurcation is an important property to homoclinic connections to a fixed point, and is identified with the so called Belyakov point: a point at which a saddle focus of the linearized fixed point becomes a saddle.

Acknowledgements This work was supported by the US-Israel Binational Science Foundation (BSF) and by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel. We thank O. Nekhamkina for helpful discussions and one of us (A. Y.) also thanks Department of Chemical Physics, Weizmann Institute of Science, where some of this work was carried out. M. S. is a member of the Minerva Center of Nonlinear Dynamics and Complex Systems. 9222 | Phys. Chem. Chem. Phys., 2009, 11, 9210–9223

References 1 M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys., 1993, 65, 851. 2 L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics, Springer, Berlin, 2006. 3 P. K. Maini, K. J. Painter and H. N. P. Chau, J. Chem. Soc., Faraday Trans., 1997, 93, 3601. 4 J. D. Murray, Mathematical Biology, Springer-Verlag, NY, 2002. 5 A. Turing, Philos. Trans. R. Soc. London, Ser. B, 1952, 237, 37. 6 R. B. Hoyle, Pattern Formation: An Introduction to Methods, Cambridge University Press, Cambridge, 2006. 7 E. Knobloch, in Nonlinear Dynamics and Chaos: Where Do We Go From Here? ed. S. J Hogan, A. R. Champneys, B. Krauskopf, M. di Bernardo, R. E. Wilson, H. M. Osinga and M. E. Homer, IOP, 2002. 8 A. R. Champneys, Phys. D, 1998, 112, 158. 9 P. D. Woods and A. R. Champneys, Phys. D, 1999, 129, 147. 10 J. Burke and E. Knobloch, Chaos, 2007, 17, 037102. 11 D. Gomila, A. J. Scroggie and W. J. Firth, Phys. D, 2007, 227, 70. 12 J. Burke, A. Yochelis and E. Knobloch, SIAM J. Appl. Dyn. Syst., 2008, 7, 651. 13 A. Yochelis and A. Garfunkel, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 77, 035204. 14 D. J. B. Lloyd, B. Sandstede, D. Avitabile and A. R. Champneys, SIAM J. Appl. Dyn. Syst., 2008, 7, 1049. 15 E. Knobloch, Nonlinearity, 2008, 21, T45, and the references therein. 16 J. Sneyd, A. LeBeaub and D. Yule, Phys. D, 2000, 145, 158. 17 M. Or-Guil, I. G. Kevrekidis and M. Ba¨r, Phys. D, 2000, 135, 154. 18 G. Bordiougov and H. Engel, Phys. Rev. Lett., 2003, 90, 148302. 19 A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd, SIAM J. Appl. Dyn. Syst., 2007, 6, 663. 20 A. Yochelis, E. Knobloch, Y. Xie, Z. Qu and A. Garfinkel, Europhys. Lett., 2008, 83, 64005. 21 M. Kærn and M. Menzinger, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2002, 65, 46202. 22 H. Sevcˇı´ kova´, M. Marek and S. C. Mu¨ller, Science, 1992, 257, 951. 23 A. B. Rovinsky and M. Menzinger, Phys. Rev. Lett., 1993, 70, 778. 24 M. Kærn and M. Menzinger, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 1999, 60, R3471. 25 M. Sheintuch and O. Nekhamkina, AIChE J., 1999, 45, 398. 26 F. Zhang, M. Mangold and A. Kienle, Chem. Eng. Sci., 2006, 61, 7161. 27 M. Kærn, M. Menzinger, R. A. Satnoianu and A. Hunding, Faraday Discuss., 2002, 120, 295. 28 H. Yamada, T. Nakagaki, R. E. Baker and P. K. Maini, J. Math. Biol., 2007, 54, 745. 29 Y. Khazan and L. Pismen, Phys. Rev. Lett., 1995, 75, 4318. 30 F. Borgogno, P. D’Odorico, F. Laio and L. Ridolfi, Rev. Geophys., 2009, 47, RG1005. 31 S. P. Kuznetsov, E. Mosekilde, G. Dewel and P. Borckmans, J. Chem. Phys., 1997, 106, 7609. 32 R. A. Satnoianu, J. H. Merkin and S. K. Scott, Phys. D, 1998, 124, 345. 33 C. A. Klausmeier, Science, 1999, 284, 1826. 34 P. Andrese´n, M. Bache, E. Mosekilde, G. Dewel and P. Borckmanns, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 1999, 60, 297.

This journal is

 c

the Owner Societies 2009

35 O. Nekhamkina, B. Y. Rubinstein and M. Sheintuch, AIChE J., 2000, 46, 1632. 36 R. A. Satnoianu and M. Menzinger, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2000, 62, 113. 37 J. R. Bamforth, S. Kalliadasis, J. H. Merkin and S. K. Scott, Phys. Chem. Chem. Phys., 2000, 2, 4013. 38 R. A. Satnoianu, M. Menzinger and P. K. Maini, J. Math. Biol., 2000, 41, 493. 39 R. A. Satnoianu, P. K. Maini and M. Menzinger, Phys. D, 2001, 160, 79. 40 J. R. Bamforth, J. H. Merkin, S. K. Scott, R. To´th and V. Ga´spa´r, Phys. Chem. Chem. Phys., 2001, 3, 1435. 41 O. Nekhamkina and M. Sheintuch, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2002, 66, 016204. 42 M. Sheintuch and O. Nekhamkina, AIChE J., 2003, 49, 1241. 43 R. A. Satnoianu, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2003, 68, 032101. 44 O. Nekhamkina and M. Sheintuch, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2003, 68, 036207. 45 P. N. McGraw and M. Menzinger, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2005, 72, 026210. 46 D. G. Mı´ guez, R. A. Satnoianu and A. P. Munuzuri, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2006, 73, 025201. 47 E. H. Flach, S. Schnell and J. Norbury, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2007, 76, 036216. 48 D. A. Vasquez, J. Meyer and H. Suedhoff, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 78, 036109. 49 S. M. Houghton, S. M. Tobias, E. Knobloch and M. R. E. Proctor, Phys. D, 2009, 238, 184. 50 M. C. Cross, Phys. Rev. Lett., 1986, 57, 2935. 51 M. C. Cross and E. Y. Kuo, Phys. D, 1992, 59, 90. 52 J. M. Chomaz, Phys. Rev. Lett., 1992, 69, 1931. 53 H. W. Mu¨ller and M. Tveitereid, Phys. Rev. Lett., 1995, 74, 1582. 54 A. Couairon and J. M. Chomaz, Phys. Rev. Lett., 1997, 79, 2666. 55 S. M. Tobias, M. R. E. Proctor and E. Knobloch, Phys. D, 1998, 113, 43. 56 O. A. Nekhamkina, A. A. Nepomnyashchy, B. Y. Rubinstein and M. Sheintuch, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2000, 61, 2436.

This journal is

 c

the Owner Societies 2009

57 58 59 60 61 62 63

64 65 66 67 68 69 70 71 72 73 74 75 76 77

A. Yochelis and M. Sheintuch, e-print arXiv: 0810.4690, 2009. M. Sheintuch and O. Nekhamkina, AIChE J., 2005, 51, 224. A. Uppal, W. H. Ray and A. Poore, Chem. Eng. Sci., 1974, 29, 967. E. Knobloch, Phys. Rev. A: At., Mol., Opt. Phys., 1986, 34, 1538. J. D. Crawford and E. Knobloch, Annu. Rev. Fluid Mech., 1991, 23, 341. P. Huerre and P. A. Monkewitz, Annu. Rev. Fluid Mech., 1990, 22, 473. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang, AUTO2000: Continuation and bifurcation software for ordinary differential equations (with HOMCONT), http://indy.cs.concordia.ca/auto/. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, NY, 1995. I. Mercader, A. Alonso and O. Batiste, Eur. Phys. J. E, 2004, 15, 311. A. Bergeon, J. Burke, E. Knobloch and I. Mercader, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 78, 046201. A. Yochelis, Y. Tintut, L. L. Demer and A. Garfinkel, New J. Phys., 2008, 10, 55002. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, NY, 1983. L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part II, World Scientific, Singapore, 2001. J. H. P. Dawes, SIAM J. Appl. Dyn. Syst., 2008, 7, 186. Y. Pomeau and P. Manneville, J. Phys., Lett., 1979, 40, 609. H. G. Paap and H. Riecke, Phys. Fluids A, 1991, 3, 1519. B. Janiaud, A. Pumir, D. Bensimon, V. Croquette, H. Richter and L. Kramer, Phys. D, 1992, 55, 269. Y. Liu and R. E. Ecke, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 1999, 59, 4091. C. Elphick, E. Meron, J. Rinzel and E. A. Spiegel, J. Theor. Biol., 1990, 146, 249. D. A. Smith and R. M. Simmons, Biophys. J., 2001, 80, 45. F. Y. G. A. Truskey and D. F. Katz, Transport Phenomena in Biological Systems, Pearson Prentice Hall, NJ, 2004.

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