# (186l) Analytical Solution of the Period of Belousov-Zhabotinsky Reaction Using a Feedback Mechanism

#### AIChE Annual Meeting

#### 2017

#### 2017 Annual Meeting

#### Computing and Systems Technology Division

#### Interactive Session: Applied Mathematics and Numerical Analysis

#### Monday, October 30, 2017 - 3:15pm to 4:45pm

solution of the period of Belousov-Zhabotinsky reaction using a feedback

mechanism

Chi Zhai^{a,b},

Wei

Sun^{a},

Ahmet Palazoglu^{b*}

^{a} Beijing Key Lab of

Membrane Science and Technology, College of Chemical Engineering, Beijing

University of Chemical Technology, 100029 Beijing, China

^{b} Department of

Chemical Engineering, University of California, Davis,

CA 95616, USA

**Abstract**

One

of the most important models for the development of Prigogine¡¯s dissipative

structure theory is the ¡°Bursselator¡±,^{ 1} whose prototype is the *Belousov–Zhabotinsky*^{2}

(BZ) reaction. Since the BZ reaction is far-from-equilibrium, the dynamics of

the system does not obey the Onsager reciprocal relations and symmetry breaking

bifurcation may cause the system to generate self-organized patterns. Figure 1 depicts

the generation of the self-oscillatory waveform observed in experiments.

Figure

1. The

snapshot (and amplification) of the diffusive BZ reaction, which can generate

waveform of color change from blue to red periodically.^{3}

By the

dissipative structure theory, the BZ reaction is viewed as an open system with

constant negative entropy consumption, and a portion of the overall reaction

entropy-change is consumed for the maintenance of the periodic color-change

structure. Figure 2 is the schematic of the reaction kinetics, indicating a

periodically dissipative system. From the viewpoint of a dynamic system, the

nonlinearity of the intermediate terms (*X*, *Y*, *Z*) bring

about a Hopf bifurcation where increasing one of the parameters beyond the

critical point may cause a periodical color-change waveform to emerge.

It

is clear that the period of the self-oscillatory pattern is relevant to the

input parameter, i.e., the input entropy flux. Knowing how the period is

related to the parameter change would be critically beneficial in identifying

the characteristics of the self-oscillatory structure. Our goal in this study

is to develop mathematical methods to compute the period of the limit cycles as

a function of parametric changes.

Figure 2. The Oregonator^{4} model kinetics and the

abstracted structure. Species identification

with respect to the FKN mechanism^{5} of the BZ reaction are *X*=HBrO^{2},

*Y*=Br^{-}, *Z*=Ce(IV), *A*=BrO^{3}, *B*=Organic

species, *P*=HOBr. The reactant and product species *A*, *B* and

*P* are normally present in much higher concentrations than the dynamic

intermediate species *X*, *Y* and *Z* and are assumed to be

constant on the time scale of a few oscillations. The oscillatory exchange of

the intermediates causes *Z* to vary between Ce(IV) and Ce(III) back and

forth, and with the presence of the ferroin indicator, the media would change

color between blue to red repeatedly.

Often

one can use a numerical continuation method^{6} or a shooting method^{7}

to obtain the period of the limit cycles as a function of changes in the

parameters, but it would be much more effective to find analytical relationships

as they would reveal the specific characteristics of the self-oscillatory system,

i.e., the maintenance entropy flux of the self-oscillatory structure.

Figure 3. The closed-loop equivalence of the dynamic system and

the criterion of self-oscillation on the Nyquist diagram.

Mees

and Chua^{8} have proposed a method to solve the period of the limit

cycle analytically based on the frequency domain Hopf bifurcation theory. A

self-oscillatory dynamic system can be reformulated as a feedback system as shown

in Figure 3. Here, *G*(*s*) in Figure 3 is the Laplace transform of

the linear part and *f* is the memoryless nonlinear part. The generalized

Nyquist criterion provides the necessary condition for the closed-loop system

with *v *= *d *= 0 to generate oscillatory outputs, which is clear in

the right hand side of Figure 3. The intersection point satisfies the following

condition: *y* = *G*(*i¦Ø*)*u* ¡Ö *G*(*i¦Ø*)*N*(*A*)*y*,

which leads to *G*(*i¦Ø*) = -1/*N*(*A*). Here, *N*(*A*)

is the amplitude correlated approximation of the nonlinear term *f*, and *N*(*A*)

is identified by the harmonic balance method. For example, if the system is

approximated by the 2^{nd}-order harmonics, *N*(*A*) could be

written as 1+*A*^{2}*¦Î*(*¦Ø*).^{ 9}

However,

the BZ reaction is a multivariable system, and, when the method above is

applied, tensor operations^{8} will render the calculations tedious and

impractical. Since the BZ reaction is a highly nonlinear system, higher order

harmonics^{10} may also be needed to approximate the oscillatory

behavior accurately, which would complicate the calculation process further.

Figure

4. Block diagram for the numerical computation of functional expansions.

In

the current study, we propose to use the Laplace-Borel (LB) transfer function

representation to express the feedback system. The LB transform is an extension

of the Laplace transform to the nonlinear polynomial terms by an infinite

series of iterated integrals, and the transformed system obeys the shuffle

algebraic operation. ^{11, 12} The functional expansion (FEx) method

approximates the analytical solution of the system which can be presented

graphically as in Figure 4. However, for a specific order of expansion *X*_{p}(*t*),

the approximated solution by the FEx method does not promise the solution to be

a closed cycle as time approaches infinity.

Similar

to Figure 3, we propose to close the system by adding dash lines to Figure 4,

and set *u*(*t*) = 0. By utilizing the shuffle algebra, each variable

of the nonlinear part *f _{i}* (

*i*¡¯s expansion) could be

decoupled if the order of the harmonics is specified, which is especially attractive

for multivariable systems. The approximation procedure is progressive and the residuals

of each expansion are used to identify the parameters of the harmonics. Because

(1). the nonlinear blocks in Figure 4 may introduce higher order harmonics than

the input ones; (2). the output orders of the harmonics is predictable if the

structure of the system is given, then, the formula of

*N*(

*A*) is known

and the parameter of

*N*(

*A*) can be identified by setting higher

order residuals as zero. This method makes

*N*(

*A*) being flexible

for different systems and expendable to higher order approximations.

**Keywords:** dissipative

structure; self-organization; Laplace-Borel transform; harmonic balance method;

frequency domain Hopf bifurcation.

**References:**

1. Prigogine, I., 1978. Time, structure,

and fluctuations. Science, 201(4358), 777-785.

2. Winfree, A. T., 1984. The prehistory

of the Belousov-Zhabotinsky oscillator. J. Chem. Educ., 61(8), 661.

3. http://hopf.chem.brandeis.edu/anatol.htm

4. Field, R. J., Noyes, R. M., 1974.

Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real

chemical reaction. The Journal of Chemical Physics, 60(5), 1877-1884.

5. Field, R. J.,

Koros, E., Noyes, R. M., 1972. Oscillations in chemical systems. II. Thorough

analysis of temporal oscillation in the bromate-cerium-malonic acid system.

Journal of the American Chemical Society, 94(25), 8649-8664.

6. Zhai, C.,

Palazoglu, A., Wang, S., et al. 2017. Strategies for the Analysis of Continuous

Bio-ethanol Fermentation under Periodical Forcing. I&ECR, in press.

7. Doedel, E. J.,

1981. AUTO: A program for the automatic bifurcation analysis of autonomous

systems. Congr. Numer, 30, 265-284.

8. Mees, A.,

Chua, L., 1979. The Hopf bifurcation theorem and its applications to nonlinear

oscillations in circuits and systems. IEEE Transactions on Circuits and Systems,

26(4), 235-254.

9. Allwright, D.

J., 1977. Harmonic balance and the Hopf bifurcation. Mathematical Proceedings

of the Cambridge Philosophical Society. Cambridge University Press, 82(03), 453-467.

10. Moiola, J.,

Chen, G., 1993. Computations of limit cycles via higher-order harmonic balance

approximation. IEEE Transactions on Automatic Control, 38(5), 782-790.

11. Harris, K.

R., Palazoglu, A., 1997. Studies on the analysis of nonlinear processes via

functional expansions-II. Forced dynamic responses. Chem. Eng. Sci. 52,

3197-3207.

12. Batigun, A, Harris, K. R., Palazoglu,

A., 1997. Studies on the analysis of nonlinear processes via functional

expansions-I. Solution of nonlinear ODEs. Chem. Eng. Sci. 1997, 52(18):

3183-3195.