# American Institute of Mathematical Sciences

June  2019, 24(6): 2955-2969. doi: 10.3934/dcdsb.2018294

## Discontinuous phenomena in bioreactor system

 1 Department of Mathematics, Faculty of Science, Taibah University, Yanbu 41911, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt

* Corresponding author: Hany A. hosham

Received  November 2017 Revised  April 2018 Published  October 2018

This paper critically examines discontinuous bifurcation and stability issues in model of methane gas production from organic waste via decaying process in two cases, namely sliding and non-sliding flow. The presence of certain types of discontinuities in Monod curve lead to discontinuous system and therefore the criteria for the existence and stability of equilibrium points are established. The analysis highlights the presence of several types of border collision bifurcations depending upon the effect of the dilution factor, biomass concentration and solid-liquid-gas separator efficiency, like nonsmooth fold, persistence and grazing-sliding scenarios. In addition, numerical simulations are carried out to illustrate and validate the results.

Citation: Hany A. Hosham, Eman D Abou Elela. Discontinuous phenomena in bioreactor system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2955-2969. doi: 10.3934/dcdsb.2018294
##### References:
 [1] A. H. Ajbar, M. ALAhmad and E. Ali, On the dynamics of biodegradation of wastewater in aerated continuous bioreactors, Mathl. Comput. Model., 54 (2011), 1930-1942. doi: 10.1016/j.mcm.2011.04.035. [2] R. T. Alqahtani, Modelling of Biological Wastewater Treatment, Ph.D. Thesis. University of Wollongong, Australia, 2013. [3] J. Awrejcewicz and C. Lamarque, Bifurcation and Chaos in Nonsmooth Mechanical Systems, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812564801. [4] A. Bornhöft, R. Hanke-Rauschenbach and K. Sundmacher, Steady-state analysis of the anaerobic digestion model no. 1 (ADM1), Nonlinear Dyn., 73 (2013), 535-549. doi: 10.1007/s11071-013-0807-x. [5] B. Benyahia, T. Sari, B. Cherki and J. Harmand, Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes, Journal of Process Control, 22 (2012), 1008-1019. [6] M. di Bernardo, C. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008. [7] M. Fečkan and M. Pospíšil, Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems, Academic Press is an imprint of Elsevier, London, 2016. [8] A. F. Filippov, Differential equations with discontinuous right-hand side, American Mathematical Society Translations, 2 (1964), 199-231. [9] Y. Gao, X. Meng and Q. Lu, Border collision bifurcations in 3D piecewise smooth chaotic circuit, Appl. Math. Mech.-Engl. Ed., 37 (2016), 1239-1250. doi: 10.1007/s10483-016-2129-6. [10] H. A. Hosham, Cone-like Invariant Manifolds for Nonsmooth Systems, Ph.D. Thesis. Universität zu Köln, Germany, 2011. [11] H. A. Hosham, Bifurcation of periodic orbits in discontinuous systems, Nonlinear Dyn., 87 (2017), 135-148. doi: 10.1007/s11071-016-3031-7. [12] T. Küpper and H. A. Hosham, Reduction to invariant cones for nonsmooth systems, Math. Comput. Simul., 81 (2011), 980-995. doi: 10.1016/j.matcom.2010.10.004. [13] T. Küpper, H. A. Hosham and K. Dudtschenko, The dynamics of bells as impacting system, J. Mech. Eng. Sci., 225 (2011), 2436-2443. [14] T. Küpper, H. A. Hosham and D. Weiss, Bifurcation for nonsmooth dynamical systems via reduction methods, in: Recent Trends in Dynamical Systems, Proceedings in Mathematics and Statistics, Springer-Verlag, 35 (2013), 79-105. doi: 10.1007/978-3-0348-0451-6_5. [15] R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Springer-Verlag, Berlin, Germany, 2004. doi: 10.1007/978-3-540-44398-8. [16] Y. Li, L. Yuan and Z. Du, Bifurcation of nonhyperbolic limit cycles in piecewise smooth planar systems with finitely many zones, Int. J. Bifurcation and Chaos, Appl. Sci. Engrg., 27 (2017), 1750162, 14 pp. doi: 10.1142/S0218127417501620. [17] L. A. Melo-Varela, S. Casanova-Trujillo and G. Olivar-Tost, Dynamics of a bioreactor with a bacteria piecewise-linear growth model in a methane-producing process, Math. Prob. in Engin, 2013 (2013), Art. ID 685452, 8 pp. doi: 10.1155/2013/685452. [18] R. Muñoz, Design and Implementation of a COD Control System of a Prototype UASB Reactor for Treating Leachates, M.S. thesis. National University of Colombia, 2006. University of Colombia, 2006. [19] S. Shen, G. C. Premier, A. Guwy and R. Dinsdale, Bifurcation and stability analysis of an anaerobic digestion model, Nonlinear Dyn., 48 (2007), 391-408. doi: 10.1007/s11071-006-9093-1. [20] D. Weiss, T. Küpper and H. A. Hosham, Invariant manifolds for nonsmooth systems, Physica D: Nonlinear Phenomena, 241 (2012), 1895-1902. doi: 10.1016/j.physd.2011.07.012. [21] D. Weiss, T. Küpper and H. A. Hosham, Invariant manifolds for nonsmooth systems with sliding mode, Math. Comput. Simul., 110 (2015), 15-32. doi: 10.1016/j.matcom.2014.02.004.

show all references

##### References:
 [1] A. H. Ajbar, M. ALAhmad and E. Ali, On the dynamics of biodegradation of wastewater in aerated continuous bioreactors, Mathl. Comput. Model., 54 (2011), 1930-1942. doi: 10.1016/j.mcm.2011.04.035. [2] R. T. Alqahtani, Modelling of Biological Wastewater Treatment, Ph.D. Thesis. University of Wollongong, Australia, 2013. [3] J. Awrejcewicz and C. Lamarque, Bifurcation and Chaos in Nonsmooth Mechanical Systems, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812564801. [4] A. Bornhöft, R. Hanke-Rauschenbach and K. Sundmacher, Steady-state analysis of the anaerobic digestion model no. 1 (ADM1), Nonlinear Dyn., 73 (2013), 535-549. doi: 10.1007/s11071-013-0807-x. [5] B. Benyahia, T. Sari, B. Cherki and J. Harmand, Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes, Journal of Process Control, 22 (2012), 1008-1019. [6] M. di Bernardo, C. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008. [7] M. Fečkan and M. Pospíšil, Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems, Academic Press is an imprint of Elsevier, London, 2016. [8] A. F. Filippov, Differential equations with discontinuous right-hand side, American Mathematical Society Translations, 2 (1964), 199-231. [9] Y. Gao, X. Meng and Q. Lu, Border collision bifurcations in 3D piecewise smooth chaotic circuit, Appl. Math. Mech.-Engl. Ed., 37 (2016), 1239-1250. doi: 10.1007/s10483-016-2129-6. [10] H. A. Hosham, Cone-like Invariant Manifolds for Nonsmooth Systems, Ph.D. Thesis. Universität zu Köln, Germany, 2011. [11] H. A. Hosham, Bifurcation of periodic orbits in discontinuous systems, Nonlinear Dyn., 87 (2017), 135-148. doi: 10.1007/s11071-016-3031-7. [12] T. Küpper and H. A. Hosham, Reduction to invariant cones for nonsmooth systems, Math. Comput. Simul., 81 (2011), 980-995. doi: 10.1016/j.matcom.2010.10.004. [13] T. Küpper, H. A. Hosham and K. Dudtschenko, The dynamics of bells as impacting system, J. Mech. Eng. Sci., 225 (2011), 2436-2443. [14] T. Küpper, H. A. Hosham and D. Weiss, Bifurcation for nonsmooth dynamical systems via reduction methods, in: Recent Trends in Dynamical Systems, Proceedings in Mathematics and Statistics, Springer-Verlag, 35 (2013), 79-105. doi: 10.1007/978-3-0348-0451-6_5. [15] R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Springer-Verlag, Berlin, Germany, 2004. doi: 10.1007/978-3-540-44398-8. [16] Y. Li, L. Yuan and Z. Du, Bifurcation of nonhyperbolic limit cycles in piecewise smooth planar systems with finitely many zones, Int. J. Bifurcation and Chaos, Appl. Sci. Engrg., 27 (2017), 1750162, 14 pp. doi: 10.1142/S0218127417501620. [17] L. A. Melo-Varela, S. Casanova-Trujillo and G. Olivar-Tost, Dynamics of a bioreactor with a bacteria piecewise-linear growth model in a methane-producing process, Math. Prob. in Engin, 2013 (2013), Art. ID 685452, 8 pp. doi: 10.1155/2013/685452. [18] R. Muñoz, Design and Implementation of a COD Control System of a Prototype UASB Reactor for Treating Leachates, M.S. thesis. National University of Colombia, 2006. University of Colombia, 2006. [19] S. Shen, G. C. Premier, A. Guwy and R. Dinsdale, Bifurcation and stability analysis of an anaerobic digestion model, Nonlinear Dyn., 48 (2007), 391-408. doi: 10.1007/s11071-006-9093-1. [20] D. Weiss, T. Küpper and H. A. Hosham, Invariant manifolds for nonsmooth systems, Physica D: Nonlinear Phenomena, 241 (2012), 1895-1902. doi: 10.1016/j.physd.2011.07.012. [21] D. Weiss, T. Küpper and H. A. Hosham, Invariant manifolds for nonsmooth systems with sliding mode, Math. Comput. Simul., 110 (2015), 15-32. doi: 10.1016/j.matcom.2014.02.004.
Structural frame of Upflow anaerobic sludge blanket(UASB)
Equilibrium transition due to the effect of SLG separator deficiency $\alpha_6$, admissible (solid line) and virtual (dashed line): (a) Two equilibrium points of $\ominus$-system (b)Two equilibrium points of $\oplus$-system.
Persistence bifurcation of CPWS (8) when $\alpha_6 = 0.1213$
Persistence bifurcation of CPWS (8) when: (a) $m = 0$ where $\alpha_1^{max} = 2.639$ and (b) $m = 200$ where $\alpha_1 = 0.7572$
Existence of nonsmooth bifurcation of sliding flow (5) at $\lambda = 0, \alpha_6 = 0.04802$
Existence of nonsmooth bifurcation of sliding flow (5) at $\lambda = 1, \alpha_6 = 0.04905$
Existence of persistence bifurcation of sliding flow (5) at $\lambda = 0, \alpha_6 = 0.0255$
Existence of persistence bifurcation of sliding flow (5) at $\lambda = 1, \alpha_6 = 0.02496$
Numerical simulation illustrating a grazing-sliding bifurcation occurring at $\alpha_6 = \alpha_6^{graz}$ in DS (5).
 [1] Gian-Italo Bischi, Laura Gardini, Fabio Tramontana. Bifurcation curves in discontinuous maps. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 249-267. doi: 10.3934/dcdsb.2010.13.249 [2] Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 [3] Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203 [4] Sabri Bensid, Jesús Ildefonso Díaz. Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1757-1778. doi: 10.3934/dcdsb.2017105 [5] Luca Dieci, Timo Eirola, Cinzia Elia. Periodic orbits of planar discontinuous system under discretization. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2743-2762. doi: 10.3934/dcdsb.2018103 [6] Eugenii Shustin, Emilia Fridman, Leonid Fridman. Oscillations in a second-order discontinuous system with delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 339-358. doi: 10.3934/dcds.2003.9.339 [7] Patrick Ballard. Can the 'stick-slip' phenomenon be explained by a bifurcation in the steady sliding frictional contact problem?. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 363-381. doi: 10.3934/dcdss.2016001 [8] Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955 [9] Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063 [10] Dan Liu, Shigui Ruan, Deming Zhu. Bifurcation analysis in models of tumor and immune system interactions. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 151-168. doi: 10.3934/dcdsb.2009.12.151 [11] G. Bonanno, Salvatore A. Marano. Highly discontinuous elliptic problems. Conference Publications, 1998, 1998 (Special) : 118-123. doi: 10.3934/proc.1998.1998.118 [12] Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159 [13] Patrick M. Fitzpatrick, Jacobo Pejsachowicz. Branching and bifurcation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1955-1975. doi: 10.3934/dcdss.2019127 [14] Fei Chen, Boling Guo, Xiaoping Zhai. Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density. Kinetic & Related Models, 2019, 12 (1) : 37-58. doi: 10.3934/krm.2019002 [15] Michel Cristofol, Patricia Gaitan, Kati Niinimäki, Olivier Poisson. Inverse problem for a coupled parabolic system with discontinuous conductivities: One-dimensional case. Inverse Problems & Imaging, 2013, 7 (1) : 159-182. doi: 10.3934/ipi.2013.7.159 [16] Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031 [17] Ling-Hao Zhang, Wei Wang. Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 591-604. doi: 10.3934/dcds.2017024 [18] Rushun Tian, Zhi-Qiang Wang. Bifurcation results on positive solutions of an indefinite nonlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 335-344. doi: 10.3934/dcds.2013.33.335 [19] Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507 [20] Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301

2018 Impact Factor: 1.008