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Chaos control for Willamowski–Rössler model of chemical reactions
Chaos, Solitons and Fractals 78 (2015) 1–9
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Chaos control for Willamowski–Rössler model of chemical reactions Ilie Bodale a, Victor Andrei Oancea b,∗ a b
“Alexandru Ioan Cuza” University, Department of Physics, Iasi, Romania “Petru Poni” Institute of Macromolecular Chemistry, Iasi, Romania
a r t i c l e
i n f o
Article history: Received 28 February 2015 Accepted 15 June 2015
Keywords: Minimal Willamowski–Rössler system Synchronization of chaotic systems Chaos control in chemical reactions
a b s t r a c t Real systems evolving towards complex state encounter chaotic behavior. This behavior is very important in chemical processes or in biological structures because it defines the direction of the evolution of the system. From this point of view, the capability of deliberate control of these phenomena has a great practical impact despite the fact that it is very difficult; this is the reason why theoretical models are useful in these situations. In order to obtain chaos control in chemical reactions, the analysis of the dynamics of Willamowski–Rössler system involving the synchronization of two Minimal Willamowski–Rössler (MWR) systems based on the adaptive feedback method of control is presented in this work. As opposed to previous studies where in order to obtain synchronization 3 controllers were used, implying from a practical point of view the control of the concentrations of three chemical species, in this study we showed that the use of just one is sufficient which in practice is important as controlling the concentration of a single chemical species would be much easier. We also showed that the transient time until synchronization depends on initial conditions of two systems, the strength and number of the controllers and we attempted to identify the best conditions for a practical synchronization. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Chemical reaction systems have become one of the favorite domains in the study of nonlinear systems, both experimentally and theoretically. These systems can exhibit nonlinear dynamic characteristics and for this reason they are very important for chemical processes and for biological structures. Among such chemical reactions, the classical Belousov–Zhabotinsky (BZ) reaction is one of the best known systems. The BZ reaction is a family of oscillating chemical reactions; in these reactions bromate ions are reduced in an acidic medium by an organic compound (usually malonic acid) with or without a catalyst (usually cerous and/or ferrous ions).
∗
Corresponding author. Tel.: +40754986215. E-mail address: [email protected] (V.A. Oancea).
http://dx.doi.org/10.1016/j.chaos.2015.06.019 0960-0779/© 2015 Elsevier Ltd. All rights reserved.
Due to the fact that these reactions are autocatalytic, the rate equations are fundamentally nonlinear. This nonlinearity can lead to the spontaneous generation of order and chaos. Ilya Prigogine argued that, far from thermodynamic equilibrium, qualitatively new behaviors appear as the system enters new dynamical regimes. Studies of oscillations, patterns, and chaos in chemical systems constitute a new challenge of chemistry, mathematics and physics. Rössler [1] studied the chaotic kinetics of some chemical reactions mentioned far from the thermodynamic equilibrium. The Rössler system is a minimal model of chaos and the attractor is a prototype for a great variety of chaotic systems, including chemical chaotic systems. Finding a simple model underlying the complex reality of a typical chemical mechanism is a very interesting and difficult experience. On the other hand, the deliberate control of these phenomena has a great practical impact despite the fact that it is very difficult; this is the reason the theoretical models are
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useful in these situations. The objective of control can be the attainment of the reaction steady mode which is suppression of oscillations or the excitation of an oscillatory or even the chaotic mode. For example, chaotic behavior is desirable for combustion processes because it enhances acceleration of the process. Since chaos leads to a better stirring, the reaction is often more uniform and the product is less polluted. In addition, the control of chemical reactions using these models can give the information about the self-control inside the biological structures where the behavior of the dynamic systems is realized by a feedback mechanism. Over the last decade, there has been considerable progress in generalizing the concept of synchronization to include the case of coupled chaotic oscillators especially from technical reasons. When the complete synchronization is achieved, the states of both systems become practically identical, while their dynamics in time remain chaotic. Many examples of synchronization have been documented in the literature, but currently theoretical understanding of the phenomena lags behind experimental studies [2–8]. The main aim of this paper is to study the synchronization of two chemical chaotic systems based on the adaptive feedback method of control. One of the famous ideal chemical models is the Willamowski–Rössler model. Willamowski and Rössler were the first which suggested that the deterministic chaos can be generated by a chemical reaction [9,10]. The Willamowski–Rössler model (WR model) represents a series of chemical reactions in the open system and its mechanism consists in the following elementary steps [11]:
(R1) (R2) (R3) (R4) (R5)
k1
A1 +X ←→ 2X k2
X + Y ←→ 2Y A4 +Y ←→ A2 k4
X + Z ←→ A3 k5
(3)
for this situation the constants are:
A5 +Z ←→ 2Z
These reactions have two catalytic steps: reaction R1 which involves the intermediary compounds X, respectively the reactant Z in reaction R5 as well as the second step, (reaction R2) that involves the specie Y. These reaction mechanisms can be abbreviated as following:
(R6) A1 +A4 +A5 ↔ A2 +A3 where the concentrations of the species A1 …A5 are maintained constant in a big reservoir. As such the reservoir has to be an open system. The time evolution of the intermediary species X, Y and Z can be described by a nonlinear system of equations in its dimensionless form: (1)
A system characterized by the above equations has a chaotic behavior in few situations. One of them occurs for the following constants:
k1 = 30, k−1 = 0.25, k2 = 1, k−2 = 10−4 , k3 = 10, k−3 = 10−3 , k4 = 1, k−4 = 10−3 , k5 = 16.5, k−5 = 0.5,
In order to study the evolution of this complex system we simulated the strange attractor of WR model. The initial conditions for the system (1) were established forx1 (0) = 0.21, x2 (0) = 0.01, x3 (0) = 0.12 and the constants as in (2). In this case, the strange attractor is shown in Fig. 1. Already in their original paper Willamowski and Rössler noted that some of the rate constants could be set to very low values without much compromising the dynamics. They named this type of system the minimal version of this model which still allows chaotic behavior. This consists in setting k−2 = 0 k−3 = 0 and k−4 = 0. In this case, we have the Minimal Willamowski–Rössler model (MWR model) described by the next three equations [12]:
x˙ 1 = k1 x1 − k−1 x21 − k2 x1 x2 − k4 x1 x3 x˙ 2 = k2 x1 x2 − k3 x2 x˙ 3 = −k4 x1 x3 + k5 x3 − k−5 x23
k3
x˙ 1 = k1 x1 − k−1 x21 − k2 x1 x2 + k−2 x22 − k4 x1 x3 + k−4 x˙ 2 = k2 x1 x2 − k−2 x22 − k3 x2 + k−3 x˙ 3 = −k4 x1 x3 + k5 x3 − k−5 x23 + k−4
Fig. 1. The 3D attractor for x1 (0) = 0.21, x2 (0) = 0.01 and x3 (0) = 0.12 initial conditions.
(2)
k1 = 30, k2 = 1, k3 = 10, k4 = 1, k5 = 16.5 and k−5 = 0.5 (4) 2. Chaotic dynamics of the Minimal Willamowski–Rössler system We obtained certain conditions in which the system is in a steady state and in which it has an oscillatory or chaotic behavior based on the Lyapunov exponents. The behavior of the chemical system can be controlled through the modification of control parameters. In order to determine the dynamics of this system we selected k_1 as a control parameter. In this case, for k_1 = 0.5, the Minimal Willamowski–Rössler system is stationary ifx1 = 10, x2 = 12 and x3 = 13. For the same initial conditions as in Fig. 1 and k_1 = 0.5, the 2D attractor for MWR system has the form as in Fig. 2. We can predict the status of the reaction at any time by analyzing the dynamics of the MWR system. For the above parameters we display the evolution of the MWR system in Fig. 3 where the evolution in time is represented in time units (t.u.). The chaotic behavior of the system is sustained by the evolution of Lyapunov exponents [13]. Under the same condition, the MWR model has two positive Lyapunov exponents:
I. Bodale, V.A. Oancea / Chaos, Solitons and Fractals 78 (2015) 1–9
Fig. 2. The 2D attractor for MWR system for x1 (0) = 0.21, x2 (0) = 0.01 and x3 (0) = 0.12 initial conditions and control parameter k-1 = 0.5.
3
Fig. 4. The Lyapunov exponents for the model (3) and the control parameter k-1 = 0.5.
Fig. 3. The time evolution of x1 (t) for control parameter k-1 = 0.5.
λ1 = 0.45818 and λ2 = 0.0081414 (which is very close to 0). This satisfies the condition that in order to become chaotic system (3) has at least one positive Lyapunov exponent as can be seen from Fig. 4. Increasing the control parameter at k-1 = 1 we noticed that the 2D attractor has another behavior described by a limit cycle (Fig. 5). Fig. 6 shows the changes of the variable x1 as a function of time for k-1 = 1 when the system was located at the periodic1 oscillation. This result is in accordance with the bifurcation diagram obtained by Lei et al. [14,15] which highlighted the period-1 of oscillation for the value of control parameter k-1 = 1. For this system an interesting behavior is obtained (Figs. 7 and 8) for k-1 = 0.3 that wasn’t presented by Lei’s group. The trajectory of the 2D attractor is near axis and scanning the inside of diagram (see Fig. 7). From Fig. 8 we can see that only two Lyapunov exponents are positive. This fact means that the system is still chaotic videlicet the chemical reaction is in progress.
Fig. 5. The 2D attractor in (x1 , x2 ) plane of MWR system for control parameter k-1 = 1.
Fig. 6. The evolution of x1 (t) for control parameter k-1 = 1.
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Fig. 7. The 2D attractor for control parameter k-1 = 0.3.
Fig. 9. The 2D attractor for control parameter k-1 = 0.2.
Fig. 10. The evolution of Lyapunov exponents for control parameter k-1 = 0.2. Fig. 8. The Lyapunov exponents for control parameter k-1 = 0.3.
system is: A specific situation is obtained for k-1 = 0.2, also missing in study of Lei’s group (Figs. 9 and 10). Fig. 10 shows that all the Lyapunov exponents are negative. In this situation the system goes to a steady state. This behavior is supported by the time evolutions of the concentrations of the intermediary species given in Fig. 11 (a) and (b). These results show the transition from chaos to order for a time over 10 t.u. The complex chemical chaotic system has the ability to self-regulate through a simple feedback loop. Some work has denoted that feedback interactions could impart precision, robustness and versatility to intercellular signals during animal development [16]. 3. The modified Minimal Willamowski–Rössler system The Minimal Willamowski–Rössler system was modified by inversing the variable x2 and x3 . In this case the non-linear
x˙ 1 = 30x1 − 0.5x21 − x1 x2 − x1 x3 x˙ 2 = 16.5x2 − x1 x2 − 0.5x22 x˙ 3 = x1 x3 − 10x3
(5)
The 3D attractor of this system has the shape as in Fig. 12 for initial conditions x1 (0) = 0.21,x2 (0) = 0.01 and x3 (0) = 0.12. We also analyzed the dynamics of the new system in order to predict the direction of the reaction in any moment of chemical transformation (Fig. 13). 4. Synchronization of two chaotic systems To synchronize two identical chemical systems we followed the method, proposed by Huang [17], Hu and Xu [18], Guo et al. [19], and Guo and Li [20] and used by Oancea et al. [21], based on Lyapunov–Lasalle theory. Let the driver (master) system be given as:
x˙ = f (x, t ), where x = (x1 , x2 ...)T ∈ Rn
(6)
I. Bodale, V.A. Oancea / Chaos, Solitons and Fractals 78 (2015) 1–9
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Fig. 11. (a) Time evolution of x1 (t) for k-1 = 0.2 and (b) the time evolution of x3 (t) for k-1 = 0.2.
is the state vector of the system and
f = ( f1 , f2 ..)T ∈ Rn
(7)
is the non-linear vector function. The receiver (slave) system, with the variable y ∈ Rn , will be described by:
y˙ = f (y, t ) + z(y − x), where z(z1 , z2 , ...) is the controller (8) If the error vector is
e = y − x,
(9)
the objective of synchronization is to make Fig. 12. The 3D attractor for the modified MWR system at initial conditions x1 (0) = 0.21, x2 (0) = 0.01 and x3 (0) = 0.12.
lim e(t ) → 0 t → +∞ The feedback strength will be duly adapted according to the following law:
z˙ i = −γi e2i , i = 1, 2 . . . n
(10)
where γ i , i = 1,2, … , n are arbitrary positive constants and, in general, we select γ i = 1. 5. Synchronization of two MWR systems Using this method of synchronization, the slave system for MWR according to (8) will be:
y˙ 1 = 30y1 − 0.5y21 − y1 y2 − y1 y3 + z1 (y1 − x1 ) y˙ 2 = y1 y2 − 10y2 + z2 (y2 − x2 )
(11)
y˙ 3 = 16.5y3 − y1 y3 + z3 (y3 − x3 ) and the control strength, according to (10):
Fig. 13. The time evolution of x1 (t), for initial conditions x1 (0) = 0.21, x2 (0) = 0.01 and x3 (0) = 0.12.
z˙ 1 = −(y1 − x1 )
2
z˙ 2 = −(y2 − x2 )
2
z˙ 3 = −(y3 − x3 )
2
(12)
Figs. 14–17 demonstrate the synchronization of the two MWR systems was achieved for x1 (0) = 1, x2 (0) = 1, x3 (0) = 1,
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Fig. 14. The synchronization of two MWR systems [x3 (t) – black, y3 (t) – blue] for x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1, z3 (0) = 1 as initial conditions. Synchronization was obtained at about 2.5 t.u. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 16. Phase portrait of (x1 , x2 ,)-black and (x1 ,y1 )-red for two MWR systems [with x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1 and z3 (0) = 1 as initial conditions]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 15. Synchronization errors between master and slave systems [with x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1 and z3 (0) = 1 as initial conditions].
Fig. 17. The control strength z1 [ with x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1, and z3 (0) = 1 as initial conditions].
y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1, and z3 (0) = 1 initial conditions. In these figures we colored the evolution of e1 with black, e2 with green and e3 with red. The feedback gain tends to a negative constant (z→z0 , for t→ ∞ i.e. y→x and the two systems are synchronized according Lyapunov–Lasalle theory) as shown in Fig. 17. If different initial conditions are used, even in the case that the values are very close, the time until synchronization is achieved can be longer than in the previous case. In Fig. 18 we can observe synchronization using values close to the initial takes a longer time than in the previous situation presented in Fig. 14. The synchronization time depends on the strength of the controllers. If for example the control functions in the initial conditions are given as:
dz1 2 = −0, 1∗ (y1 − x1 ) dt
dz2 2 = −0, 1∗ (y2 − x2 ) dt dz3 2 = −0, 1∗ (y3 − x3 ) dt Then synchronization is achieved much later as it can be observed from Fig. 19. On the contrary if the intensity of the control functions is taken as:
dz1 2 = −10∗ (y1 − x1 ) dt dz2 2 = −10∗ (y2 − x2 ) dt dz3 2 = −10∗ (y3 − x3 ) dt Then the time until synchronization is decreased, as it can be observed by comparing Figs. 14, 19 and 20.
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Fig 18. The synchronization of two MWR systems [x3 (t) – black, y3 (t) – blue] for x1 (0) = 0.21, x2 (0) = 0.01, x3 (0) = 0.12, y1 (0) = 0.2, y2 (0) = 0.011, y3 (0) = 0.1, z1 (0) = 1, z2 (0) = 1, z3 (0) = 1. Synchronization was obtained at about 3.5 t.u. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig 19. The synchronization of two MWR systems [x3 (t) – black, y3 (t) – blue] for x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1, z3 (0) = 1 as initial conditions. Synchronization was obtained at about 3.5 t.u. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
From a practical point of view, the synchronization using a single controller is of great interest. Huang [17], by testing the chaotic systems including the Lorenz system, Rossler system, Chua’s circuit, and the Sprott’s collection of the simplest chaotic flows, found that a single controller can be used to achieve identical synchronization of a three-dimensional system. For the Lorenz system this is possible only by adding the controller in the second equation. For two MWR systems we achieved the synchronization if one controller is applied only in the first or in the third equation. The synchronization errors between master and slave when one controller is applied in the first equation of the system (11) are shown in Fig. 21 and when one controller is applied in the second equation in Fig. 22.
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Fig 20. The synchronization of two MWR systems [x3 (t) – black, y3 (t) – blue] for x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1, z3 (0) = 1 as initial conditions. Synchronization was obtained at about 1.5 t.u. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 21. Synchronization errors between master and slave MWR systems [with x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, and z1 (0) = 1 as initial conditions].
In the case of the first and the third equation that correspond to reactions R1 and R5 we encounter a single intermediary chemical species and in order to achieve synchronization we intervene with just one controller. In the case of the second equation, which corresponds to reaction R2, we encounter two intermediate chemical species x and y and as a result in order to obtain synchronization the use of at least two controllers would be required, as it would be unachievable by using just one. Finally, in Fig. 23 is shown the synchronization of the two modified MWR systems. The evolutions of the three parameters reach the steady state in a relatively short interval (2.3 t.u.). After this point the two modified MWR systems are synchronized.
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Fig. 22. Synchronization errors between master and slave MWR systems [with x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, and z2 (0) = 1 as initial conditions].
be applied in the first or in the third equation. The control method described in this paper is very easy and might be useful in the case of the other chaotic systems which will be treated in future. In the current paper we focused on the synchronization of two Willamowski–Rössler systems with the aim of determining the favorable conditions for a practical implementation of this synchronization. In the previous studies of the Willamowski–Rössler model, three controllers were used in obtaining synchronization which implies the simultaneous control of the concentrations of three chemical species. In this study in order to achieve synchronization we used a single controller which implies from a practical point of view the control of the concentration of a single chemical species and which would be easier to implement. Experimentally the synchronization of electric oscillators was achieved but not for chemical ones. It is our belief that this can be accomplished and that is why we focused on chemical oscillators and in the future we intend to achieve this type of synchronization in a practical setup. Acknowledgment This work was supported by the European Social Fund in Romania, under the responsibility of the Managing Authority for the Sectored Operational Program for Human Resources Development 2007–2013 [grant POSDRU/159/1.5/S/137750, project “Doctoral and Postdoctoral programs support for increased competitiveness in Exact Sciences research”]. References
Fig. 23. Synchronization errors for two modified MWR systems [with x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1, and z3 (0) = 1 as initial conditions].
6. Conclusions In order to obtain chaos control in chemical reactions, first we analyzed the dynamics of the Willamowski–Rössler system and compared our results with literature. Afterwards we realized the synchronization of two chaotic Minimal Willamowski–Rössler (MWR) systems using an adaptive feedback method. The transient time until synchronization depends on initial conditions of two systems, the strength of the controllers and their number. As such we can control this chemical system in accordance with recent debates of Wang and Chen [22] about full global synchronization and partial synchronization in a system of two or three coupled chemical chaotic oscillators. Using three controllers for the three differential equations we were able to quickly obtain the synchronization. By reducing the controllers from three to one, the controller must
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Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Chaos control for Willamowski–Rössler model of chemical reactions Ilie Bodale a, Victor Andrei Oancea b,∗ a b
“Alexandru Ioan Cuza” University, Department of Physics, Iasi, Romania “Petru Poni” Institute of Macromolecular Chemistry, Iasi, Romania
a r t i c l e
i n f o
Article history: Received 28 February 2015 Accepted 15 June 2015
Keywords: Minimal Willamowski–Rössler system Synchronization of chaotic systems Chaos control in chemical reactions
a b s t r a c t Real systems evolving towards complex state encounter chaotic behavior. This behavior is very important in chemical processes or in biological structures because it defines the direction of the evolution of the system. From this point of view, the capability of deliberate control of these phenomena has a great practical impact despite the fact that it is very difficult; this is the reason why theoretical models are useful in these situations. In order to obtain chaos control in chemical reactions, the analysis of the dynamics of Willamowski–Rössler system involving the synchronization of two Minimal Willamowski–Rössler (MWR) systems based on the adaptive feedback method of control is presented in this work. As opposed to previous studies where in order to obtain synchronization 3 controllers were used, implying from a practical point of view the control of the concentrations of three chemical species, in this study we showed that the use of just one is sufficient which in practice is important as controlling the concentration of a single chemical species would be much easier. We also showed that the transient time until synchronization depends on initial conditions of two systems, the strength and number of the controllers and we attempted to identify the best conditions for a practical synchronization. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Chemical reaction systems have become one of the favorite domains in the study of nonlinear systems, both experimentally and theoretically. These systems can exhibit nonlinear dynamic characteristics and for this reason they are very important for chemical processes and for biological structures. Among such chemical reactions, the classical Belousov–Zhabotinsky (BZ) reaction is one of the best known systems. The BZ reaction is a family of oscillating chemical reactions; in these reactions bromate ions are reduced in an acidic medium by an organic compound (usually malonic acid) with or without a catalyst (usually cerous and/or ferrous ions).
∗
Corresponding author. Tel.: +40754986215. E-mail address: [email protected] (V.A. Oancea).
http://dx.doi.org/10.1016/j.chaos.2015.06.019 0960-0779/© 2015 Elsevier Ltd. All rights reserved.
Due to the fact that these reactions are autocatalytic, the rate equations are fundamentally nonlinear. This nonlinearity can lead to the spontaneous generation of order and chaos. Ilya Prigogine argued that, far from thermodynamic equilibrium, qualitatively new behaviors appear as the system enters new dynamical regimes. Studies of oscillations, patterns, and chaos in chemical systems constitute a new challenge of chemistry, mathematics and physics. Rössler [1] studied the chaotic kinetics of some chemical reactions mentioned far from the thermodynamic equilibrium. The Rössler system is a minimal model of chaos and the attractor is a prototype for a great variety of chaotic systems, including chemical chaotic systems. Finding a simple model underlying the complex reality of a typical chemical mechanism is a very interesting and difficult experience. On the other hand, the deliberate control of these phenomena has a great practical impact despite the fact that it is very difficult; this is the reason the theoretical models are
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useful in these situations. The objective of control can be the attainment of the reaction steady mode which is suppression of oscillations or the excitation of an oscillatory or even the chaotic mode. For example, chaotic behavior is desirable for combustion processes because it enhances acceleration of the process. Since chaos leads to a better stirring, the reaction is often more uniform and the product is less polluted. In addition, the control of chemical reactions using these models can give the information about the self-control inside the biological structures where the behavior of the dynamic systems is realized by a feedback mechanism. Over the last decade, there has been considerable progress in generalizing the concept of synchronization to include the case of coupled chaotic oscillators especially from technical reasons. When the complete synchronization is achieved, the states of both systems become practically identical, while their dynamics in time remain chaotic. Many examples of synchronization have been documented in the literature, but currently theoretical understanding of the phenomena lags behind experimental studies [2–8]. The main aim of this paper is to study the synchronization of two chemical chaotic systems based on the adaptive feedback method of control. One of the famous ideal chemical models is the Willamowski–Rössler model. Willamowski and Rössler were the first which suggested that the deterministic chaos can be generated by a chemical reaction [9,10]. The Willamowski–Rössler model (WR model) represents a series of chemical reactions in the open system and its mechanism consists in the following elementary steps [11]:
(R1) (R2) (R3) (R4) (R5)
k1
A1 +X ←→ 2X k2
X + Y ←→ 2Y A4 +Y ←→ A2 k4
X + Z ←→ A3 k5
(3)
for this situation the constants are:
A5 +Z ←→ 2Z
These reactions have two catalytic steps: reaction R1 which involves the intermediary compounds X, respectively the reactant Z in reaction R5 as well as the second step, (reaction R2) that involves the specie Y. These reaction mechanisms can be abbreviated as following:
(R6) A1 +A4 +A5 ↔ A2 +A3 where the concentrations of the species A1 …A5 are maintained constant in a big reservoir. As such the reservoir has to be an open system. The time evolution of the intermediary species X, Y and Z can be described by a nonlinear system of equations in its dimensionless form: (1)
A system characterized by the above equations has a chaotic behavior in few situations. One of them occurs for the following constants:
k1 = 30, k−1 = 0.25, k2 = 1, k−2 = 10−4 , k3 = 10, k−3 = 10−3 , k4 = 1, k−4 = 10−3 , k5 = 16.5, k−5 = 0.5,
In order to study the evolution of this complex system we simulated the strange attractor of WR model. The initial conditions for the system (1) were established forx1 (0) = 0.21, x2 (0) = 0.01, x3 (0) = 0.12 and the constants as in (2). In this case, the strange attractor is shown in Fig. 1. Already in their original paper Willamowski and Rössler noted that some of the rate constants could be set to very low values without much compromising the dynamics. They named this type of system the minimal version of this model which still allows chaotic behavior. This consists in setting k−2 = 0 k−3 = 0 and k−4 = 0. In this case, we have the Minimal Willamowski–Rössler model (MWR model) described by the next three equations [12]:
x˙ 1 = k1 x1 − k−1 x21 − k2 x1 x2 − k4 x1 x3 x˙ 2 = k2 x1 x2 − k3 x2 x˙ 3 = −k4 x1 x3 + k5 x3 − k−5 x23
k3
x˙ 1 = k1 x1 − k−1 x21 − k2 x1 x2 + k−2 x22 − k4 x1 x3 + k−4 x˙ 2 = k2 x1 x2 − k−2 x22 − k3 x2 + k−3 x˙ 3 = −k4 x1 x3 + k5 x3 − k−5 x23 + k−4
Fig. 1. The 3D attractor for x1 (0) = 0.21, x2 (0) = 0.01 and x3 (0) = 0.12 initial conditions.
(2)
k1 = 30, k2 = 1, k3 = 10, k4 = 1, k5 = 16.5 and k−5 = 0.5 (4) 2. Chaotic dynamics of the Minimal Willamowski–Rössler system We obtained certain conditions in which the system is in a steady state and in which it has an oscillatory or chaotic behavior based on the Lyapunov exponents. The behavior of the chemical system can be controlled through the modification of control parameters. In order to determine the dynamics of this system we selected k_1 as a control parameter. In this case, for k_1 = 0.5, the Minimal Willamowski–Rössler system is stationary ifx1 = 10, x2 = 12 and x3 = 13. For the same initial conditions as in Fig. 1 and k_1 = 0.5, the 2D attractor for MWR system has the form as in Fig. 2. We can predict the status of the reaction at any time by analyzing the dynamics of the MWR system. For the above parameters we display the evolution of the MWR system in Fig. 3 where the evolution in time is represented in time units (t.u.). The chaotic behavior of the system is sustained by the evolution of Lyapunov exponents [13]. Under the same condition, the MWR model has two positive Lyapunov exponents:
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Fig. 2. The 2D attractor for MWR system for x1 (0) = 0.21, x2 (0) = 0.01 and x3 (0) = 0.12 initial conditions and control parameter k-1 = 0.5.
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Fig. 4. The Lyapunov exponents for the model (3) and the control parameter k-1 = 0.5.
Fig. 3. The time evolution of x1 (t) for control parameter k-1 = 0.5.
λ1 = 0.45818 and λ2 = 0.0081414 (which is very close to 0). This satisfies the condition that in order to become chaotic system (3) has at least one positive Lyapunov exponent as can be seen from Fig. 4. Increasing the control parameter at k-1 = 1 we noticed that the 2D attractor has another behavior described by a limit cycle (Fig. 5). Fig. 6 shows the changes of the variable x1 as a function of time for k-1 = 1 when the system was located at the periodic1 oscillation. This result is in accordance with the bifurcation diagram obtained by Lei et al. [14,15] which highlighted the period-1 of oscillation for the value of control parameter k-1 = 1. For this system an interesting behavior is obtained (Figs. 7 and 8) for k-1 = 0.3 that wasn’t presented by Lei’s group. The trajectory of the 2D attractor is near axis and scanning the inside of diagram (see Fig. 7). From Fig. 8 we can see that only two Lyapunov exponents are positive. This fact means that the system is still chaotic videlicet the chemical reaction is in progress.
Fig. 5. The 2D attractor in (x1 , x2 ) plane of MWR system for control parameter k-1 = 1.
Fig. 6. The evolution of x1 (t) for control parameter k-1 = 1.
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Fig. 7. The 2D attractor for control parameter k-1 = 0.3.
Fig. 9. The 2D attractor for control parameter k-1 = 0.2.
Fig. 10. The evolution of Lyapunov exponents for control parameter k-1 = 0.2. Fig. 8. The Lyapunov exponents for control parameter k-1 = 0.3.
system is: A specific situation is obtained for k-1 = 0.2, also missing in study of Lei’s group (Figs. 9 and 10). Fig. 10 shows that all the Lyapunov exponents are negative. In this situation the system goes to a steady state. This behavior is supported by the time evolutions of the concentrations of the intermediary species given in Fig. 11 (a) and (b). These results show the transition from chaos to order for a time over 10 t.u. The complex chemical chaotic system has the ability to self-regulate through a simple feedback loop. Some work has denoted that feedback interactions could impart precision, robustness and versatility to intercellular signals during animal development [16]. 3. The modified Minimal Willamowski–Rössler system The Minimal Willamowski–Rössler system was modified by inversing the variable x2 and x3 . In this case the non-linear
x˙ 1 = 30x1 − 0.5x21 − x1 x2 − x1 x3 x˙ 2 = 16.5x2 − x1 x2 − 0.5x22 x˙ 3 = x1 x3 − 10x3
(5)
The 3D attractor of this system has the shape as in Fig. 12 for initial conditions x1 (0) = 0.21,x2 (0) = 0.01 and x3 (0) = 0.12. We also analyzed the dynamics of the new system in order to predict the direction of the reaction in any moment of chemical transformation (Fig. 13). 4. Synchronization of two chaotic systems To synchronize two identical chemical systems we followed the method, proposed by Huang [17], Hu and Xu [18], Guo et al. [19], and Guo and Li [20] and used by Oancea et al. [21], based on Lyapunov–Lasalle theory. Let the driver (master) system be given as:
x˙ = f (x, t ), where x = (x1 , x2 ...)T ∈ Rn
(6)
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Fig. 11. (a) Time evolution of x1 (t) for k-1 = 0.2 and (b) the time evolution of x3 (t) for k-1 = 0.2.
is the state vector of the system and
f = ( f1 , f2 ..)T ∈ Rn
(7)
is the non-linear vector function. The receiver (slave) system, with the variable y ∈ Rn , will be described by:
y˙ = f (y, t ) + z(y − x), where z(z1 , z2 , ...) is the controller (8) If the error vector is
e = y − x,
(9)
the objective of synchronization is to make Fig. 12. The 3D attractor for the modified MWR system at initial conditions x1 (0) = 0.21, x2 (0) = 0.01 and x3 (0) = 0.12.
lim e(t ) → 0 t → +∞ The feedback strength will be duly adapted according to the following law:
z˙ i = −γi e2i , i = 1, 2 . . . n
(10)
where γ i , i = 1,2, … , n are arbitrary positive constants and, in general, we select γ i = 1. 5. Synchronization of two MWR systems Using this method of synchronization, the slave system for MWR according to (8) will be:
y˙ 1 = 30y1 − 0.5y21 − y1 y2 − y1 y3 + z1 (y1 − x1 ) y˙ 2 = y1 y2 − 10y2 + z2 (y2 − x2 )
(11)
y˙ 3 = 16.5y3 − y1 y3 + z3 (y3 − x3 ) and the control strength, according to (10):
Fig. 13. The time evolution of x1 (t), for initial conditions x1 (0) = 0.21, x2 (0) = 0.01 and x3 (0) = 0.12.
z˙ 1 = −(y1 − x1 )
2
z˙ 2 = −(y2 − x2 )
2
z˙ 3 = −(y3 − x3 )
2
(12)
Figs. 14–17 demonstrate the synchronization of the two MWR systems was achieved for x1 (0) = 1, x2 (0) = 1, x3 (0) = 1,
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Fig. 14. The synchronization of two MWR systems [x3 (t) – black, y3 (t) – blue] for x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1, z3 (0) = 1 as initial conditions. Synchronization was obtained at about 2.5 t.u. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 16. Phase portrait of (x1 , x2 ,)-black and (x1 ,y1 )-red for two MWR systems [with x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1 and z3 (0) = 1 as initial conditions]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 15. Synchronization errors between master and slave systems [with x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1 and z3 (0) = 1 as initial conditions].
Fig. 17. The control strength z1 [ with x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1, and z3 (0) = 1 as initial conditions].
y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1, and z3 (0) = 1 initial conditions. In these figures we colored the evolution of e1 with black, e2 with green and e3 with red. The feedback gain tends to a negative constant (z→z0 , for t→ ∞ i.e. y→x and the two systems are synchronized according Lyapunov–Lasalle theory) as shown in Fig. 17. If different initial conditions are used, even in the case that the values are very close, the time until synchronization is achieved can be longer than in the previous case. In Fig. 18 we can observe synchronization using values close to the initial takes a longer time than in the previous situation presented in Fig. 14. The synchronization time depends on the strength of the controllers. If for example the control functions in the initial conditions are given as:
dz1 2 = −0, 1∗ (y1 − x1 ) dt
dz2 2 = −0, 1∗ (y2 − x2 ) dt dz3 2 = −0, 1∗ (y3 − x3 ) dt Then synchronization is achieved much later as it can be observed from Fig. 19. On the contrary if the intensity of the control functions is taken as:
dz1 2 = −10∗ (y1 − x1 ) dt dz2 2 = −10∗ (y2 − x2 ) dt dz3 2 = −10∗ (y3 − x3 ) dt Then the time until synchronization is decreased, as it can be observed by comparing Figs. 14, 19 and 20.
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Fig 18. The synchronization of two MWR systems [x3 (t) – black, y3 (t) – blue] for x1 (0) = 0.21, x2 (0) = 0.01, x3 (0) = 0.12, y1 (0) = 0.2, y2 (0) = 0.011, y3 (0) = 0.1, z1 (0) = 1, z2 (0) = 1, z3 (0) = 1. Synchronization was obtained at about 3.5 t.u. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig 19. The synchronization of two MWR systems [x3 (t) – black, y3 (t) – blue] for x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1, z3 (0) = 1 as initial conditions. Synchronization was obtained at about 3.5 t.u. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
From a practical point of view, the synchronization using a single controller is of great interest. Huang [17], by testing the chaotic systems including the Lorenz system, Rossler system, Chua’s circuit, and the Sprott’s collection of the simplest chaotic flows, found that a single controller can be used to achieve identical synchronization of a three-dimensional system. For the Lorenz system this is possible only by adding the controller in the second equation. For two MWR systems we achieved the synchronization if one controller is applied only in the first or in the third equation. The synchronization errors between master and slave when one controller is applied in the first equation of the system (11) are shown in Fig. 21 and when one controller is applied in the second equation in Fig. 22.
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Fig 20. The synchronization of two MWR systems [x3 (t) – black, y3 (t) – blue] for x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1, z3 (0) = 1 as initial conditions. Synchronization was obtained at about 1.5 t.u. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 21. Synchronization errors between master and slave MWR systems [with x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, and z1 (0) = 1 as initial conditions].
In the case of the first and the third equation that correspond to reactions R1 and R5 we encounter a single intermediary chemical species and in order to achieve synchronization we intervene with just one controller. In the case of the second equation, which corresponds to reaction R2, we encounter two intermediate chemical species x and y and as a result in order to obtain synchronization the use of at least two controllers would be required, as it would be unachievable by using just one. Finally, in Fig. 23 is shown the synchronization of the two modified MWR systems. The evolutions of the three parameters reach the steady state in a relatively short interval (2.3 t.u.). After this point the two modified MWR systems are synchronized.
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Fig. 22. Synchronization errors between master and slave MWR systems [with x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, and z2 (0) = 1 as initial conditions].
be applied in the first or in the third equation. The control method described in this paper is very easy and might be useful in the case of the other chaotic systems which will be treated in future. In the current paper we focused on the synchronization of two Willamowski–Rössler systems with the aim of determining the favorable conditions for a practical implementation of this synchronization. In the previous studies of the Willamowski–Rössler model, three controllers were used in obtaining synchronization which implies the simultaneous control of the concentrations of three chemical species. In this study in order to achieve synchronization we used a single controller which implies from a practical point of view the control of the concentration of a single chemical species and which would be easier to implement. Experimentally the synchronization of electric oscillators was achieved but not for chemical ones. It is our belief that this can be accomplished and that is why we focused on chemical oscillators and in the future we intend to achieve this type of synchronization in a practical setup. Acknowledgment This work was supported by the European Social Fund in Romania, under the responsibility of the Managing Authority for the Sectored Operational Program for Human Resources Development 2007–2013 [grant POSDRU/159/1.5/S/137750, project “Doctoral and Postdoctoral programs support for increased competitiveness in Exact Sciences research”]. References
Fig. 23. Synchronization errors for two modified MWR systems [with x1 (0) = 1, x2 (0) = 1, x3 (0) = 1, y1 (0) = 1.5, y2 (0) = 1.5, y3 (0) = 1.5, z1 (0) = 1, z2 (0) = 1, and z3 (0) = 1 as initial conditions].
6. Conclusions In order to obtain chaos control in chemical reactions, first we analyzed the dynamics of the Willamowski–Rössler system and compared our results with literature. Afterwards we realized the synchronization of two chaotic Minimal Willamowski–Rössler (MWR) systems using an adaptive feedback method. The transient time until synchronization depends on initial conditions of two systems, the strength of the controllers and their number. As such we can control this chemical system in accordance with recent debates of Wang and Chen [22] about full global synchronization and partial synchronization in a system of two or three coupled chemical chaotic oscillators. Using three controllers for the three differential equations we were able to quickly obtain the synchronization. By reducing the controllers from three to one, the controller must
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