doi: 10.3934/dcdss.2020433

Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions

1. 

Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

2. 

Department of Medical Research, China Medical University Hospital, Taichung 40204, Taiwan

3. 

Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan

4. 

Department of Mathematics, Faculty of Science, Cairo University, Al Orman, Giza 12613, Egypt

5. 

Department of Mathematics, College of Arts and Sciences, Najran University, Najran, Kingdom of Saudi Arabia

6. 

Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen

* Corresponding author: H. M. Srivastava

Received  November 2019 Revised  June 2020 Published  November 2020

The approximate solutions of a two-cell reaction-diffusion model equation subjected to the Dirichlet conditions are obtained. The reaction is assumed to occur in the presence of cubic autocatalyst which decays to an inert compound in the first cell. Coupling with the reactant is assumed to be cubic in the concentrations. A linear exchange in the concentration of the reactant is taken between the two cells. The formal exact solution is found analytically. Here, in this investigation, use is made of the Picard iterative scheme which is constructed and applied after the exact one. The results obtained are compared with those found by means of a numerical method. It is observed that the solution obtained here is symmetric with respect to the mid-point of the container.The travelling wave is expected due to the parity of the space operator and the symmetric boundary conditions. Symmetric patterns, including among them a parabolic one, are observed for a large time.

When the initial conditions are periodic, the most dominant modes travel at a constant speed for a large time. This phenomenon is highly affected by the rate of decay of the autocatalyst to an inert compound. The present work is of remarkably significant interest in chemical engineering as well as in other physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product, which is carried by controlling the initial concentration of the reactant. Furthermore, in the last section on conclusions, we have cited many potentially useful recent works related to the subject-matter of this investigation in order to provide incentive and motivation for making further advances by using space-time fractional derivatives along the lines of the problem of finding approximate analytical solutions of the reaction-diffusion model equations which we have discussed in this article.

Citation: H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020433
References:
[1]

H. I. Abdel-Gawad and K. M. Saad, On the behaviour of soultions of the two-cell cubic autocatalator, ANZIAM J., 44 (2002), E1–E32. doi: 10.1017/S1446181100007859.  Google Scholar

[2]

H. I. Abdel-Gawad and H. A. Abdusalam, Approximate solutions of the Kuramoto-Sivashinsky equation for periodic boundary value problems and chaos, Chaos Solitons Fract., 12 (2001), 2039-2050.  doi: 10.1016/S0960-0779(00)00142-9.  Google Scholar

[3]

P. Arcuri and J. D. Murray, Pattern sensitivity to boundary conditions in reaction-diffusion models, J. Math. Biol., 24 (1986), 141-165.  doi: 10.1007/BF00275996.  Google Scholar

[4] N. F. Britton, Reaction-Diffusion Equations and Their Applications to Biology, Academic Press, New York, 1986.   Google Scholar
[5]

L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäuser, Basel and Boston, 1997. doi: 10.1007/978-1-4899-2846-7.  Google Scholar

[6]

I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns and Chaos, Clarendon (Oxford University) Press, Oxford, London and New York, 1998. Google Scholar

[7]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[8]

L. K. Forbes, On stability and uniqueness of stationary one-dimensional patterns in the Belousov-Zhabotinskii reaction, Physica D: Nonlinear Phenom., 50 (1991), 42-58.  doi: 10.1016/0167-2789(91)90077-M.  Google Scholar

[9]

W. Jager, J. Moser and R. Renmert, Modelling of Patterns in Space and Time, Springer Lectures in Biomathematics, Springer-Verlag, Berlin, Heidelberg and New York, 1984. Google Scholar

[10]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.  Google Scholar

[11]

A. KolmogoroffI. Petrovsky and N. Piscounoff, Etude de l'equation de la diffusion avec croissance de la quantité de matie'er et son application a un problém biologique, Moscow Univ. Bull. Math., 1 (1937), 1-25.   Google Scholar

[12]

J. A. LeachJ. H. Merkin and S. K. Scott, Oscillations and waves in the Belousov-Zhabotinskii reaction in a finite medium, J. Math. Chem., 16 (1994), 115-124.  doi: 10.1007/BF01169200.  Google Scholar

[13]

D. Luss, M. Golubitsky and S. Strogatz, Pattern Formation in Continuous and Coupled Systems, IMA Volumes in Mathematics and Its Applications, Springer-Verlag, Berlin, Heidelberg and New York, 1999. doi: 10.1007/978-1-4612-1558-5.  Google Scholar

[14]

P. K. Maini, Spatial and spatiotemporal pattern formation in generalised Turing systems, Comput. Math. Appl., 32 (1996), 71-77.  doi: 10.1016/S0898-1221(96)00198-8.  Google Scholar

[15]

T. R. Marchant, Cubic autocatalytic reaction-diffusion equations$:$ Semi-analytical solutions, Proc. Roy. Soc. London Ser. A Math. Phys. Engrg. Sci., 458 (2002), 1-16.  doi: 10.1098/rspa.2001.0899.  Google Scholar

[16]

J. H. MerkinD. J. Needham and S. K. Scott, Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system, IMA J. Appl. Math., 50 (1993), 43-76.  doi: 10.1093/imamat/50.1.43.  Google Scholar

[17]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, Heidelberg and New York, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[18]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative system. Ⅱ, AIP J. Chem. Phys., 48 (1968), 1-7.  doi: 10.1063/1.1668896.  Google Scholar

[19]

K. M. Saad and A. M. El-Shrae, Travelling waves in a cubic autocatalytic reaction, Adv. Appl. Math. Sci., 8 (2011), 87-104.   Google Scholar

[20]

K. M. Saad and E. H. F. Al-Sharif, Comparative study of a cubic autocatalytic reaction via different analysis methods, Discrete Continuous Dyn. Syst. Ser. S, 12 (2019), 665-684.  doi: 10.3934/dcdss.2019042.  Google Scholar

[21]

K. M. SaadH. M. Srivastava and J. F. Gómez-Aguilar, A fractional quadratic autocatalysis associated with chemical clock reactions involving linear inhibition, Chaos Solitons Fractals, 132 (2020), 1-9.  doi: 10.1016/j.chaos.2019.109557.  Google Scholar

[22]

R. A. SatnoianuM. Menzinger and P. K. Maini, Turing instabilities in general systems, J. Math. Biol., 41 (2000), 493-512.  doi: 10.1007/s002850000056.  Google Scholar

[23]

E. E. Sel'kov, Self-oscillations in glycolysis. $1:$ A simple kinetic model, European J. Biochem., 4 (1968), 79-86.  doi: 10.1111/j.1432-1033.1968.tb00175.x.  Google Scholar

[24]

H. M. Srivastava and K. M. Saad, Some new models of the time-fractional gas dynamics equation, Adv. Math. Models Appl., 3 (2018), 5-17.   Google Scholar

[25]

H. M. Srivastava and K. M. Saad, New approximate solution of the time-fractional Nagumo equation involving fractional integrals without singular kernel, Appl. Math. Inform. Sci., 14 (2020), 1-8.  doi: 10.18576/amis/140101.  Google Scholar

[26]

H. M. SrivastavaK. M. Saad and E. H. F. Al-Sharif, New analysis of the time-fractional and space-time fractional-order Nagumo equation, J. Inform. Math. Sci., 10 (2018), 545-561.   Google Scholar

[27]

H. M. SrivastavaH. I. Abdel-Gawad and K. M. Saad, Stability of traveling waves based upon the Evans function and Legendre polynomials, Appl. Sci., 10 (2020), 1-16.  doi: 10.3390/app10030846.  Google Scholar

[28]

H. M. Srivastava, Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J., 60 (2020), 73-116.   Google Scholar

[29]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B Biol. Sci., 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[30]

J. J. Tyson, Classification of instabilities in chemical reaction systems, AIP J. Chem. Phys., 62 (1975), 1-7.  doi: 10.1063/1.430567.  Google Scholar

show all references

References:
[1]

H. I. Abdel-Gawad and K. M. Saad, On the behaviour of soultions of the two-cell cubic autocatalator, ANZIAM J., 44 (2002), E1–E32. doi: 10.1017/S1446181100007859.  Google Scholar

[2]

H. I. Abdel-Gawad and H. A. Abdusalam, Approximate solutions of the Kuramoto-Sivashinsky equation for periodic boundary value problems and chaos, Chaos Solitons Fract., 12 (2001), 2039-2050.  doi: 10.1016/S0960-0779(00)00142-9.  Google Scholar

[3]

P. Arcuri and J. D. Murray, Pattern sensitivity to boundary conditions in reaction-diffusion models, J. Math. Biol., 24 (1986), 141-165.  doi: 10.1007/BF00275996.  Google Scholar

[4] N. F. Britton, Reaction-Diffusion Equations and Their Applications to Biology, Academic Press, New York, 1986.   Google Scholar
[5]

L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäuser, Basel and Boston, 1997. doi: 10.1007/978-1-4899-2846-7.  Google Scholar

[6]

I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns and Chaos, Clarendon (Oxford University) Press, Oxford, London and New York, 1998. Google Scholar

[7]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[8]

L. K. Forbes, On stability and uniqueness of stationary one-dimensional patterns in the Belousov-Zhabotinskii reaction, Physica D: Nonlinear Phenom., 50 (1991), 42-58.  doi: 10.1016/0167-2789(91)90077-M.  Google Scholar

[9]

W. Jager, J. Moser and R. Renmert, Modelling of Patterns in Space and Time, Springer Lectures in Biomathematics, Springer-Verlag, Berlin, Heidelberg and New York, 1984. Google Scholar

[10]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.  Google Scholar

[11]

A. KolmogoroffI. Petrovsky and N. Piscounoff, Etude de l'equation de la diffusion avec croissance de la quantité de matie'er et son application a un problém biologique, Moscow Univ. Bull. Math., 1 (1937), 1-25.   Google Scholar

[12]

J. A. LeachJ. H. Merkin and S. K. Scott, Oscillations and waves in the Belousov-Zhabotinskii reaction in a finite medium, J. Math. Chem., 16 (1994), 115-124.  doi: 10.1007/BF01169200.  Google Scholar

[13]

D. Luss, M. Golubitsky and S. Strogatz, Pattern Formation in Continuous and Coupled Systems, IMA Volumes in Mathematics and Its Applications, Springer-Verlag, Berlin, Heidelberg and New York, 1999. doi: 10.1007/978-1-4612-1558-5.  Google Scholar

[14]

P. K. Maini, Spatial and spatiotemporal pattern formation in generalised Turing systems, Comput. Math. Appl., 32 (1996), 71-77.  doi: 10.1016/S0898-1221(96)00198-8.  Google Scholar

[15]

T. R. Marchant, Cubic autocatalytic reaction-diffusion equations$:$ Semi-analytical solutions, Proc. Roy. Soc. London Ser. A Math. Phys. Engrg. Sci., 458 (2002), 1-16.  doi: 10.1098/rspa.2001.0899.  Google Scholar

[16]

J. H. MerkinD. J. Needham and S. K. Scott, Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system, IMA J. Appl. Math., 50 (1993), 43-76.  doi: 10.1093/imamat/50.1.43.  Google Scholar

[17]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, Heidelberg and New York, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[18]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative system. Ⅱ, AIP J. Chem. Phys., 48 (1968), 1-7.  doi: 10.1063/1.1668896.  Google Scholar

[19]

K. M. Saad and A. M. El-Shrae, Travelling waves in a cubic autocatalytic reaction, Adv. Appl. Math. Sci., 8 (2011), 87-104.   Google Scholar

[20]

K. M. Saad and E. H. F. Al-Sharif, Comparative study of a cubic autocatalytic reaction via different analysis methods, Discrete Continuous Dyn. Syst. Ser. S, 12 (2019), 665-684.  doi: 10.3934/dcdss.2019042.  Google Scholar

[21]

K. M. SaadH. M. Srivastava and J. F. Gómez-Aguilar, A fractional quadratic autocatalysis associated with chemical clock reactions involving linear inhibition, Chaos Solitons Fractals, 132 (2020), 1-9.  doi: 10.1016/j.chaos.2019.109557.  Google Scholar

[22]

R. A. SatnoianuM. Menzinger and P. K. Maini, Turing instabilities in general systems, J. Math. Biol., 41 (2000), 493-512.  doi: 10.1007/s002850000056.  Google Scholar

[23]

E. E. Sel'kov, Self-oscillations in glycolysis. $1:$ A simple kinetic model, European J. Biochem., 4 (1968), 79-86.  doi: 10.1111/j.1432-1033.1968.tb00175.x.  Google Scholar

[24]

H. M. Srivastava and K. M. Saad, Some new models of the time-fractional gas dynamics equation, Adv. Math. Models Appl., 3 (2018), 5-17.   Google Scholar

[25]

H. M. Srivastava and K. M. Saad, New approximate solution of the time-fractional Nagumo equation involving fractional integrals without singular kernel, Appl. Math. Inform. Sci., 14 (2020), 1-8.  doi: 10.18576/amis/140101.  Google Scholar

[26]

H. M. SrivastavaK. M. Saad and E. H. F. Al-Sharif, New analysis of the time-fractional and space-time fractional-order Nagumo equation, J. Inform. Math. Sci., 10 (2018), 545-561.   Google Scholar

[27]

H. M. SrivastavaH. I. Abdel-Gawad and K. M. Saad, Stability of traveling waves based upon the Evans function and Legendre polynomials, Appl. Sci., 10 (2020), 1-16.  doi: 10.3390/app10030846.  Google Scholar

[28]

H. M. Srivastava, Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J., 60 (2020), 73-116.   Google Scholar

[29]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B Biol. Sci., 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[30]

J. J. Tyson, Classification of instabilities in chemical reaction systems, AIP J. Chem. Phys., 62 (1975), 1-7.  doi: 10.1063/1.430567.  Google Scholar

Figure 1.  The concentration of the reactant $ \alpha_{1} $ displayed against $ x $ for $ \gamma = 0.1 $, $ k = 0.01 $ and $ L = 100 $ in (a), (b) and (c) for $ t = 30 $, and $ t = 100 $, respectively
Figure 2.  The concentration of the autocatalyst $ \beta $ displayed against $ x $ for $ \gamma = 0.1 $, $ k = 0.01 $ and $ L = 100 $ in (a), (b) and (c) for $ t = 30 $, and $ t = 100 $, respectively
Figure 3.  The concentration of the reactant $ \alpha_{2} $ displayed against $ x $ for $ \gamma = 0.1 $, $ k = 0.01 $ and $ L = 100 $ in (a), (b) and (c) for $ t = 30 $, and $ 100 $, respectively
Figure 4.  The concentration of $ \alpha_{1} $, $ \beta $ and $ \alpha_{2} $ displayed against $ x $ for $ \gamma = 0.6 $, $ k = 0.09 $ and $ L = 100 $. $ (\cdots) $ for $ t = 0 $, $ (–) $ for $ t = 30 $ and $ (-) $ for $ t = 100 $, respectively
Figure 5.  Contour plots of $ \alpha_{1} $, $ \beta $ and $ \alpha_{2} $, respectively, for $ \gamma = 0.09 $, $ k = 0.01 $ and $ L = 100 $
Figure 6.  Approximate analytical and numerical solutions of $ \alpha_{i}(x,t) $ and $ \beta(x,t) $ displayed against $ x $ for $ t = 50 $, $ k = 0.09 $, $ \gamma = 0.03 $ and $ L = 10 $. $ (-) $ for approximate analytical solutions; $ (–) $ for numerical solutions. The error estimate is of order $ ( $$ 10^{-8}) $
Figure 7.  Approximate analytical and numerical solutions of $ \alpha_{i}(x,t) $ and $ \beta(x,t) $ displayed against $ x $ for $ t = 50 $, $ k = 0.01 $, $ \gamma = 0.001 $ and $ L = 100 $. $ (-) $ for approximate analytical solutions; $ (–) $ for numerical solutions.Th error estimate is of order $ (10^{-4}) $
Figure 8.  The absolute error between the approximate analytical and numerical solutions of $ \alpha_{i}(x,t) $ and $ \beta(x,t) $ displayed against $ x $, $ \alpha_{1}(x,t) $ (solid line), $ \beta(x,t) $ (doted-line) and $ \alpha_{2}(x,t) $ (dashed line) for $ k = 0.09 $, $ \gamma = 0.1 $ and $ L = 100 $. (a) $ t = 30 $, (b) $ t = 50 $ and (c) $ t = 100 $. The error estimate is of order $ (10^{-3} $)
[1]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[2]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005

[3]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

[4]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[5]

Dmitry Treschev. Travelling waves in FPU lattices. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 867-880. doi: 10.3934/dcds.2004.11.867

[6]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004

[7]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[8]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[9]

José Raúl Quintero, Juan Carlos Muñoz Grajales. On the existence and computation of periodic travelling waves for a 2D water wave model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 557-578. doi: 10.3934/cpaa.2018030

[10]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[11]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[12]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190

[13]

Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042

[14]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298

[15]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

[16]

Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329

[17]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91

[18]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027

[19]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[20]

Pascal Noble, Sebastien Travadel. Non-persistence of roll-waves under viscous perturbations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 61-70. doi: 10.3934/dcdsb.2001.1.61

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (49)
  • HTML views (129)
  • Cited by (0)

[Back to Top]