# Suppressing Chaos for a Fractional-Order Chaotic Chemical Reaction Model viaPDζ Controller

Authors: Hui Wang
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this work, based on the earlier publications, we build a new fractional-order chemical reaction model. Computer simulations manifest that the fractional-order chemical reaction model presents chaotic behavior under a certain parameter condition. To eliminate the chaotic dynamical property, a suitable fractional-orderPDζ controller with time delay is designed. Regarding the time delay as a bifurcation parameter, we set up a novel delay-independent stability and bifurcation criterion guaranteeing the stability and the creation of Hopf bifurcation of the controlled fractional-order chemical reaction model. The influence of time delay on the stability and Hopf bifurcation of the controlled fractional-order chemical reaction model is revealed. At last, numerical simulations are performed to sustain the rationality of the designed PDζ controller. The obtained conclusions of this work are completely novel and have immense application prospects in the chaos control of chemical reaction systems. Furthermore, the research idea can also be utilized to suppress the chaos of a lot of fractional-order chaotic models.


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## Body

## 1. Introduction

Chaos exists widely in many areas such as climate, physics, chemistry, engineering, economy and finance, complex networks, and population systems [1–6]. The chaotic phenomenon depends on the initial condition of the original system. The chaotic behavior occurring in the nonlinear dynamical systems owns the very complex property and unpredictability. In many cases, chaotic behavior is not what we want in our practical life. Thus, a natural problem arises: how to suppress the chaotic behavior of the original system has been an important theme in many disciplines. During the past several decades, suppressing chaos has received great attention from many scholars. For example, Chen [7] controlled the chaos via a simple adaptive feedback control technique. Du et al. [8] applied the phase space compression method to control the chaos of an economic model. In 1990, Yorke [9] utilized Ott, Grebogi, and Yorke (OGY) control approach to control the chaos of a chaotic model. Paula and Savi [10] designed an extended time-delayed feedback control approach to suppress the chaos of a nonlinear pendulum. For more detailed literature on this aspect, one can refer [11–14].In 1996, Geysermans and Baras [15] proposed a homogeneous chaotic Wilamowski–Rossler model. The balance equations of this model have a well-defined microscopic counterpart and all the reaction follows the following “elementary” steps:(1)A1+X⟶κ12X,A1+X⟵κ−12X,X+Y⟶κ22Y,A5+Y⟶κ3A2,X+Z⟶κ4A3,A4+Z⟶κ52Z,A4+Z⟵κ−52Z.System (1) includes two autocatalytic steps involving constituents X and Z, coupled via three other steps involving the three constituents X, Z, and Y. The initial A1,A4,A5 and final (A2,A3) product concentrations remain fixed. The distance from thermodynamic equilibrium is controlled by the values of A1,A2,A3,A4,A5. κ±ii=1,2,3,4,5 stands for the rate constant. In model (1), there are 15 free parameters. To reduce the number of free parameters, Geysermans and Baras [15] selected the rate coefficients κ−2=0,κ−3=0, and κ−4=0. Note that the last two relations imply that A2 and A3 are continuously removed from the reactor [15, 16].Assuming that there exists an ideal mixture and a well-stirred reactor, then the macroscopic rate equations of model (1) can be expressed as follows:(2)du1tdt=β1u1t−κ−1u12t−u1tu2t−u1tu3t,du2tdt=u1tu2t−β5u2t,du3tdt=β4u3t−u1tu3t−k−5u32t,where u1t,u2t, and u3t stand for the mole fractions of X,Y, and Z at the time t. The rate constants κ1,κ3, and κ5 are incorporated in the parameters β1,β5, and β4 (e.g., β1=κ1A1,…,o) and β1>0,β4>0,β5>0,κ−1>0,κ−5>0 stand for the constants. In detail, one can refer [15, 16]. In 2015, Xu and Wu [17] dealt with the bifurcation control of chaos for model (2) via three-time delay feedback controllers. Namely, they considered the following controller chemical model:(3)du1tdt=β1u1t−κ−1u12t−u1tu2t−u1tu3t+μ1u1t−u1t−θ,du2tdt=u1tu2t−β5u2t+μ2u2t−u2t−θ,du3tdt=β4u3t−u1tu3t−k−5u32t+μ3u3t−u3t−θ,where μii=1,2,3 stands for a real constant and θ is a delay.It is worth mentioning that all the literature above (see [15–17]) are only concerned with the integer-order chemical models and they are not concerned with the fractional-order chemical models. Recent studies have shown that fractional-order differential equation is deemed as a more effective tool to portray natural phenomena than the classical integer-order ones since it has great advantages in memory trait and hereditary property of numerous materials and development processes [18–20]. Recently, fractional-order dynamical systems have displayed great application in lots of areas such as network systems, intelligent control, physical science, biological engineering, chemistry, finance, and so on [21–26]. Rich achievements on fractional-order dynamical models have been obtained. For example, Ke [27] dealt with the Mittag–Leffler stability and asymptotic ω-periodicity for a class of fractional-order inertial delayed neural networks. Du and Lu [28] focused on the finite-time stability for fractional-order fuzzy delayed cellular neural networks. Xiao et al. [29] probed into the bifurcation control problem for fractional-order small-world networks; Huang et al. [30] studied the bifurcation problem of fractional-order delayed neural networks. In detail, we refer the readers to [31–34].Inspired by the exploration above and on the basis of system (2), in order to describe the continuous change process of the mole fractions of X,Y, and Z and characterize the memory trait and hereditary property of the variables X,Y, and Z, we modify system (2) as the following fractional-order form:(4)dζu1tdtζ=β1u1t−κ−1u12t−u1tu2t−u1tu3t,dζu2tdtζ=u1tu2t−β5u2t,dζu3tdtζ=β4u3t−u1tu3t−k−5u32t,where ζ∈0,1. The research indicates that when ζ=0.97,β1=30,κ−1=0.55,β5=9.5,β4=16.5,κ−5=0.5, then there is a chaotic phenomenon in system (3). The software simulation figures are presented in Figure 1.Figure 1
Software simulation figures of system (4) with ζ=0.97,β1=30,κ−1=0.55,β5=9.5,β4=16.5,κ−5=0.5.In this current work, we are to deal with the chaos control of system (4) by virtue of a fractional-order PDζ controller. The key contributions of this study are as follows:(i)
On the basis of the earlier studies, a new fractional-order chaotic chemical reaction model is set up(ii)
The chaotic phenomenon of system (4) is suppressed by means of an appropriate fractional-order PDζ controller(iii)
The study approach can be utilized to suppress the chaos of lots of fractional-order dynamical models in many subjectsThis manuscript can be arranged as follows: some prerequisite theory on the fractional-order differential equation is prepared in Section2; in Section 3, we prove the existence and uniqueness of the solution of system (4); in Section 4, the chaos of system (4) is suppressed via fractional-order PDζ controller and a delay-independent sufficient condition that ensures the stability and the creation of Hopf bifurcation of the fractional-order controlled chaotic chemical reaction model is built; in Section 5, software simulation results are presented to sustain the established conclusions; and Section 6 completes this article.


## 2. Preliminary Knowledge

In this part, we present some indispensable theories on a fractional-order differential equation.Definition 1 (see [35]).
The fractional type integral of the orderζ of the function uξ is given by(5)ℐζuξ=1Γζ∫ξ0ξξ−νζ−1uνdν,where ξ>ξ0,ζ>0 and Γν=∫0∞sν−1e−sds.Definition 2 (see [35]).
The Caputo fractional order derivative of the orderζ of the function uν∈ν0,∞,R is defined as follows:(6)Dζuν=1Γκ−ζ∫ν0νuκsν−sζ−κ+1ds,where ν≥ν0 and κ represents a positive integer ζ∈κ−1,κ. In particular, if ζ∈0,1, then(7)Dζuν=1Γ1−ζ∫ν0νu′sν−sζds.Lemma 1 (see [36]).
Consider the fractional-order systemDζw=ℱw,w0=w0 where ζ∈0,1,w∈Rl,ℱ∈Rl×l. Assuming that χii=1,2,…,l is the root of the characteristic equation of Dζw=ℱw, then the equilibrium point of the system Dζw=ℱw is locally asymptotically stable if argχi>ζπ/2i=1,2,…,l and the equilibrium point of the system Dζw=ℱw is stable if argχi>ζπ/2i=1,2,…,l and all critical eigenvalues that satisfy argχi=ζπ/2i=1,2,…,l own geometric multiplicity one.


## 3. Existence and Uniqueness of the Solution of System (4)

In this section, we will prove the existence and uniqueness of the solution of system (4).Theorem 1.
LetΛ=u1,u2,u3∈R3:maxu1,u2,u3≤A, where A>0 is a constant. ∀u10,u20,u30∈Λ, system (4) with the initial value u10,u20,u30 has a unique solution U=u1,u2,u3∈Λ.Proof.
Define the following mapping:(8)fU=f1U,f2U,f3U,where(9)f1U=β1u1t−κ−1u12t−u1tu2t−u1tu3t,f2U=u1tu2t−β5u2t,f3U=β4u3t−u1tu3t−k−5u32t.∀U,U¯∈Λ, one obtains(10)fU−fU˜=|β1u1t−κ−1u12t−u1tu2t−u1tu3t−β1u¯1t−κ−1u¯12t−u¯1tu¯2t−u¯1tu¯3t|+u1tu2t−β5u2t−u¯1tu¯2t−β5u¯2t+|β4u3t−u1tu3t−k−5u32t−β4u¯3t−u¯1tu¯3t−k−5u¯32t|≤β1+2κ−1A+Au1t−u¯1t+Au2t−u¯2t+Au3t−u¯3t+Au1t−u¯1t+A+β5u2t−u¯2t+Au1t−u¯1t+β4+A+2κ−5Au3t−u¯3t=A1u1t−u¯1t+A2u2t−u¯2t+A3u3t−u¯3t≤A0U−U¯,where(11)A1=β1+2κ−1A+3A,A2=2A+β5,A3=β4+2A+2κ−5A,(12)A0=maxA1,A2,A3.
ThenfU satisfies Lipschitz condition with respect to U (see [39, 40]). According to Banach fixed point theorem, we know that Theorem 1 is true.


## 4. Suppressing Chaos via Fractional-OrderPDζ Controller

In this part, we are to apply a suitable controller to eliminate the chaotic phenomenon of system (4). By virtue of the idea of Tang et al. [37], the fractional-order PDζ controller can be designed as follows:(13)ψt=ϱpu1t−θ+ϱddζu1tdtζ,where ϱp and ϱd≠1 present the proportional control parameter and the derivative control parameter, respectively, and θ denotes a delay. Adding (13) to the first equation of system (4), we get(14)dζu1tdtζ=β1u1t−κ−1u12t−u1tu2t−u1tu3t+ψt,dζu2tdtζ=u1tu2t−β5u2t,dζu3tdtζ=β4u3t−u1tu3t−k−5u32t.That is(15)dζu1tdtζ=β1u1t−κ−1u12t−u1tu2t−u1tu3t+ϱpu1t−θ+ϱddζu1tdtζ,dζu2tdtζ=u1tu2t−β5u2t,dζu3tdtζ=β4u3t−u1tu3t−k−5u32t.System (15) can be rewritten as the following form:(16)dζu1tdtζ=β11−ϱdu1t−κ−11−ϱdu12t−11−ϱdu1tu2t−11−ϱdu1tu3t+ϱp1−ϱdu1t−θ,dζu2tdtζ=u1tu2t−β5u2t,dζu3tdtζ=β4u3t−u1tu3t−k−5u32t.It is not difficult to obtain that if the following condition(17)Q1β4>β5,β1−κ−1κ−5>κ4−κ5,holds, then system (16) owns the following unique positive equilibrium Uu1∗,u2∗,u3∗, where(18)u1∗=β5,u2∗=β1−κ−1β5κ−5−β4+β5κ−5,u3∗=β4−β5κ−5.The linear system of (16) around the positive equilibrium Uu1∗,u2∗,u3∗ takes the following form:(19)dζu1tdtζ=a1u1t+a2u2t+a2u3t+a3u1t−θ,dζu2tdtζ=a4u1t+a5u2t,dζu3tdtζ=a6u1t+a7u3t,where(20)a1=β1−2κ−1u1∗−u2∗−u3∗1−ϱd,a2=−u1∗1−ϱd,a3=ϱp1−ϱd,a4=u2∗,a5=u1∗−β5,a6=−u3∗,a7=β4−u1∗−2κ5u3∗.The characteristic equation of system (19) takes the form(21)detsζ−a1−a3e−sθ−a2−a2−a4sζ−a50−a60sζ−a7=0.Then,(22)s3ζ+b1s2ζ+b2sζ+b3+c1s2ζ+c2sζ+c3e−sθ=0,where(23)b1=−a1+a5+a7,b2=a5a7+a1a5+a1a7−a2a6−a2a4,b3=a2a5a6+a2a4a7−a1a5a7,c1=−a3,c2=a3a5+a7,c3=−a3a5a7.Whenθ=0, then (22) becomes(24)λ3+b1+c1λ2+b2+c2λ+b3+c3=0.Assuming that(25)Q2b1+c1>0,b1+c1b2+c2>b3+c3,b3+c3b1+c1b2+c2−b3+c3>0,is true, then the three roots λ1,λ2,λ3 of (24) satisfy argλ1>ζπ/2,argλ2>ζπ/2, and argλ3>ζπ/2. By virtue of Lemma 1, we can conclude that the positive equilibrium point Uu1∗,u2∗,u3∗ of system (14) is locally asymptotically stable when θ=0.Assume thats=iρ=ρcosζπ/2+isinπ/2 is the root of equation (22). It follows from (22) that(26)ρ3ζcos3ζπ2+isin3ζπ2+b1ρ2ζcosζπ+isinζπ+b2ρζcosζπ2+isinζπ2+b3+c1ρ2ζcosζπ+isinζπ+c2ρζcosζπ2+isinζπ2+c3×cosρθ−isinρθ=0.Then,(27)G1cosρθ+G2sinρθ=ℋ1,G2cosρθ−G1sinρθ=ℋ2,where(28)G1=d1ρ2ζ+d2ρζ+d3,G2=d4ρ2ζ+d5ρζ,ℋ1=e1ρ3ζ+e2ρ2ζ+e3ρζ+e4,ℋ2=e5ρ3ζ+e6ρ2ζ+e7ρζ,where(29)d1=c1cosζπ,d2=c2cosζπ2,d3=c3,d4=c1sinζπ,d5=c2sinζπ2,e1=−cosζπ2,e2=−b1cosζπ,e3=−b2cosζπ2,e4=−b3,e5=−sinζπ2,e6=−b1sinζπ,e7=−b2sinζπ2.It follows from (27) that(30)cosρθ=ℋ1G1+ℋ2G2G12+G22,(31)G12+G22=ℋ12+ℋ22.By virtue of (28) and (31), one gets(32)d1ρ2ζ+d2ρζ+d32+d4ρ2ζ+d5ρζ2=e1ρ3ζ+e2ρ2ζ+e3ρζ+e42+e5ρ3ζ+e6ρ2ζ+e7ρζ2,which leads to(33)ϵ1ρ6ζ+ϵ2ρ5ζ+ϵ3ρ4ζ+ϵ4ρ3ζ+ϵ5ρ2ζ+ϵ6ρζ+ϵ7=0,where(34)ϵ1=e12+e52,ϵ2=2e1e2+e5e6,ϵ3=e22+e62−d12−d42+2e1e3+e5e7,ϵ4=2e1e4+e2e3+e6e7−d1d2−d4d5,ϵ5=e32+e72−d22−d52+2e2e4−d1d3,ϵ6=2e3e4−d2d3,ϵ7=e42−d32.Set(35)Θρ=ϵ1ρ6ζ+ϵ2ρ5ζ+ϵ3ρ4ζ+ϵ4ρ3ζ+ϵ5ρ2ζ+ϵ6ρζ+ϵ7.Assuming that(36)Q3e4<d3is true, since limρ⟶∞Θρ=+∞, then equation (33) owns at least one real positive root. So equation (22) has at least one pair of pure roots. Making use of Sun et al. [38], one can easily establish the conclusion as follows.Lemma 2.
(a) Supposing thatϵk>0k=1,2,3,4,5,6, equation (22) owns no root with zero real parts for θ≥0. (b) Supposing that Q3 holds and ϵk>0k=1,2,3,4,5, then equation (22) owns a pair of purely imaginary roots ±iρ0 if θ=θ0ll=1,2,…, where(37)θ0l=1ρ0arccosℋ1G1+ℋ2G2G12+G22+2lπ,where l=0,1,…, and ρ0>0 represents the unique zero of Θρ.Denoteθ0=θ00. Now the following hypothesis is given:(38)Q4C1RC2R+C1IC2I>0,where(39)C1R=3ζρ03ζ−1cos3ζ−1π2+2ζc1ρ02ζ−1cos2ζ−1π2+ζc2ρ0ζ−1cosζ−1π2+2ζc1ρ02ζ−1cos2ζ−1π2+ζc2ρ0ζ−1cosζ−1π2cosρ0θ0+sinρ0θ0×2ζc1ρ02ζ−1sin2ζ−1π2+ζc2ρ0ζ−1sinζ−1π2,C1I=3ζρ03ζ−1sin3ζ−1π2+2ζc1ρ02ζ−1sin2ζ−1π2+ζc2ρ0ζ−1sinζ−1π2−2ζc1ρ02ζ−1cos2ζ−1π2+ζc2ρ0ζ−1cosζ−1π2sinρ0θ0+cosρ0θ0×2ζc1ρ02ζ−1sin2ζ−1π2+ζc2ρ0ζ−1sinζ−1π2,C2R=c1ρ02ζcosζπ+c2ρ0ζcosζπ2+c3ρ0sinρ0θ0−c1ρ02ζsinζπ+c2ρ0ζsinζπ2+c3ρ0cosρ0θ0,C2I=c1ρ02ζcosζπ+c2ρ0ζcosζπ2+c3ρ0cosρ0θ0+c1ρ02ζsinζπ+c2ρ0ζsinζπ2+c3ρ0sinρ0θ0.Lemma 3.
Letsθ=ϕ1θ+iϕ2θ be the root of (22) at θ=θ0 satisfying ϕ1θ0=0,ϕ2θ0=ρ0, then Reds/dθ|θ=θ0,ρ=ρ0>0.Proof.
Making use of (22), we get(40)3ζs3ζ−1+2ζb1s2ζ−1+ζb2sζ−1dsdθ+2ζc1s2ζ−1+ζc2sζ−1e−sθdsdθ−e−sθdsdθθ+sc1s2ζ+c2sζ+c3=0,which leads to(41)3ζs3ζ−1+2ζb1s2ζ−1+ζb2sζ−1+2ζc1s2ζ−1+ζc2sζ−1e−sθ−θe−sθc1s2ζ+c2sζ+c3dsdθ=se−sθc1s2ζ+c2sζ+c3.
Then,(42)dsdθ−1=C1sC2s−θs,where(43)C1s=3ζs3ζ−1+2ζb1s2ζ−1+ζb2sζ−1+2ζc1s2ζ−1+ζc2sζ−1e−sθ,C2s=se−sθc1s2ζ+c2sζ+c3.
Then,(44)Redsdθ−1θ=θ0,ρ=ρ0=ReC1sC2sθ=θ0,ρ=ρ0=C1RC2R+C1IC2IC2R2+C2I2.
In view ofQ4, we have(45)Redsdθ−1θ=θ0,ρ=ρ0>0,which completes the proof.
Making use of Lemma1, we can easily obtain the following conclusion.Theorem 2.
Supposing thatQ1–Q4 hold, then the positive equilibrium point Uu1∗,u2∗,u3∗ of system (16) is locally asymptotically stable if the time delay θ lies in the interval 0,θ0 and the Hopf bifurcation phenomenon of system (16) will arise near the positive equilibrium point Uu1∗,u2∗,u3∗ if θ=θ0.Remark 1.
Xu and Wu [17] dealt with the chaos control for an integer-order chaotic chemical reaction model by time-delay feedback control technique. This manuscript deals with the chaos control issue for a fractional-order chaotic chemical reaction model via a fractional-order PDζ controller. The model and the research approach is very different from those in [17]. From this viewpoint, we think that the obtained results and the research method of this manuscript supplement the work of [17] and promote the development of the chaos control theory of fractional-order differential equation to some degree.Remark 2.
In this paper, we use the fractional-orderPDζ controller to control the chaos of the fractional-order chaotic chemical reaction model (4). Compared with the time delay feedback controller, the fractional-order PDζ controller has more adjustable parameters and then can control the chaos of model (4) neatly.


## 5. Example

Consider the following controlled fractional-order chaotic chemical reaction model:(46)dζu1tdtζ=β1u1t−κ−1u12t−u1tu2t−u1tu3t+ϱpu1t−θ+ϱddζu1tdtζ,dζu2tdtζ=u1tu2t−β5u2t,dζu3tdtζ=β4u3t−u1tu3t−k−5u32t,where ζ=0.97,β1=30,κ−1=0.55,β5=9.5,β4=16.5,κ−5=0.5. Let ϱp=0.5,ϱd=0.9. It is not difficult to obtain the unique positive equilibrium point of system (46) is U9.5,10.775,14. By direct computation via MATLAB software, we can easily get ρ0=4.0239 and θ0=0.25. The three assumptions Q1–Q4 of Theorem 2 are easily verified to be right. So we can conclude that the positive equilibrium point U9.5,10.775,14 of system (46) is locally asymptotically stable if the time delay θ lies in the interval 0,0.25 and the Hopf bifurcation phenomenon for system (46) will arise near the positive equilibrium point U9.5,10.775,14 if θ=0.25. In this paper, we use the predictor-correctors approach [39, 41, 42] to discretize system (46) and by virtue of the MATLAB software to carry out numerical simulations. In order to display these results, we select two sets of different delay parameters. Firstly, we choose θ=0.20<θ0=0.25, and the software simulation plots are presented in Figure 2, which implies that u1⟶9.5,u2⟶10.775,u3⟶14 as the time t tends to infinity. From the chemical point of view, the mole fraction of the constituent X will be close to 9.5, the mole fraction of the constituent Y will be close to 10.775, and the mole fraction of the constituent Z will be close to 14. Secondly, we choose θ=0.28>θ0=0.25, and the software simulation plots are presented in Figure 3, which implies that a Hopf bifurcation periodic solution of system (46) will arise near the positive equilibrium point U9.5,10.775,14 as the time t tends to infinity. From the chemical point of view, the mole fraction of the constituent X, the mole fraction of the constituent Y, and the mole fraction of the constituent Z will remain periodically oscillatory situations near the values 9.5, 10.775, 14, respectively. Furthermore, we give the bifurcation plot, which can be seen in Figure 4, to indicate that the bifurcation value of system (46) is 0.25.Figure 2
Computer simulation figures of the controlled fractional-order chaotic chemical reaction model (46) with θ=0.20<θ0=0.25. The blue line represents u1t, the red line represents u2t, and the green line represents u3t.Figure 3
Computer simulation figures of the controlled fractional-order chaotic chemical reaction model (46) with θ=0.28>θ0=0.25. The blue line represents u1t, the red line represents u2t, and the green line represents u3t.Figure 4
Bifurcation plot of the controlled fractional-order chaotic chemical reaction model (46): θ-u1.


## 6. Conclusions

Suppressing the chaotic behavior of nonlinear dynamical systems has been a significant and classic issue in many disciplines. For a long time, the suppression of chaos has attracted much attention from many scholars in mathematics, physics, chemistry, engineering, and numerous other areas. In the present manuscript, based on the earlier publications, we set up a novel fractional-order chaotic chemical reaction model. Taking advantage of an appropriate fractional-orderPDζ controller, we can effectively eliminate the chaotic phenomenon of the involved fractional-order chaotic chemical reaction model. A delay-independent sufficient condition to guarantee the stability and the creation of Hopf bifurcation of the fractional-order controlled chaotic chemical reaction model is built. The exploration manifests that the delay occurring in fractional-order PDζ controller is the key factor in suppressing the chaotic phenomenon of the fractional-order chaotic chemical reaction model. The derived conclusions of this manuscript are entirely new and the exploration approach of this manuscript can also be utilized to inquire into numerous chaos control problem of lots of fractional-order chaotic dynamical systems.



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