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

Analytical
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–Zhabotinsky²
(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 Oregonator⁴ model kinetics and the
abstracted structure. Species identification
with respect to the FKN mechanism⁵ of the BZ reaction are X=HBrO²,
Y=Br^-, Z=Ce(IV), A=BrO³, 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 method⁶ or a shooting method⁷
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 Chua⁸ 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+A²¦Î(¦Ø). ⁹

However,
the BZ reaction is a multivariable system, and, when the method above is
applied, tensor operations⁸ will render the calculations tedious and
impractical. Since the BZ reaction is a highly nonlinear system, higher order
harmonics¹⁰ 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.

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