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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 & 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.   Google Scholar

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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.   Google Scholar

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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.   Google Scholar

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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. Google Scholar

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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.  Google Scholar

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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. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[18]

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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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

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

[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.  Google Scholar

[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.   Google Scholar

[34]

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

[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.  Google Scholar

[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.   Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

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

[24]

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

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[31]

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

[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.  Google Scholar

[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.   Google Scholar

[34]

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

[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.  Google Scholar

[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.   Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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