Multiplicity and oscillations in a model for catalyzed oxidation of carbon monoxide

PHYSICAL REVIEW BVOLUME 49, NUMBER 20 1& MAY 1994-II Multiplicity and oscillations in amodel for catalyzed oxidation of carbon monoxide M. C. Lemos and A. Cordoba Departamento de Falsi ea de la Materia Condensada, Universidad de Sevilla, Apartado 1065, 41080Sevi lla, Spain (Received 16 November 1993j We extend amodel proposed for explaining multiplicity and oscillations of concentrations and temperature in catalyzed oxidation of carbon monoxide; the importance of the dimension of the system and the closure approximation applied to the results, and, especially to the oscillatory behavior, is analyzed. Kinetic phase transitions, namely, single state multiplicity, single state~ oscillations, and multiplicity~ oscillations are found, depending on the reaction heat and the temperature relaxation parameter. Also, the role played by desorption of reactants is considered. When there is no desorption, temperature oscillations take place around room temperature, but if desorption is operative, oscillations occur about ahigher temperature. For the one-dimensional case aspurious kinetic phase transition is obtained when the singlet closure approximation is applied. I. INTRODUCTION Multiplicity and oscillations are phenomena often observed in reactions catalyzed on asurface. Oxidation of carbon monoxide on platinum or other metals is one of the most representative of these reactions and has been the object of agreat number of experimental and theoretical studies. Anumber of models, based on different mechanisms, have been proposed for explaining multiplicity and oscillations in this reaction, easy models as well as models with increasing difhculty. ' Recently some papers have analyzed kinetic phase transitions, bistability, hysteresis, "poisoning, "and so on, for this or similar reactions, on the basis of lattice mode1s, using the Monte Carlo method or applying different closure approximations to obtain rate equations. On the other hand, the importance of the dimension of the system and the approximation applied for obtaining the kinetic equations from amaster equation has been revealed in the study of kinetic phase transitions. Specifically, closure approximations that are too simple can give rise to spurious kinetic phase transitions in onedimensional systems where competitive adsorption of two species takes place, with or without further chemical reaction, when there is interaction between nearest neighbors'6' (something similar to that happens in the equilibrium Ising model when the Bragg-Williams or the Bethe-Peierls approximations are applied). Significant differences among results obtained using different approximations have been detected in oscillatory situations. ' Moreover, models where, because of reaction heats, a catalytic surface can keep adifferent temperature from that of its surroundings and temperature oscillations can originate jointly with concentration oscillations have been considered. ''Temperature fluctuations give rise to astrong nonlinearity of kinetic equations owing to the Arrhenius law. In this paper, the model initially proposed by Lagos, Sales, and Suhl' is extended to consider other features of the problem and to analyze the influence of the system dimension and the kind of closure approximation on the reII. SQUARE LATTICE. SINGLET CLOSURE APPROXIMATION Temperature and concentration oscillations in chemical reactions catalyzed on asurface constitute arepresentative case of oscillatory states in far from thermodynamic equilibrium systems. Catalyzed oxidation of carbon monoxide is agood example ofthis kind ofbehavior. Aknown model, by Lagos, Sales, and Suhl, 'assumes that the reaction takes place according to three elemental mechanisms: Oz(g)+2 V(s)~20(s), CO(g)+ V(s)~CO(s), CO(s)+0(s) ~COz(g)+2V(s), (2.l) (2.2) (2.3) where Oz(g), CO(g), and COz(g) indicate, respectively, oxygen, carbon monoxide, and carbon dioxide molecules in gaseous phase; 0(s) and CO(s) denote an oxygen atom and acarbon monoxide molecule adsorbed on the surface and V(s) is avacant site on the surface. This simple scheme constitutes the Langmuir-Hinshelwood (LH) mechanism. The surface where adsorption and reaction processes take place is assumed to be like asquare lattice and each oxygen atom (0) and each carbon monoxide molecule (CO) can occupy one lattice site at most. Adsorption of an oxygen molecule (Oz) needs two adjacent vacant sites, dissociation in two atoms being allowed. On the other hand, aCO molecule and an 0atom, both adsorbed next suits and especially on oscillatory behavior and its characteristics. In Sec. II the two-dimensional model is analyzed by applying asinglet closure approximation (SCA), and in Sec. III desorption of reactives is also considered; in Sec. IV a doublet closure approximation (DCA) is applied; finally, in Sec. Vthe case of alinear chain is studied for comparison with the two-dimensional system, the inAuence of the system dimension in this kind of kinetic phase transition being made clear. 0163-1829/94/49{20)/14648(9}/$06.00 49 1994 The American Physical Society 49 MULTIPLICITY AND OSCILLATIONS IN AMODEL FOR. ..14 649 — k3(T)nonco (2.4) deco =Wz — W3 kp(T)(1 no "— co) k3(T)nonco . (2.5) The rate constant of the process (2.1}is assumed in the Arrhenius form E1 1 k(T)=k (T )exp TT B where TB is room temperature, Eis the activation energy for the dissociative adsorption of the Oz molecule (assumed independent of the coverage), and Ris the gas constant. Also, processes (2.2) and (2.3) are not considered activates and, therefore, kz( T)=kz( T~ )and k3(T)=k3(Ta ). If thermal diffusivity of the surface is much higher than that of the surroundings, the surface temperature, due to adsorption and reaction heats, can be different from the room temperature, thus affecting the process rates. Equations (1) and (2) must be completed with an equation for the surface temperature, dT 3 C=L(T— T~)+ — ghH; W;, i=1 (2.6) where Cis the heat capacity of the system, L/C (L is approximately equal to thermal conductivity times a geometric factor) is the relaxation rate of Tto Ttt, and hH; are the reaction heats of the processes (2.1)— (2.3), including asuitable geometric factor. Kinetic equations (2.4)— (2.6) can be written in terms of the dimensionless constants and variables EI. v.=k3t, c=,y= RTB 'k3C ' k, (T~) k~(T~) a=2, b= k3(T~) k3(T~) hH, 6Hz gg 2CTB 'CTB 'CTB where bQ=AH, /2+ 6Hz+ EH3 )0is the net reaction neighbors, can react, constituting aCO2 molecule, which leaves the surface immediately. The transition probabilities for the processes (2.1)— (2.3) are chosen as W, =k,(T)n„„, Wz=kz(T)n„, W3 k3(T) noco where k,.(T) (i =1,2, 3) denote the rate constants of the processes (2.1)— (2.3); ns is the vacant fraction, nzv the vacant pair fraction, and nQcQ the density of oxygen-carbon monoxide pairs. Kinetic equations for the singlet densities np and nco are formulated from amean 6eld approximation, where all the cluster densities are written in terms of singlet densities (SCA) de p=2W) — W3 =2k )(T)(1 no — nc— o) heat. Moreover, for simplicity, h, =h~ =h is assumed. Kinetic equations are dnp =ag (z)(1 no neo )n— o"co d'T d"cp d7. =b(1 — no — nco) n— o"co ~ (2.7} (2.8) dz =— yz+h [ag(z)(1 no—n— co) +b(1 n— o— nco) 2 2nonco]+qnonco (2.9) where g(z)=exp[ez/(z +1)]. Equations (2.7}— (2.9) can give rise to oscillatory states for certain values ofthe parameters I"=[a,b, e, y, h, q]. Four steady solutions, x;=[no,nco, z"],are obtained from dnoldr=dncoldr=dz/d~=0, which are denoted x+,=(1,0,0), xi=(0,1,0), (n st+ nst+ zst+ } 0+ 0~CO & st —st —stxo—=(no,nco,z), no =— '(1 v0+tt0), nco =T~(1 v0+ =(q/y )n o*nco where v0 =[b /ag (z")]and u0 =[(1— v0)z 4bv0]'~— z. Solutions x0+ and x0 are for the meaningful range (0&no &1,0&neo &1,0&no+neo &1} for values of vc &1 and 4bv0&(1 — v0} .Solutions x0+ and x0 are symmetrical with each other. The stability analysis is performed in the usual way, from the eigenvalues of the stability matrix. x+1 is a saddle point and x, is astable node (corresponding to the poisoning of the surface by CO at room temperature}; x0+ and x0can be unstable, marginally stable, or stable, depending on the values ofthe parameter I. With the aim of studying the oscillatory behavior of the system and the existence of limit cycles three representative cases have been analyzed. (i) q=0 and y~ ~. For this case only Eqs. (2.7) and (2.8}must be considered because as y~ sc the system relaxes instantaneously to room temperature (z =0). Now x0 is asaddle point and x0+ is astable node or aspiral point (state marginally stable), depending on the values of I=[a,b].As can be seen, there are no selfsustained oscillations. The Runge-Kutta method has been applied for different initial conditions. In Fig. 1, for a=1 and b=0.2, some trajectories in the phase space np-nco are shown. Steady solutions are x0+ =(0.7464,0.0536) (stable node), x0 =(0.0536,0.7464) (saddle point}, x+,=(1,0) (saddle point), and x,=(0,1) (stable node). The phase space is clearly split into two basins, corresponding to x0+ and x„respectively. The separatrix crosses the saddle point x0 (ii) q=0 and yPnite. In this case, the steady solutions are only functions of I=[a,b], since z"=0, and can be obtained as in case (i}.However, for analyzing stability of solutions the set of parameters I=Ia,b, e,y, h]must be 14 650 M. C. LEMOS AND A. CORDOBA 49 CO 00 0.3- &=07. FIG. 1. Trajectories on the Phase Plane no-neo for the case q=0and y~~,with a=1,b=0.2. odenotes stable node and 6saddle point. considered. Steady solution xo is also asaddle point, and x, a stable node, and the trajectories starting from points in the basin of x&tend to x,', the remaining trajectories tend to the stable node xo+ or, if xo+ is unstable, come near to alimit cycle. For 0&y&y, (with the other parameter fixed) limit cycles increase in size as ydecreases in relation to y, .Thus, for a=1, b=0.2a=10 and h=5, y, is 1.11 and limit cycles can be found for the range 0&y&1.11. Steady solutions for a=1and b=0.2 (the ey do not depend on e, h, and y) are x+, =(1,0,0), x&=(0, 10) xpi =(0.7464 0.0536 0) and x ,0.7464,0). In general, xp+ is an unstable focus, marginally stable state, stable focus, and stable node as yincreases, for the above values of the other parameters. In Fig. 2(a) two limit cycles are shown for y=0.7and 1(both for a=1, b=0.2, e=10 h=5 =0) IF' . ~j, oscillations in concentrations and temperature are shown for y=1. (iii) q%0 and yPnite No.w, although an analytical solution is not possible and numerical methods must be applied, the distribution of singular points, stability, and separatrix are similar to those of case (ii). An analysis has been performed considering different values of hand y, choosing qlb =2 and fixing the other parameters. Now, xo+ is astable node, stable focus, marginally stable state, and an unstable focus as yincreases. For certain values of yxand d' o+ an xo disappear, x+, a,nd x,being the only steady states. In Fig. 3alimit cycle is shown for a=0.0027, b=0.8, 0.9- () 0.1— co 0 0.3n n 0.7" 0.2" 0 0" 0.9' no 0.7" 0.1" neo 0" (b) 1.5" Z~ 0.50 0.2VVVV 3x1P 0.2-- nCO (b) -0.1090 FIG. 2. (a) Pro'ection on jnon the nQ-ncQ plane of several limit cycles: y=0.7and 1both with a=1, b=0.2, v=10, h=5 q=0. The unstable f aeocus is located in the state nQ=0. 7464 n«=0.0536, z=0 for Q eratur or both cases. (b) Time evolution ft pure and concentrations for a=1, b=0.2c= 'no emq=0, y=1. 8=10, h=5, 0i 0.7no 0.9 FIG 3Oscillatory state nQ=0 8003 n« =0 0998 z=0.9988 obtained for a=0.0027, b=0.8, c, =16, h=12.5, q=,y= .Case of large h(large surface to volume ratio).=25 =2. Temporary oscillations are shown in (a) and the projection on the nQ-ncQ plane ofthe limit cycle obtained is shown in (b). SIN AMODEL FOR ~ND OSCILLATIONS IN MULTIPLICITY 49 — 2ere and con- =2. The temperature 3(b) Here Ii i1(1 drawn in 'g. in Fig. ues of the other parameters, i;. b...... y„„'0+,.„„hure decreases as yincr 005 c20 i'apear or L' '1 in between 377 an am litudeofosci ainfrequency and amp ca be see, de endon g™p litudes an tan erature (T&3 than for lower temper CO(g )+V(s)— +CO(s), +2V(s), CO s}+0(s)~C02(g 20(s)~02(g)+ +2V(s), CO(s)~CO(g) +V(s), (3.2) (3.3) (3.4) (3.5) ow kineti q nQnCQ — a'nQ, =ag (z)[1 no — neo] dv b'g— '(z)neo )"o"co =b(1— no nco d7 2 z+h [ag(z)(1 no — n— co (3.6) odel analyzed in Sec. II, by (3.4) dbo thus extending tep in desorption proce id .f ide (3.5rom 'c euations are N III. REEACTIVE DESORPTION. SCA e'kes lace accord' g in to five ereaction ta es p Now we assume elemen a t1mechanisms: Oz(g )+2V(s)~20(s), (3.1) 0.8" "o ", Qgi. ~UIUIUUUUUUU 0.2i Qi. Q.5' 'JJJ JJ 0.)JJJJJJJ03 (b) Q 0.20.8 =o.4568, neo=- =0.0536, ~4. Ost:iHatory state no =. FIG. 4. s «the oscillations is so lane of the limit cyle o tion on the no-n«plane ote (b). )2nonco] +b (1 no—"co — h'a no +bg(z)neo ], +qnonco fthe preexponent ntial fac- 'and b' are teratios fhrocesses 1''is aquan 'y Arrhenius ntity propo es ective y; c' ' 3.5); h' is aq e''ener yo 'nheats inpr al to the ratio ofhit of the system;'and hheat capacity o nd (3.5}to terarameters +)]. eother p the set of parameters co del analyzed in. Sec I""0 ta eunt desorption, go to desorption proc 0. This case assum "p tion does puelower ta0coverage. that for a kinetic go ythe paramete rate oox ters propo ainly by eand, re dd dso pt.nan +r .Additionally, up to soluti ez"=0 and the o dthree steady sol tions =0.2b' gfi '=0 (thmb'= these so u' 1tions disapb'=0.011. For b'= are x solutions a =0.01 steady 0=0236 0 stable focus or no e le focus g addle point, and o+ stable 1) gto alimit cyc e,(giving rise o 14 652 M. C. LEMOS AND A. CORDOBA focus, and stable node. For b'=0.012, x, and xp disappear and xo+ =(0.7516,0.0517,0). Now, the formation of aCO packed structure, x limits the reaction steady rate by an end, and the formation of astate where oxygen atoms are closely bound (oxygen adsorption predominates), xo+, limits it at the other end. Thus the trajectories in the phase space tend to x,or xp+ and if xp+ is not stable they come near alimit cycle, for 0&y&y,.The size of the limit cycle decreases as y increases. For c.'=0.10 there are limit cycles for the range 0&y&1.08; for c'=20 in 0&@&1. 09, and for c.'=50, 0&y&1.10, i.e.,the value of c' scarcely influences the value of y, .In Fig. 5alimit cycle is shown, for b'=0.01, c'=5, and y=0.7. (b) h'&h. For values h'~4. 5, with b'=0. 1, s'=5, and y=0.7, four steady solutions can be found, x+„x xp, xp+, and their characteristics are similar to those of case (a). However, unlike case h=h' (where the system relaxed to room temperature, z"=0), now the desorption mechanism makes z"&0. Thus, for h'=4. 5and the values of the other parameters fixed, x+& =(1,0,0) (saddle point), x,=(0.0092,0.9025,0.0067) (stable focus), xo =(0.0155,0.8718,0.0064) (saddle point), and x0+ =(0.7517,0.0516,0.0004) (unstable focus). For h' &4.5, x&and xp disappear. In Fig. 6alimit cycle corresponding to this case is shown, with h'=3, b' =0.1, c,'=5, and y=0.7, resulting in xp+ =(0.8057,0.0350,0.0105). If yvaries with the other parameters fixed, limit cycles can be obtained for the range 0&7&0.90. (ii) a'%0,b'=0. For this case the steady state x,=(0,1,0) is always obtained; it is astable node. Depending on the values of h', two additional steady solutions can be obtained, but the state x+, =(1,0,0) never appears. %'hen h'=h or A. '&h there are limit cycles for the range 0&y&y, (with asuitable value for y, )but z"=0for the former case and z"&0for the latter. (iii) a'AO, O'%0. Now, some of the states x+, =(1,0,0) or x,=(0,1,0) are not obtained, unlike the previous cases. %ith regard to existence and characteristics of limit cycles, results are similar to those previously obtained. Summarizing, with regard to oscillations, when desorption exists, the behavior of the system is essentially similar to the case when desorption is not operative. The most remarkable feature is z"&0 (for h'&h), i.e.,temperature fluctuates around atemperature greater than room temperature. IV. DOUBLET CLOSURE APPROXIMATION In order to analyze the influence of the closure approximation applied on the existence ofmultiplicity and, espeno 1~ 03l. 0.3" n 0.6' 0' 2.1 Opl02x)0 -0.2' 02x10 0.3.- 00.3no 0.4 CO 00.6 (b) no FIG. 5. Oscillatory state no =0.7507, neo =0.0519,z=0obtained for a=1, b=0.2, c.=10, h=5, q=0, y=0.7, a'=0, b'=0.01, c'=5, h'=5. Case h=h'. Temporary oscillations of coverage fractions and temperature are shown in (a). The projection on the no-neo plane ofthe limit cycle is in {b). FIG. 6. Oscillatory state no =0.8057, nco =0.0350, z=0.0105 obtained for a=1, b=0.2, @=10, h=5, q=0, @=0.7, a'=0, b'=0. 1, c'=5, h'=3. Case h&h'. Temporary oscillations are shown in (a} and the projection on the nQ-n+Q plane ofthe limit cycle in (b). 49 MULTIPLICITY AND OSCILLATIONS IN AMODEL FOR. ..14 653 cially, on oscillations, in this section kinetic equations have been obtained for the kinetic model of Sec. II by applying aclosure approximation consisting of closing the hierarchy of kinetic equations at the level of doublets (DCA). In general, the densities of clusters including more than two sites are written in terms of densities of singlets, n, ,and doublets, n,.: n)J nJ'k n1J J where i, j,and kcan be V(vacancy), CO (site occupied by acarbon monoxide molecule), and 0(site occupied by an oxygen atom}. Now we assume that the surface is asquare twodimensional lattice. Following the guideline of Ref. 27, in adoublet closure approximation the state of the surface is determined giving the density of single particles n, and the density of pairs, n" and n~, where i,j=V, CO, 0, and x(y) indicates that the pair is situated on the x axis (y axis). Thus, for acarbon monoxide molecule, CO, adsorbed on the surface, the four nearest neighbors are considered, leaving the surface when aneighbor oxygen atom is found to react, forming acarbon dioxide molecule. For an oxygen molecule, 02, adsorbed on the surface, its six nearest neighbors are considered; one oxygen atom leaves the surface when meeting aneighbor carbon monoxide molecule to form acarbon dioxide molecule. By dependence relationship among the variables n;, n,"", and n,J, the number of independent variables can be reduced. Choosing as independent variables no, nco, no o, nQco ncQ cQ and the scaled temperature z, kinetic equations are no =ag(z)[1— 2no 2"co+no-o+2"o-co+ "co-coj"o-co 7 dnco d~ =b(1 no — nco) no-c— o no-0 3("o no-o no-co) =— ag(z)[1 — 2no 2nco+"o-o+2"o-co+neo-co] + d7 221no — nco 3no Qno co 2no (4.1) no-co 3ag(z) [1 2no 2"co+no-o+2"o-co+"co-co)["co "o-co nco-coj 41 0co 0-co 1+ 0-co +o-co n3n 3n no neo neo-co 3n Q-conco-co =2b["co "o-co "co-coj dr 2nco dz =yz+— h[ag(z}[1 2"o — 2"co— +~oo+2"oco+&cocol+b(1 no +co} 2no-COI+qnoco, I where nQQ, nQcQ and ncQ cQ are the densities of pairs 0-0, O-CO, and CO-CO, respectively; I=Ia,b, s,y, h, qI is the set of relevant parameters ofthe model. In general, four steady solutions, x=(no, nco, noo, noco, ncoco, z}, are obtained, at most; they are labeled, for analogy with Secs. II and III, as x+, =(1,0, 1,0,0,0), xi=(0,1,0,0, 1,0), xo+ =(no+, st+ st+ st+ st+ st+ ~st+ st+ nco "o-o "o-co nco-co, z), with no &"co and st —st —st —st —stst — itstxo—("0 ~co no-o no-co +co-co z)with no (nco.,now xo is not symmetrical with xo+. Thus the number of steady solutions in the DCA is the same as in the SCA. The previous cases analyzed in Sec. II are now considered: q=0, yahoo; q=0,yfinite;q%0, yfinite. The results obtained are qualitatively similar to those of Sec. II, but there are some quantitative differences. Thus the range 0(y &y, allowing oscillations is shortened; for example, in the case q=0, yfinite, for a=1, b=0.2, a=10, and h=5, y, =0.34 for the DCA, but y, =1.11 for the SCA. Some quantitative differences for the values of no, neo, and so on, and reaction rate can be appreciated. Figures 7and 8, both with a=0.002, b=0.05, c.=20, Q.2r CO 0~ 0.2n FIG. 7. Projection on the phase plane no-nco of the limit cycle obtained for a=0.002, b=0.05, a=20, h=0.5, q=1, y=0.08. The unstable focus is no =0.5381* ~co =0-0617 +o-o =0 3354 &o-co =0-0» ~co-co =0.0071 and ~=0 2501. h=0.5, q=1, and y=0.08, show case qAO and yfinite for DCA. Figure 7shows the projection of limit cycle on the phase plane no-neo,'and temperature and concentration oscillations for the same limit cycle are shown in Fig. 8. Both figures can be compared with Fig. 4, which represents an analogous case in SCA. Summarizing, in this two-dimensional model, the kind of approximation applied does not essentially affect the results, although some quantitative difFerences appear. Considering the kinetic phase transitions taking place in 14 654 M. C. LEMOS AND A. CORDOBA 49 TABLE I. Kinetic phase transitions in this model. Transitions %0 Finite Finite Multiplicity YES Oscillations NO Multiplicity YES Oscillations YES (Temperature fluctuates around the room temperature) Multiplicity Oscillations (Temperature fluctuates around atemperature greater than the room temperature) 1stable node + 1stable node 2stable nodes 2stable nodes 1stable node and 1limit cycle this system, depending on the "thermal" parameters q and y, Table Ican be written. d"o1 d7. 2 =— ag (z)(1 no —— nco) —none— o, 2 V. LINEAR CHAIN. SCA AND DCA Catalyzed oxidation of carbon monoxide takes place on asurface and, therefore, atwo-dimensional model must be applied. However, as has been demonstrated for other cases, '''some closure approximations can introduce spurious kinetic transitions when they have been applied to one-dimensional systems. For checking this possibility in this model, the adsorption and reaction processes described in Sec. II are assumed to take place on alinear chain. Kinetic equations applying the SCA are dnco 1 =b (1 np — ncp— )—np— ncp, d~ 2(5.1) dz =— yz +h — ag (z)(1 no — neo— ) 12 +b(1 no —— nco) no"co +"onco and the ones resulting from the DCA are dno 1 =— ag(z)[1 — 2no 2nco+no-o+2no-co+neo-co] — — no co, d7 2-co 2o-co ~ neo =b(1 no nco) — — no-co d7. dn o-o 1("o "o-o "o-co) ag (z)[1 2"o2"co+ "oo+2"o-co-+ "co-co]+ d1. 221no nc— o "o-o"o-co 2np (5.2) «o-co d7 ag (z) "o 2"co+"oo+2"oco+ "co-co][" co no cp — ncp cp] "o "co 1"o-co "o-co +b ("o "o-o "o-co)no-co 1— ++ 4no nco «co-co 1no-conco-co d=2b(nco — no-co — nco-co) d72neo pz+hag(z)[1 2n2pn(p+noo+2np co+neo co]+b(1 no neo)no+(lno 1 49 MULTIPLICITY ANDAND OSCILLATIONS I SIN AMODEL FOR .. 14 655 Q9" np 0.2" 0.2" 'to 0" 07" "pp ' 0.1" n0"' p-cp 0 0$ ~~ CO Q~ 0.5" 0.1T FIG. 8. Tem ora po ry oscillatco fcoverage ,y=0.08. co 0.0617, o- ~co-co =0071,z=0.2501. unstable, dep d u, epending on the ep dhe values of the he teparameters n1i ar results to bt}1 od eanalysis of th k' rom the DCA einetic eq fro h and in this se 1s ct ofor th SCA ectively, for q=0and, yteste nesaddle 'anpoints or stabl —a, J. There aenode, de e state s. pg ip cecity of s stable steady en the possible exist 1o .Awide sweep in th f on yone stable ste of ''g, enonlineari 1td bg asystems osci ations therefore tut not in one-dim d' ional system is s one-dimensi ent eSCA' nce othe approxim t' ason applied. a First, kinetic e ic equations from hd'dimensional cas f e, our stead ysolutions bl' n'd no e; xo+ and xepoint xo can be stable or ACKNOWLEDGMENT This work was ppp ed by G Tofthe Spanish G is overnment. ~L. F. Razon and R. A. Sc, at R .Schmitz, Cat l. R.ci at .R.ci. Eng. 28, 89 i, .Gulari, and n .G', Y. Barshad Ph .e R. Dickman Ph ..3 hys. Re h.ev. Lett. 56 n, .iys. Rev. 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