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doi: 10.3934/naco.2021025
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Adaptive controllability of microscopic chaos generated in chemical reactor system using anti-synchronization strategy

Department of Mathematics, Jamia Millia Islamia, India

* Corresponding author: harindri20dbc@gmail.com

Received  December 2020 Revised  June 2021 Early access July 2021

In this manuscript, we design a methodology to investigate the anti-synchronization scheme in chaotic chemical reactor system using adaptive control method (ACM). Initially, an ACM has been proposed and analysed systematically for controlling the microscopic chaos found in the discussed system which is essentially described by employing Lyapunov stability theory (LST). The required asymptotic stability criterion of the state variables of the discussed system having unknown parameters is derived by designing appropriate control functions and parameter updating laws. In addition, numerical simulation results in MATLAB software are performed to illustrate the effective presentation of the considered strategy. Simulations outcomes correspond that the primal aim of chaos control in the given system have been attained computationally.

Citation: Taqseer Khan, Harindri Chaudhary. Adaptive controllability of microscopic chaos generated in chemical reactor system using anti-synchronization strategy. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2021025
References:
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K. Bouallegue, A new class of neural networks and its applications, Neurocomputing, 249 (2017), 28-47. 

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M. Chen and Z. Han, Controlling and synchronizing chaotic genesio system via nonlinear feedback control, Chaos, Solitons & Fractals, 17 (2003), 709-716.  doi: 10.1016/S0960-0779(02)00487-3.

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H. Delavari and M. Mohadeszadeh, Hybrid complex projective synchronization of complex chaotic systems using active control technique with nonlinearity in the control input, Journal of Control Engineering and Applied Informatics, 20 (2018), 67-74. 

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Z. Ding and Y. Shen, Projective synchronization of nonidentical fractional-order neural networks based on sliding mode controller, Neural Networks, 76 (2016), 97-105. 

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J.-P. Eckmann, S. O. Kamphorst, D. Ruelle and S. Ciliberto, Liapunov exponents from time series, Physical Review A, 34 (1986), 4971. doi: 10.1103/PhysRevA.34.4971.

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D. GhoshA. MukherjeeN. R. Das and B. N. Biswas, Generation & control of chaos in a single loop optoelectronic oscillator, Optik, 165 (2018), 275-287. 

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S. K. Han, C. Kurrer and Y. Kuramoto, Dephasing and bursting in coupled neural oscillators, Physical Review Letters, 75 (1995), 3190.

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M. HuY. YangZ. Xu and L. Guo, Hybrid projective synchronization in a chaotic complex nonlinear system, Mathematics and Computers in Simulation, 79 (2008), 449-457.  doi: 10.1016/j.matcom.2008.01.047.

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A. W. Hubler, Adaptive control of chaotic system, Helv. Phys. Acta, 62 (1989), 343-346. 

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T. Khan and H. Chaudhary, Estimation and identifiability of parameters for generalized lotka-volterra biological systems using adaptive controlled combination difference anti-synchronization, Differential Equations and Dynamical Systems, Special Issue, 28 (2020), 515-526.  doi: 10.1007/s12591-020-00534-8.

[11]

A. Khan and H. Chaudhary, Adaptive control and hybrid projective combination synchronization of chaos generated by generalized lotka-volterra biological systems, Bloomsbury India, (2019), 174.

[12]

A. Khan and H. Chaudhary, Hybrid projective combination-combination synchronization in non-identical hyperchaotic systems using adaptive control, Arabian Journal of Mathematics, 9 (2020), 597-611.  doi: 10.1007/s40065-020-00279-w.

[13]

C. Li and X. Liao, Complete and lag synchronization of hyperchaotic systems using small impulses, Chaos, Solitons & Fractals, 22 (2004), 857-867.  doi: 10.1016/j.chaos.2004.03.006.

[14]

D. Li and X. Zhang, Impulsive synchronization of fractional order chaotic systems with time-delay, Neurocomputing, 216 (2016), 39-44. 

[15]

G.-H. Li and S.-P. Zhou, Anti-synchronization in different chaotic systems, Chaos, Solitons & Fractals, 32 (2007), 516-520.  doi: 10.1016/j.chaos.2005.12.009.

[16]

G.-H. Li, Modified projective synchronization of chaotic system, Chaos, Solitons & Fractals, 32 (2007), 1786-1790.  doi: 10.1016/j.chaos.2005.12.009.

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S.-Y. LiC.-H. YangC.-T. LinL.-W. Ko and T.-T. Chiu, Adaptive synchronization of chaotic systems with unknown parameters via new backstepping strategy, Nonlinear Dynamics, 70 (2012), 2129-2143.  doi: 10.1007/s11071-012-0605-x.

[18]

Z. Li and D. Xu, A secure communication scheme using projective chaos synchronization, Chaos, Solitons & Fractals, 22 (2004), 477-481.  doi: 10.1016/j.chaos.2004.02.004.

[19]

T.-L. Liao and S.-H. Tsai, Adaptive synchronization of chaotic systems and its application to secure communications, Chaos, Solitons & Fractals, 11 (2000), 1387-1396. 

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[21]

J. MaL. MiP. ZhouY. Xu and T. Hayat, Phase synchronization between two neurons induced by coupling of electromagnetic field, Applied Mathematics and Computation, 307 (2017), 321-328.  doi: 10.1016/j.amc.2017.03.002.

[22]

B. K. PatleD. R. K. ParhiA. Jagadeesh and S. K. Kashyap, Matrix-binary codes based genetic algorithm for path planning of mobile robot, Computers & Electrical Engineering, 67 (2018), 708-728. 

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L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Physical Review Letters, 64 (1990), 821. doi: 10.1103/PhysRevLett.64.821.

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H. Poincare, Sur le probleme des trois corps et les equations de la dynamique, Acta Mathematica, 13 (1890), A3–A270.

[25]

A. ProvataP. Katsaloulis and D. A. Verganelakis, Dynamics of chaotic maps for modelling the multifractal spectrum of human brain diffusion tensor images, Chaos, Solitons & Fractals, 45 (2012), 174-180. 

[26]

S. Rasappan and S. Vaidyanathan, Synchronization of hyperchaotic liu system via backstepping control with recursive feedback, In International Conference on Eco-friendly Computing and Communication Systems, Springer, (2012), 212–221.

[27]

F. P. RussellP. D. DubenX. NiuW. Luk and T. N. Palmer, Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures, Computer Physics Communications, 221 (2017), 160-173.  doi: 10.1016/j.cpc.2017.08.011.

[28]

B. Sahoo and S. Poria, The chaos and control of a food chain model supplying additional food to top-predator, Chaos, Solitons & Fractals, 58 (2014), 52-64.  doi: 10.1016/j.chaos.2013.11.008.

[29]

N. SamardzijaL. D. Greller and E. Wasserman, Nonlinear chemical kinetic schemes derived from mechanical and electrical dynamical systems, The Journal of Chemical Physics, 90 (1989), 2296-2304.  doi: 10.1063/1.455970.

[30]

Z. ShiS. Hong and K. Chen, Experimental study on tracking the state of analog Chua's circuit with particle filter for chaos synchronization, Physics Letters A, 372 (2008), 5575-5580. 

[31]

T. Shinbrot, E. Ott, C. Grebogi and J. A. Yorke, Using chaos to direct trajectories to targets, Physical Review Letters, 65 (1990), 3215.

[32]

A. K. SinghV. K. Yadav and S. Das, Synchronization between fractional order complex chaotic systems, International Journal of Dynamics and Control, 5 (2017), 756-770.  doi: 10.1007/s40435-016-0226-1.

[33]

P. P. Singh and B. K. Roy, Microscopic chaos control of chemical reactor system using nonlinear active plus proportional integral sliding mode control technique, The European Physical Journal Special Topics, 228 (2019), 169-184. 

[34]

K. S. Sudheer and M. Sabir, Hybrid synchronization of hyperchaotic lu system, Pramana, 73 (2009), 781.

[35]

X.-J. TongM. ZhangZ. WangY. Liu and J. Ma, An image encryption scheme based on a new hyperchaotic finance system, Optik, 126 (2015), 2445-2452.  doi: 10.1007/s11071-012-0658-x.

[36]

S. Vaidyanathan and S. Sampath, Anti-synchronization of four-wing chaotic systems via sliding mode control, International Journal of Automation and Computing, 9 (2012), 274-279. 

[37]

S. Vaidyanathan, Adaptive biological control of generalized lotkavolterra three-species biological system, International Journal of PharmTech Research, 8 (2015), 622-631. 

[38]

X. WangS. VaidyanathanC. VolosV.-T. Pham and T. Kapitaniak, Dynamics, circuit realization, control and synchronization of a hyperchaotic hyperjerk system with coexisting attractors, Nonlinear Dynamics, 89 (2017), 1673-1687.  doi: 10.1007/s11071-017-3542-x.

[39]

G.-C. WuD. Baleanu and Z.-X. Lin, Image encryption technique based on fractional chaotic time series, Journal of Vibration and Control, 22 (2016), 2092-2099.  doi: 10.1177/1077546315574649.

[40]

Z. WuJ. Duan and X. Fu, Complex projective synchronization in coupled chaotic complex dynamical systems, Nonlinear Dynamics, 69 (2012), 771-779.  doi: 10.1007/s11071-011-0303-0.

[41]

M. T. Yassen, Adaptive control and synchronization of a modified Chua's circuit system, Applied Mathematics and Computation, 135 (2003), 113-128.  doi: 10.1016/S0096-3003(01)00318-6.

[42]

P. Zhou and W. Zhu, Function projective synchronization for fractional-order chaotic systems, Nonlinear Analysis: Real World Applications, 12 (2011), 811-816.  doi: 10.1016/j.nonrwa.2010.08.008.

show all references

References:
[1]

K. Bouallegue, A new class of neural networks and its applications, Neurocomputing, 249 (2017), 28-47. 

[2]

M. Chen and Z. Han, Controlling and synchronizing chaotic genesio system via nonlinear feedback control, Chaos, Solitons & Fractals, 17 (2003), 709-716.  doi: 10.1016/S0960-0779(02)00487-3.

[3]

H. Delavari and M. Mohadeszadeh, Hybrid complex projective synchronization of complex chaotic systems using active control technique with nonlinearity in the control input, Journal of Control Engineering and Applied Informatics, 20 (2018), 67-74. 

[4]

Z. Ding and Y. Shen, Projective synchronization of nonidentical fractional-order neural networks based on sliding mode controller, Neural Networks, 76 (2016), 97-105. 

[5]

J.-P. Eckmann, S. O. Kamphorst, D. Ruelle and S. Ciliberto, Liapunov exponents from time series, Physical Review A, 34 (1986), 4971. doi: 10.1103/PhysRevA.34.4971.

[6]

D. GhoshA. MukherjeeN. R. Das and B. N. Biswas, Generation & control of chaos in a single loop optoelectronic oscillator, Optik, 165 (2018), 275-287. 

[7]

S. K. Han, C. Kurrer and Y. Kuramoto, Dephasing and bursting in coupled neural oscillators, Physical Review Letters, 75 (1995), 3190.

[8]

M. HuY. YangZ. Xu and L. Guo, Hybrid projective synchronization in a chaotic complex nonlinear system, Mathematics and Computers in Simulation, 79 (2008), 449-457.  doi: 10.1016/j.matcom.2008.01.047.

[9]

A. W. Hubler, Adaptive control of chaotic system, Helv. Phys. Acta, 62 (1989), 343-346. 

[10]

T. Khan and H. Chaudhary, Estimation and identifiability of parameters for generalized lotka-volterra biological systems using adaptive controlled combination difference anti-synchronization, Differential Equations and Dynamical Systems, Special Issue, 28 (2020), 515-526.  doi: 10.1007/s12591-020-00534-8.

[11]

A. Khan and H. Chaudhary, Adaptive control and hybrid projective combination synchronization of chaos generated by generalized lotka-volterra biological systems, Bloomsbury India, (2019), 174.

[12]

A. Khan and H. Chaudhary, Hybrid projective combination-combination synchronization in non-identical hyperchaotic systems using adaptive control, Arabian Journal of Mathematics, 9 (2020), 597-611.  doi: 10.1007/s40065-020-00279-w.

[13]

C. Li and X. Liao, Complete and lag synchronization of hyperchaotic systems using small impulses, Chaos, Solitons & Fractals, 22 (2004), 857-867.  doi: 10.1016/j.chaos.2004.03.006.

[14]

D. Li and X. Zhang, Impulsive synchronization of fractional order chaotic systems with time-delay, Neurocomputing, 216 (2016), 39-44. 

[15]

G.-H. Li and S.-P. Zhou, Anti-synchronization in different chaotic systems, Chaos, Solitons & Fractals, 32 (2007), 516-520.  doi: 10.1016/j.chaos.2005.12.009.

[16]

G.-H. Li, Modified projective synchronization of chaotic system, Chaos, Solitons & Fractals, 32 (2007), 1786-1790.  doi: 10.1016/j.chaos.2005.12.009.

[17]

S.-Y. LiC.-H. YangC.-T. LinL.-W. Ko and T.-T. Chiu, Adaptive synchronization of chaotic systems with unknown parameters via new backstepping strategy, Nonlinear Dynamics, 70 (2012), 2129-2143.  doi: 10.1007/s11071-012-0605-x.

[18]

Z. Li and D. Xu, A secure communication scheme using projective chaos synchronization, Chaos, Solitons & Fractals, 22 (2004), 477-481.  doi: 10.1016/j.chaos.2004.02.004.

[19]

T.-L. Liao and S.-H. Tsai, Adaptive synchronization of chaotic systems and its application to secure communications, Chaos, Solitons & Fractals, 11 (2000), 1387-1396. 

[20]

E. N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[21]

J. MaL. MiP. ZhouY. Xu and T. Hayat, Phase synchronization between two neurons induced by coupling of electromagnetic field, Applied Mathematics and Computation, 307 (2017), 321-328.  doi: 10.1016/j.amc.2017.03.002.

[22]

B. K. PatleD. R. K. ParhiA. Jagadeesh and S. K. Kashyap, Matrix-binary codes based genetic algorithm for path planning of mobile robot, Computers & Electrical Engineering, 67 (2018), 708-728. 

[23]

L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Physical Review Letters, 64 (1990), 821. doi: 10.1103/PhysRevLett.64.821.

[24]

H. Poincare, Sur le probleme des trois corps et les equations de la dynamique, Acta Mathematica, 13 (1890), A3–A270.

[25]

A. ProvataP. Katsaloulis and D. A. Verganelakis, Dynamics of chaotic maps for modelling the multifractal spectrum of human brain diffusion tensor images, Chaos, Solitons & Fractals, 45 (2012), 174-180. 

[26]

S. Rasappan and S. Vaidyanathan, Synchronization of hyperchaotic liu system via backstepping control with recursive feedback, In International Conference on Eco-friendly Computing and Communication Systems, Springer, (2012), 212–221.

[27]

F. P. RussellP. D. DubenX. NiuW. Luk and T. N. Palmer, Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures, Computer Physics Communications, 221 (2017), 160-173.  doi: 10.1016/j.cpc.2017.08.011.

[28]

B. Sahoo and S. Poria, The chaos and control of a food chain model supplying additional food to top-predator, Chaos, Solitons & Fractals, 58 (2014), 52-64.  doi: 10.1016/j.chaos.2013.11.008.

[29]

N. SamardzijaL. D. Greller and E. Wasserman, Nonlinear chemical kinetic schemes derived from mechanical and electrical dynamical systems, The Journal of Chemical Physics, 90 (1989), 2296-2304.  doi: 10.1063/1.455970.

[30]

Z. ShiS. Hong and K. Chen, Experimental study on tracking the state of analog Chua's circuit with particle filter for chaos synchronization, Physics Letters A, 372 (2008), 5575-5580. 

[31]

T. Shinbrot, E. Ott, C. Grebogi and J. A. Yorke, Using chaos to direct trajectories to targets, Physical Review Letters, 65 (1990), 3215.

[32]

A. K. SinghV. K. Yadav and S. Das, Synchronization between fractional order complex chaotic systems, International Journal of Dynamics and Control, 5 (2017), 756-770.  doi: 10.1007/s40435-016-0226-1.

[33]

P. P. Singh and B. K. Roy, Microscopic chaos control of chemical reactor system using nonlinear active plus proportional integral sliding mode control technique, The European Physical Journal Special Topics, 228 (2019), 169-184. 

[34]

K. S. Sudheer and M. Sabir, Hybrid synchronization of hyperchaotic lu system, Pramana, 73 (2009), 781.

[35]

X.-J. TongM. ZhangZ. WangY. Liu and J. Ma, An image encryption scheme based on a new hyperchaotic finance system, Optik, 126 (2015), 2445-2452.  doi: 10.1007/s11071-012-0658-x.

[36]

S. Vaidyanathan and S. Sampath, Anti-synchronization of four-wing chaotic systems via sliding mode control, International Journal of Automation and Computing, 9 (2012), 274-279. 

[37]

S. Vaidyanathan, Adaptive biological control of generalized lotkavolterra three-species biological system, International Journal of PharmTech Research, 8 (2015), 622-631. 

[38]

X. WangS. VaidyanathanC. VolosV.-T. Pham and T. Kapitaniak, Dynamics, circuit realization, control and synchronization of a hyperchaotic hyperjerk system with coexisting attractors, Nonlinear Dynamics, 89 (2017), 1673-1687.  doi: 10.1007/s11071-017-3542-x.

[39]

G.-C. WuD. Baleanu and Z.-X. Lin, Image encryption technique based on fractional chaotic time series, Journal of Vibration and Control, 22 (2016), 2092-2099.  doi: 10.1177/1077546315574649.

[40]

Z. WuJ. Duan and X. Fu, Complex projective synchronization in coupled chaotic complex dynamical systems, Nonlinear Dynamics, 69 (2012), 771-779.  doi: 10.1007/s11071-011-0303-0.

[41]

M. T. Yassen, Adaptive control and synchronization of a modified Chua's circuit system, Applied Mathematics and Computation, 135 (2003), 113-128.  doi: 10.1016/S0096-3003(01)00318-6.

[42]

P. Zhou and W. Zhu, Function projective synchronization for fractional-order chaotic systems, Nonlinear Analysis: Real World Applications, 12 (2011), 811-816.  doi: 10.1016/j.nonrwa.2010.08.008.

Figure 1.  Phase diagram of 3-D chaotic chemical reactor system in $ x_{m1}-x_{m2}-x_{m3} $ space
Figure 2.  Time series of 3-D chaotic chemical reactor system
Figure 3.  Anti-synchronization state trajectories of 3-D chaotic chemical system (A) between $ x_{m1}(t)-x_{s1}(t) $, (B) between $ x_{m2}(t)-x_{s2}(t) $, (C) between $ x_{m3}(t)-x_{s3}(t) $, (D) synchronization error of the system, (E) Parameter estimation
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