Modelling of the complex nonlinear processes: Determination of the instability region by the stoichiometric network analysis
Abstract
Modeling of a complex nonlinear process whatever it describes is a serious task from mathematical point of view. Generally, in a complex nonlinear reaction system there is possibility to find region in parameter space where the main steady state is unstable. In that region numerous self-organized dynamic states, such as multistability, periodicity and chaos can be established. Although mentioned dissipative structures are common in nature (many biochemical processes are in the oscillatory states), the region of parameters where they appear is often very narrow. Therefore, mathematical modeling of the process under consideration, with clear and precise determination of instability region is desirable. If the model can be reduced to two or three variables, the locus of unstable steady (nonequilibrium stationary) states can be easily obtained. Models with more variables must be explored by some general method, such as the stoichiometric network analysis, a known powerful method for the ex...amination of complex reaction systems, the possible pathways in them, and corresponding stability analysis. However, in the form proposed by B. Clarke, the general analytical expression for the instability condition related to experimental information was not achieved, although geometrical solutions of the problem were suggested. In last papers, we have offered the procedure for obtaining the instability condition in the function of reaction rates. Our aim here is to present the mathematical derivation of instability condition with application of theory to selected models.
Source:
Mathematical Modelling, 2013, 111-178Publisher:
- Nova Science Publishers, Inc.
Funding / projects:
- Nanostructured Functional and Composite Materials in Catalytic and Sorption Processes (RS-MESTD-Integrated and Interdisciplinary Research (IIR or III)-45001)
- Dynamics of nonlinear physicochemical and biochemical systems with modeling and predicting of their behavior under nonequilibrium conditions (RS-MESTD-Basic Research (BR or ON)-172015)
Scopus: 2-s2.0-84892914730
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Institution/Community
IHTMTY - CHAP AU - Čupić, Željko AU - Marković, V. AU - Ivanović, Ana AU - Kolar-Anić, Ljiljana PY - 2013 UR - https://cer.ihtm.bg.ac.rs/handle/123456789/1383 AB - Modeling of a complex nonlinear process whatever it describes is a serious task from mathematical point of view. Generally, in a complex nonlinear reaction system there is possibility to find region in parameter space where the main steady state is unstable. In that region numerous self-organized dynamic states, such as multistability, periodicity and chaos can be established. Although mentioned dissipative structures are common in nature (many biochemical processes are in the oscillatory states), the region of parameters where they appear is often very narrow. Therefore, mathematical modeling of the process under consideration, with clear and precise determination of instability region is desirable. If the model can be reduced to two or three variables, the locus of unstable steady (nonequilibrium stationary) states can be easily obtained. Models with more variables must be explored by some general method, such as the stoichiometric network analysis, a known powerful method for the examination of complex reaction systems, the possible pathways in them, and corresponding stability analysis. However, in the form proposed by B. Clarke, the general analytical expression for the instability condition related to experimental information was not achieved, although geometrical solutions of the problem were suggested. In last papers, we have offered the procedure for obtaining the instability condition in the function of reaction rates. Our aim here is to present the mathematical derivation of instability condition with application of theory to selected models. PB - Nova Science Publishers, Inc. T2 - Mathematical Modelling T1 - Modelling of the complex nonlinear processes: Determination of the instability region by the stoichiometric network analysis SP - 111 EP - 178 UR - https://hdl.handle.net/21.15107/rcub_cer_1383 ER -
@inbook{ author = "Čupić, Željko and Marković, V. and Ivanović, Ana and Kolar-Anić, Ljiljana", year = "2013", abstract = "Modeling of a complex nonlinear process whatever it describes is a serious task from mathematical point of view. Generally, in a complex nonlinear reaction system there is possibility to find region in parameter space where the main steady state is unstable. In that region numerous self-organized dynamic states, such as multistability, periodicity and chaos can be established. Although mentioned dissipative structures are common in nature (many biochemical processes are in the oscillatory states), the region of parameters where they appear is often very narrow. Therefore, mathematical modeling of the process under consideration, with clear and precise determination of instability region is desirable. If the model can be reduced to two or three variables, the locus of unstable steady (nonequilibrium stationary) states can be easily obtained. Models with more variables must be explored by some general method, such as the stoichiometric network analysis, a known powerful method for the examination of complex reaction systems, the possible pathways in them, and corresponding stability analysis. However, in the form proposed by B. Clarke, the general analytical expression for the instability condition related to experimental information was not achieved, although geometrical solutions of the problem were suggested. In last papers, we have offered the procedure for obtaining the instability condition in the function of reaction rates. Our aim here is to present the mathematical derivation of instability condition with application of theory to selected models.", publisher = "Nova Science Publishers, Inc.", journal = "Mathematical Modelling", booktitle = "Modelling of the complex nonlinear processes: Determination of the instability region by the stoichiometric network analysis", pages = "111-178", url = "https://hdl.handle.net/21.15107/rcub_cer_1383" }
Čupić, Ž., Marković, V., Ivanović, A.,& Kolar-Anić, L.. (2013). Modelling of the complex nonlinear processes: Determination of the instability region by the stoichiometric network analysis. in Mathematical Modelling Nova Science Publishers, Inc.., 111-178. https://hdl.handle.net/21.15107/rcub_cer_1383
Čupić Ž, Marković V, Ivanović A, Kolar-Anić L. Modelling of the complex nonlinear processes: Determination of the instability region by the stoichiometric network analysis. in Mathematical Modelling. 2013;:111-178. https://hdl.handle.net/21.15107/rcub_cer_1383 .
Čupić, Željko, Marković, V., Ivanović, Ana, Kolar-Anić, Ljiljana, "Modelling of the complex nonlinear processes: Determination of the instability region by the stoichiometric network analysis" in Mathematical Modelling (2013):111-178, https://hdl.handle.net/21.15107/rcub_cer_1383 .