Modeling, Analysis, and Applications of Complex SystemsView this Special Issue
Research Article | Open Access
Deterministic and Stochastic Bifurcations of the Catalytic CO Oxidation on Ir(111) Surfaces with Multiple Delays
The main purpose is to investigate both deterministic and stochastic bifurcations of the catalytic CO oxidation. Firstly, super- and subcritical bifurcations are determined by the signs of the Poincaré-Lyapunov coefficients of the center manifold scalar bifurcation equations. Secondly, we explore the stochastic bifurcation of the catalytic CO oxidation on Ir(111) surfaces with multiple delays according to the qualitative changes in the invariant measure, the Lyapunov exponent, and the stationary probability density of system response. Some new criteria ensuring stability and stochastic bifurcation are obtained.
For most of the practical cases, the dynamical systems will be disturbed by some stochastic perturbation. There are real benefits to be gained in using stochastic rather than deterministic models. It could be intuitively thought that the external noise would smear out the finer details of the nonlinear system, such as bifurcations between well-defined states. However, it could also happen that noise forces the system far away from its deterministic attractors to explore regions of phase space that are otherwise not visited or interact with the nonlinearities giving rise to new phenomena [1–10].
Now, a great care should be taken when reducing the internal/external sources of noise and uncertainties such as thermal noise, observation errors, and model misspecification. The CO oxidation on platinum group crystals offers an experimental setup, where the nonlinear mechanisms have been captured with high accuracy, and sources of internal noise are controlled. The CO oxidation on crystals shows, in the absence of noise, a wide variety of phenomena, for example, oscillations, bistability, bifurcation, excitability, spatiotemporal patterns, and turbulence. These reactions have been receiving an increasing attention as a laboratory analogy of catalysis used at the industrial level to manufacture chemical products and process harmful species. Understanding the effects of external noise on this oxidation reaction can be considered as an effort to bridge the quality gap. Besides material and pressure gap, which have been discussed in the literature before, the quality gap separates small experiments in perfectly controlled settings and the industrial processes that are always under the influence of uncertainties.
In this paper, we investigate the deterministic and stochastic bifurcations of the catalytic CO oxidation on Ir(111) surfaces with multiple delays. Mechanisms and parameters are known with a high degree of accuracy, and the relevant parameters can be controlled. These ideal properties make it possible to go back and forth between theory and experiments. Our paper focuses on homogeneous states and builds on the work of Hayase et al. [11–18] who studied experimentally and numerically the effect of noise on CO oxidation on Ir(111). They verified that the mathematical model was correct and observed probability distributions that are consistent with the presence of multiplicative noise.
The structure of the paper is as follows. In Section 2, we analyzed the single and double Hopf bifurcation of deterministic system. In Section 3, we analyzed the stochastic stability and bifurcation of stochastic system.
2. Deterministic System
In this section, we will investigate the delayed CO oxidation on Ir(111) surfaces as follows: where is the bifurcation parameter. It is obvious that there is a unique equilibrium point in system (1).
Let , then by substituting the corresponding variables in (1), we have
Linearized System. As the first step, we analyze the stability of the trivial solution of the linearized system of (1), which is given by in , and its adjoint form in with respect to the associated bilinear relation has eigenvalues satisfying the transcendental characteristic equation
By the Hopf bifurcation, we let , with, and , be the eigenvalue of (6). All the remaining eigenvalues of have negative real parts. Then, by substituting into and setting the real and imaginary parts of the resulting algebraic equations to zero, we have
Separating the real and imaginary parts of (7) gives the following equations: or equivalently Possible values of are of the form , . Thus, we obtain the bifurcation parameter as
Now, we compute Hopf’s transversality condition by differentiating implicitly (6) with respect to , obtaining
Lemma 1. The transversality conditions hold. If the choice of values for the model parameters is such that the inequality holds.
This means that the pair of eigenvalues of with , will cross the imaginary axis from left to right in the complex plane with a nonzero speed as the bifurcation parameter varies near its critical value . Also, it ensures that the steady-state stability of the system is lost at , and the instability in the sense of supercritical and subcritical bifurcations will persist for larger values of . The exact nature of such bifurcations satisfying Hopf’s conditions ((6)–(12)) will depend on the nonlinearity in the delay system.
Rewriting (10) as a quadratic in and letting , we have whose real and positive solutions are of the form if the following holds In addition, one has to show that the substitution of and into (10) will ensure Hopf’s transversality conditions, that is, . Therefore, with and , we obtain By setting and , we have
These are the minimum values separating the regimes of stable and unstable solutions of the linearized delay equation at the Hopf bifurcation point , . Below and equal to these minimum values, all the infinite numbers of eigenvalues of the transcendental characteristic equation (6) will lie in the left-hand side of the complex plane.
Next, deriving an expression for begins by further rewiring (8) as follows: thus, we obtain and by using the minimum value for , we get With these minimum values for the gain , time delay , and the response frequency , one can describe all the critical settings of the model parameters in the delay equation (1) at which the unstable-free oscillation persists. Should unstable oscillations appear at some desired frequency such that , fine-tuning the model parameters in the vicinity of their critical settings can control such oscillations. To find out whether the bifurcations are supercritical and/or subcritical, we carry out a local center manifold analysis of the nonlinear delay equation (1) as presented in the preceding section.
2.1. The Single Hopf Bifurcation
To obtain the explicit analytical expressions for the stability condition of the Hopf bifurcation solutions (limit cycles), system (1) should be reduced to its center manifold [19–21]. While studying the critical infinite dimensional problem on a two-dimensional center manifold, we begin by defining the elements of the initial continuous function and its projections onto the center and stable subspaces . For the eigenvalues , we have with , of the adjoint-generalized eigenspace and , , of the adjoint-generalized eigenspace . The values of the basis for are thus given by and, similarly, the basis for is of the form
These bases form the inner product matrix in which
The substitution of the elements of where
Consequently, the constant matrices and are equivalent, and the elements of at the Hopf bifurcation are .
With the algebraic simplifications one can easily see that and it follows that is indeed the solution operator of the linearized delay equations in . Similarly, we have , , and is the solution operator for the adjoint delay equations in .
By the change of variable with , we obtain the following as a first-order approximation in , for : and for , Therefore, we obtain (1) of the center manifold as follows: where the perturbation functions and denote the following:
Here, the lengthy expressions of are omitted here, and which in polar coordinate, , , is equivalent to The so-called Poincaré-Lyapunov coefficient is independently determined by the following formula: As is well known [19–24], the amplitude equation (35) together with (36) can determine the stability behaviour of the center manifold ODEs (33), where all the partial derivatives of , , for example, , are evaluated at the bifurcation point . Thus, the partial derivatives are given as follows: from which we readily obtain the required values for the formula in (36) as Substituting these quantities into the formulate in (36) will yield This coefficient together with the amplitude bifurcation equation (35) produces the nonlinear bifurcations for multiple delays as shown in the following: (a) supercritical bifurcation; (b) subcritical bifurcation.
2.2. The Double Hopf Bifurcation
For the eigenvalue , of the equation in (6), we have the base function , , , in and , accordingly, and they form the inner product matrix , namely, The substitution of the elements into the bilinear pairing (5) and integrating, we obtain the nonsingular matrix where the constant values of this matrix are given by Then, is normalized to the new basis , , where the elements , of are given by Again, the substitution of the elements , , of will yield the identity matrix, namely, Then, the constant matrix is given by The change of variable , will yield thereby, for and , we have and so The ODEs of the center manifold are given by as follows: where the perturbations in these equations denote and the above coefficients in these equations denote the constant values in (52). Here, the lengthy expressions of and are omitted here. These equations are further written into the desired standard form of amplitude and phase relations using the polar coordinates , , , . Again, the application of the integral averaging method to the resulting amplitude and phase equations will yield where the coefficients in these equations are the resulting constant values after averaging. Similarly, the coefficients in these averaged equations, in particular the cubic ones , , will determine whether the double Hopf interactions are of the form (i) subsupercritical bifurcation type, (ii) supersubcritical bifurcation type, (iii) subsubcritical bifurcation type, and (iv) supersupercritical bifurcation type. Like the Poincaré-Lyapunov coefficient in (36) for the single Hopf, there are also explicit formulas for computing the double Hopf coefficients , , which are extremely too long to be displayed here. For the account of these formulas for the double Hopf coefficients, again, the publications [22, 23] are excellent references. Classical studies of dynamical systems resulting to equations of the form (53) have demonstrated unprecedented dynamics ranging from global dynamic bifurcations to ultimately chaotic dynamics (see [22, 24]). There exist reliable and robust mathematical tools to examine the long-term behaviour of such dynamics. Campbell et al. , in particular, have shown numerically that equations of the form (53) exhibit four double Hopf’s interactions, namely, two super-super-, one super-sub-, and one sub-sub-critical bifurcations. Other interesting dynamics such as large limit cycles, tori, and period doubling were also characterized by the authors.
3. Stochastic System
In the section, we present some preliminary results to be used in a subsequent section to establish the stochastic stability and stochastic bifurcation. Before proving the main theorem, we give some lemmas and definitions.
Definition 3 (see  (D-bifurcation)). Dynamical bifurcation is concerned with a family of random dynamical systems which is differential and has the invariant measure . If there exists a constant satisfying in any neighborhood of , there exists another constant and the corresponding invariant measure satisfying as . Then, the constant is a point of dynamical bifurcation.
Definition 4 (see  (P-bifurcation)). Phenomenological bifurcation is concerned with the change in the shape of density (stationary probability density) of a family of random dynamical systems as the change of the parameter. If there exists a constant satisfying in any neighborhood of , there exist other two constants , , and their corresponding stationary density functions , satisfying and are not equivalent. Then, the constant is a point of phenomenological bifurcation.
Definition 5 (see  (stochastic pitchfork bifurcation)). In the viewpoint of phenomenological bifurcation, the stationary solution of the Fokker-Planck-Kolmogorov (FPK) equation which is corresponded with the stochastic differential equation changes from one peak into two peaks.
In the viewpoint of dynamical bifurcation, there exists a constant that satisfies the following conditions:(i)when , the stochastic differential equation has only one invariant measure ; moreover, is stable (i.e., the maximal Lyapunov exponent is negative).(ii)when , the invariant measure loses its stability and becomes unstable (i.e., the maximal Lyapunov exponent is positive); moreover, the rotation number is zero.(iii)when , the stochastic differential equation has three invariant measures , , and ; both and are stable (i.e., their maximal Lyapunov exponents are both negative).(iv)the global attractors of the stochastic differential equation change from a single-point set into a collection of one-dimensional (the closure of the unstable manifold of the unstable invariant measure).
If the stochastic bifurcation of a stochastic differential equation has the above characters, then the stochastic differential equation admits stochastic pitchfork bifurcation at .
Definition 6 (see  (the stochastic Hopf bifurcation)). In the viewpoint of phenomenological bifurcation, the stationary solution of the FPK equation which is corresponded to the stochastic differential equation changes from one peak into a crater.
In the viewpoint of dynamical bifurcation, the following can be noticed:(i)if one of the invariant measures of the stochastic differential equation loses its stability and becomes unstable (i.e., two Lyapunov exponents are positive), then the rotation number is not zero. Meanwhile, there at least appears one new stable invariant measure,(ii)the global attractors of the stochastic differential equation change from a single-point set into a random topological disk (the closure of the unstable manifold of the unstable invariant measure).
If the stochastic bifurcation of a stochastic differential equation has the above characters, then the stochastic differential equation admits the stochastic Hopf bifurcation.
Definition 7 (see  (stochastically stable)). The trivial solution of stochastic differential equation is said to be stochastically stable or stable in probability if for every pair of and , there exists such that whenever . Otherwise, it is said to be stochastically unstable.
In this section, taking some stochastic factors into account, we introduce randomness into the model (1) by replacing the parameters by . This is only a first step in introducing stochasticity into the model. Ideally, we would also like to introduce stochastic environmental variation into the other parameters such as the transmission coefficient and , but this would make the analysis much too difficult. Hence, we get the following system of stochastic differential equations:
Here, is the multiplicative random excitation and the external random excitation directly, and is independent, in possession of zero mean value and standard variance Gaussian white noises, that is, . Note that the intensity of the noise , is the bifurcation parameter. From (3)–(33), we obtain (55) of the stochastic center manifold: Here, the lengthy expressions of and are omitted.
Setting the coordinate transformation , and by substituting the variable in (34), we obtain