Abstract
This study provides semi-analytical solutions to the Selkov-Schnakenberg reaction-diffusion system. The Galerkin method is applied to approximate the system of partial differential equations by a system of ordinary differential equations. The steady states of the system and the limit cycle solutions are delineated using bifurcation diagram analysis. The influence of the diffusion coefficients as they change, on the system stability is examined. Near the Hopf bifurcation point, the asymptotic analysis is developed for the oscillatory solution. The semi-analytical model solutions agree accurately with the numerical results.
1 Introduction
The emergence of oscillatory and multiple steady state solutions and chaotic behaviors is an interesting phenomenon observed in many chemical, biological, and physical systems. A number of experimental studies of chemical systems have been done to examine oscillating solutions involving Briggs-Rausher [1], Belousov-Zhabotinsky [2], and Bray-Liebhafsky [3] reactions, which demonstrate periodic variations in concentrations as visually striking color changes [4]. The Selkov-Schnakenberg system is an example of an oscillating systems associated with cellular processes in biochemical system [5]. The oscillatory phenomena in these systems have been investigated using continuous-flow well-stirred tank reactors (CSTRs), governed by ordinary differential equations (ODEs) for concentrations. CSTRs have exhibited excellent results in theoretical and experimental studies of chemical oscillations. Another reactor type important for the study of oscillating reactions is the reaction-diffusion cell. It is described using partial differential equations (PDEs) that can be used to characterize the emergence of many phenomena.
In 1979, Schnakenberg introduced a simple chemical reaction model for glycolysis that showed limit cycle behavior. The reaction scheme, known as Selkov-Schnakenberg model, occurs in the three steps:
In the above reactions,
A great literature has been dedicated to the study of (1), including the Selkov model and the Schnakenberg model; see, for instance, [8,9, 10,11,12, 13,14] and the references therein. A generalized Selkov-Schnakenberg reaction-diffusion system is analyzed in [15]. The authors provide values for the stability of the unique constant steady state. Also, the effect of the diffusion coefficients on the stability of the system has been investigated in this study. In [16], the Selkov-Schnakenberg model is considered, and the steady state problem is studied. They obtained results show a change in the stability of the system, leading to the formation of different spatial patterns. In [17], the Selkov-Schnakenberg system was studied numerically, and numerical results for snaking of patterns were obtained.
One of the most important mathematical methods that has proven effective in developing semi-analytical models for reaction-diffusion systems is the Galerkin method [18,19]. It is based on using the approximation of a system of PDEs with a system of ODEs with the same behavior. This method was used in many research studies, for example, in 2002, Marchant [20] employed this method to analyze Gray-Scott model. In this study, the regions of stability and instability in the system were determined using combustion theory. This method yielded higher accuracy compared to the semi-analytical model and the PDE system (its numerical results). Also, in [21,22] this method was applied to the reversible Selkov model. The researchers found the Hopf regions and bifurcation diagram for the model, as well they studied the effect of feedback control in the boundary conditions of the system with and without precursor and final products. Moreover, [23,24] also considered a semi-analytical model for the Schnakenberg reaction-diffusion system. The two studies focused on the stability and singularity behavior of the model, showing that the model has a hysteresis bifurcation pattern. To further elaborate on the application of the Galerkin method in many reaction-diffusion problems, we recommend the reader to refer to these references [25,26,27, 28,29,30, 31,32].
The purpose of this study is to investigate the singularity behavior and stability of the Selkov-Schnakenberg system which, through previous studies, have not yet been investigated and so all the results of this work are novel and genuine. This paper is structured as follows. Section 2 shall present the mathematical model of the Selkov-Schnakenberg system and describes the methodology for application of the Galerken method to PDEs and boundary conditions. Section 3 uses steady state concentration profiles and bifurcation diagrams to illustrate the complexity of the steady state multiplicity of the system. Section 4 reports the results of the application of the singularity theory to determine the hysteresis bifurcation points and defines the region of the two bifurcation diagram patterns. In 5, stability analysis of the semi-analytical model is performed and Hopf bifurcation region is determined. In Section 6, an asymptotic analysis of the periodic solution near the Hopf bifurcation point in the semi-analytical model is performed.
2 The semi-analytical model
2.1 The mathematical model
The Selkov-Schnakenberg system (1) is governed by the following coupled nonlinear PDEs in a one-dimensional (
In this system,
2.2 The Galerkin method
The semi-analytical model of the Selkov-Schnakenberg system is obtained through the use of the Galerkin method in a 1-D reaction-diffusion cell. This converts the PDEs (2) governing the system into a system of ODEs. A function expansion
is used, where
This expansion fulfills the spatial boundary conditions (3). At the same time, it satisfies the PDEs (2) in an average sense, but not exactly. Moreover, the expansion used in (4) has the property that at the cell center the concentrations are
The system of ODEs is found by truncating the series of basis functions (4) after two terms. This truncation is sufficient to show high accuracy in the results without enlarging the expression. To ensure sufficient accuracy, it is compared to the one-term solution obtained by assuming
3 Analysis of steady state bifurcations
This section deals with the steady state bifurcations of the system (5). To obtain the steady state solutions, the time derivatives
Figure 1(a) and (b) depict the steady state concentration profiles for the autocatalyst and the reactant,
The bifurcation diagrams demonstrate the steady state concentrations of both
4 Application of singularity theory
Singularity theory is based on a theoretical study that describes cases of steady state behavior in the systems of ODEs [36]. Several studies have examined the feasibility of applying this theory to chemical reactions. One of the most famous of these studies is [37] in which the conditions for designating regions of isola and hysteresis bifurcation curves in CSTR were presented.
This section aims to apply the theory of singularity to the semi-analytical model (2) to detect the steady state bifurcation diagrams that can emerge in this system. The two primary bifurcation types are those with unique and breaking wave patterns. It is possible to write the steady state equations for the two-term semi-analytical model in the general form (4) as follows:
Hence, our selected bifurcation parameter is
In calculations, we regard
Figure 4 shows the hysteresis bifurcation curve between the autocatalyst diffusion coefficient
Figure 5 demonstrates the two-term hysteresis curve with
5 The phenomena of limit cycles and Hopf bifurcations
The limit cycles and Hopf bifurcations are phenomena considered very important for studying physical, biological, and chemical systems. The Hopf bifurcation theory was explained in a variety of sources on dynamical systems and bifurcations theory [38,39]. The Hopf bifurcation results from the emergence of a periodic solution due to local switching of the properties of the steady state solutions from stable to unstable when a simple complex conjugate pair of eigenvalues crosses over the complex plane’s imaginary axis. This section explores the local stability of the semi-analytical model to investigate the effect of the diffusion coefficients on the system stability (2). The degenerate Hopf points are computed to predict a semi-analytical map in which Hopf bifurcations and limit cycles occur. A comparison of this forecast with numerical results is then made.
First, the stability of the ODEs system (5) representing the two-term semi-analytical model is studied. The Jacobian matrix is the following:
The eigenvalues,
When one pair of the eigenvalues is completely imaginary, Hopf bifurcations are observed:
Therefore, solving the following condition yields the degenerate Hopf points:
Results in Figure 6 show the degenerate Hopf bifurcation curve for the two-term model of (11) and the numerical results of governing PDEs in the plane of the autocatalyst diffusion coefficient
Figure 7 shows another possible example. It displays the loci of Hopf bifurcation lines for both the two-term model and the numerical solution in the (
Figure 8(a) shows that the limit cycle occurs in a phase plane of the autocatalyst concentration
Figure 10 shows phase portraits of the concentrations at the center of the reactor for various choices of bifurcation parameter
6 Oscillatory solutions near the Hopf bifurcation point
This section elaborates on the periodic solutions of the semi-analytical ODEs model (5) for the Selkov-Schnakenberg system and applies asymptotic analysis to them. The Lindstedt-Poincaré method [40] perturbation theory is used to compute asymptotic solutions to both the one- and two-term semi-analytical models.
The determination of periodic solutions of power series with respect to a small oscillation amplitude parameter is considered. The method is applied to remove secular terms and obtain corrections to the bifurcation parameter and frequency [41].
6.1 Lindstedt-Poincaré perturbation method for the one-term model
One-term semi-analytical model comprises the following system of ODEs:
The Lindstedt-Poincaré method is utilized to find the
where
Then, the frequency
where
For the first two orders, the solutions of (16) and (17) may be found as
where c.c. is the complex conjugate. By substituting the solutions (19) into (16) and (17), and then splitting into real and imaginary parts, we can find the amplitudes
Both equations (20) have
Now, the solution is complete and we introduce the results in the following special case in details. The solution is obtained at
Hence, the limit cycle as well as its period may be expressed as
where the extrema of the oscillatory solutions are given by
6.2 Lindstedt-Poincaré perturbation method for the two-term model
The ODEs system for the two-term semi-analytical model is found in (5). The periodic solutions near the Hopf bifurcation point are obtained through the same procedure as for the one-term model, but with some additional complexities. The Lindstedt-Poincaré solution of (5) has the following form:
The perturbation equations for the first three orders of
To find the complex amplitudes
Again, substituting (24) and (25) into the
As a result, the solution for the special case is obtained; it is expounded below. The Hopf bifurcation arises at
The leading-order periodic solution is
where the extrema of the periodic solutions are
Figure 12 demonstrates the bifurcation diagram for the concentration of autocatalyst
7 Conclusion
Within the scope of this paper, a detailed study of the Selkov-Schnakenberg system in the reaction-diffusion cell is presented. The PDEs system of the model was approximated with the system of ODEs using the Galerkin method. The steady state and bifurcation diagrams for the system have been constructed based on the singularity theory and discussed. The Hopf bifurcation region was also studied and the limit cycles in the system have been shown. An asymptotic analysis of the periodic solution near the Hopf bifurcation point was carried out and its results were compared to the semi-analytical solution. It has been shown that the diffusion coefficients for both autocatalyst and reactant, as well as the
This method has shown its effectiveness and accuracy and may be applied to other chemical models, studying the stability of which is associated with greater complexity such as the three-component FitzHugh-Nagumo model and the three-component reversible Gray-Scott model. The effect of feedback and time delay in boundary conditions on the stability of the considered model should be further investigated.
Acknowledgments
The author expresses his sincere thanks and appreciation to Taif University Researchers Supporting Project number (TURSP-2020/271), Taif University, Taif, Saudi Arabia for supporting this paper. Also, the author thanks the referees for their careful reading and helpful suggestions that helped improve this paper.
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Conflict of interest: Authors state no conflict of interest.
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