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surface chemical reactions play an important role in the processes of heterogeneous catalysis , which are widely used in the chemical industry  @xcite .
monte carlo simulation is a powerful tool for a microscopic description of these reactions  @xcite .
kinetic phase transitions , oscillations and chaos , pattern formation , and coupling between catalytic oscillators have been studied by the monte carlo method  @xcite .
the oxidation of carbon monoxide @xmath0 is one of the most extensively studied surface catalytic reactions  @xcite . in a recent monte carlo study ,
an attenpt of a mean field modeling was made for site blocking induced surface coverage oscillations  @xcite .
the authors of ref .
@xcite studied a mathematical model based on macroscopic rate equations for the characterization of oscillations seen in the monte carlo simulations .
however , it was found that the mean field model does not lead to oscillatory dynamics in a parameter region where their monte carlo simulations show oscillations . in order to obtain oscillations ,
they numerically solved a part of their rate equations with data from the monte carlo simulation for dynamics of the free site and the inert molecule coverage .
such a combination of deterministic and stochastic time evolutions seems not to be satisfactory ; it is more desirable to find a supercritical hopf bifurcation in the model to ensure the occurrence and stability of oscillations .
the aim of this work is to show that inclusion of high order nonlinear terms , which represent interactions inherent in this system , allows us to find oscillations within the rate equations .
we find a hopf bifurcation in the extended system , and we can determine the parameter domain for oscillatory dynamics .
the model system under study is the zgb model  @xcite , which is extended by the absorption and desorption of an inert molecule :    @xmath1    where @xmath2 indicates that the molecule is absorbed on the surface and @xmath3 indicates that the molecule is in the gas phase .
@xmath4 indicates a vacant site and @xmath5 represents the site blocking inert molecule .    as was reported by jansen and nieminen  @xcite , one can observe sustained coverage oscillations in the monte carlo simulations of eqn .
( 1 ) . in fig .
1 we show results of our monte carlo simulation for these coverage oscillations .
all variables and time are dimensionless in this paper . as compared with ref .
@xcite , we have used a different set of parameters to show that these oscillations can be observed in wide range of parameters . to improve the numerical performance we have implemented _ lists _ in our monte carlo simulations .
separate lists for the locations of free sites , @xmath0 , and inert molecules were updated during the simulations .
for example , for an adsorption of a chosen adsorbent , a site was randomly selected from the list of free sites ; for a desorption , a site was randomly selected from the corresponding list of @xmath0 or @xmath5 .
the authors of ref .
@xcite studied rate equations for a macroscopic characterization of the oscillations seen in the monte carlo simulation .
the rate equations for the coverage of @xmath6 - @xmath7 , @xmath8 - @xmath9 , and @xmath10 - @xmath11 are  @xcite : @xmath12 where @xmath13 is the free site coverage .
it is known that eqn .
( 2 ) shows oscillations in a narrow parameter domain  @xcite , however , temporal pattern of these oscillations are very different from the oscillations seen in monte carlo simulations as shown in fig .
it was found that eqn .
( 2 ) does not show oscillations in the parameter domain where monte carlo simulations lead to oscillations . to obtain oscillations , the authors of ref .
@xcite used data for @xmath13 and @xmath11 from the monte carlo simulation .
we suppose that the reason why eqn .
( 2 ) does not show oscillations in the parameter domains of intereset is that it does not include important higher order nonlinear effects inherent in eqn .
( 1 ) , which are essential for oscillations shown in fig . 1 .
monte carlo simulations of eqn . ( 1 ) suggest that two nonlinear effects are crucial for oscillations in this system .
they destabilize @xmath14 and @xmath0 rich surfaces , correspondingly .    in fig .
2 we show snapshots of the surface at two different time moments . fig .
2a corresponds to the time moment when there are no @xmath6 molecules left on the surface . note that in the original zgb model , for the parameters we have chosen , the system goes to the state fully covered by @xmath8  @xcite
however , the presence of the inert molecules shown in fig . 2a as the brightest spots , prevent such a fully @xmath8 covered state .
moreover , these inert molecules allow for the presence of enough vacant sites ; these are shown as black areas in fig .
2 . in these vacant areas ,
@xmath0 molecules can be absorbed ; their coverage can grow if they are isolated by the inert molecules from @xmath8 covered areas . by taking into account this nonlinear effect , which includes interaction among @xmath6 , @xmath8 , and @xmath10 , the reaction term in eqn .
( 2 ) can be modified as @xmath15 , where @xmath16 is the blocking coefficient .
2b shows nucleation of the @xmath8 covered state on the @xmath6 rich surface .
as an adsorption of @xmath17 requires two free sites , nucleation of the @xmath8 island may seem an unlikely process , if one starts from the fully @xmath6 covered state as an initial condition .
however , there is an another nonlinear effect which may lead to explosive growth of an @xmath8 island .
suppose that the desorptions of two @xmath6 molecules have vacated a pair of adjacent sites . on these sites
, @xmath17 can be absorbed , and if this happens , @xmath6 and @xmath8 will immediately react .
as the surface is rich with @xmath6 , the reaction will free four sites . on these four sites
, two @xmath17 molecules can be absorbed .
this cascade may lead to explosive growth of the @xmath8 island .
therefore , there is an another nonlinear effect which is particularly noticeable during a nucleation of an @xmath8 island on a @xmath6 rich surface . in the lowest order of nonlinearity , due to the nonlinear increase of available adjacent free sites through the reaction
, the growth of @xmath8 can be termed as @xmath18 , where @xmath19 is the coefficient for the nonlinear growth of adjacent free sites .
we verified that @xmath20 can also serve well for this term .
the two nonlinear effects manifest themselves randomly in monte carlo simulations . for a macrospcopic description ,
we consider them deterministic processes .
thus , eqn .
( 2 ) can be modified into the form :    @xmath21    we want eqn .
( 3 ) as simple as possible .
we include the reaction term @xmath22 in the first equation only because its effect in the second equation is assumed to be small .
this assumption stems from a microscopic process : a pair of @xmath8 can not be blocked by a single inert molecule .
the reason why we include @xmath23 in the second equation only is also taken from a microscopic fact : for the adsorption of @xmath0 , the presence of two free adjacent sites is not essential .
we note that the two effects were also described by jansen and nieminen  @xcite ; however , they did not include them explicitly in the rate equations .
we also note that eqn .
( 3 ) is much simpler than the rate equations in the pair approximation of eqn .
( 1 ) , which do not show oscillations in the parameter range of interest  @xcite . eqn .
( 3 ) is an approximate model , and it is desireable to derive it systematically .
because of its nonlinear terms , analytic treatment of eqn .
( 3 ) is difficult .
nevertheless , it can be studied numerically .
for example , one may use the software `` xppaut '' by b. ermountrout  @xcite .
we found that for the occurrence of a hopf bifurcation it is not necessary that both @xmath16 and @xmath19 were nonzero in eqn .
however , we found that parameter space for oscillations are larger if both @xmath16 and @xmath19 are nonzero .      in the present case , it is difficult to find analytic expressions for the steady states of eqn .
( 3 ) . in fig .
3 we show a numerical integration of eqn .
( 4 ) . here , in contrast to its dynamics shown in fig .
1 , the inert molecule concentration remains high .
we suppose that the main reason for this is that @xmath26 is finite in fig . 3 , but @xmath27 in fig . 1 .
also , a better choice of the rate constants can be made in order to compare dynamics in the rate equations and monte carlo simulations  @xcite .
a hopf bifurcation in this system is shown in fig .
4 . thus , the nonlinear effects we outlined in the previous section may indeed lead to stable oscillations .
a two parameter bifurcation diagram is shown in fig . 5 .
we found that with the increase of @xmath26 , the parameter region for a hopf bifurcation widens .
we found that such an increase is accompanied by the emergence of saddle nodes .      in the present case ,
the steady states are given by , @xmath29 where @xmath30 is a solution to @xmath31    once again we were able to analyze stability of @xmath32 , @xmath33 and @xmath30 numerically only .
oscillations for the present case are shown in fig .
we note that in the present case , except the hopf bifurcation , saddle node bifurcations occur in the system .      as in the previous case , the steady states of eqn .
( 3 ) can be expressed through a polynomial .
again , we were able to solve the polynomial numerically only .
as a result , a linear stability analysis of these steady states has turned out to be difficult .
numerically detected oscillations for the present case are shown in fig .
note the exceptions of @xmath26 s value and @xmath25 ; these are the reaction rates used in ref .
@xcite for monte carlo simulations .
we note that for the parameters used in fig . 6 - 7 ,
a parameter domain for the hopf bifurcation was smaller in the present case then it was in the previous case .
a microscopic study of a reaction system based on monte carlo simulations can reveal the molecular mechanisms of catalytic processes . as an example , in this work , such study has allowed us to outline the essential nonlinear processes in site blocking caused oscillations .
however , fora more complete study of a reaction system , the role of diffusion should be taken into account  @xcite . in that sense ,
( 4 ) can be a useful starting point for a reaction diffusion study of site blocking induced oscillations .
the preliminary results of a reaction diffusion system in one dimension , with a diffusion of @xmath7 , @xmath9 , and @xmath11 , show synchronous oscillations , stable long lived islands and diffusion induced chemical turbulence  @xcite .
more detailed results on this , and its comparison to the monte carlo simulations , which include diffusion of adsorbents , will be reported elsewhere  @xcite .
higher order nonlinear effects of site blocking , which induces oscillations in the monte carlo simulation of co oxidation , are outlined here . 
 it is shown that the rate equations which include these effects exhibit a supercritical hopf bifurcation in parameter domains where monte carlo simulations lead to oscillations .