Nonlinear Dynamics
Most natural and engineering systems of intellectual interest and scientific/technological value are characterized by synergistic interactions, feedback loops, sensitivity to initial data, emergent group behavior etc. A mathematical description of such systems necessarily results in models consisting of non-linear equations. These nonlinear systems can show a very rich variety of behaviors like multiple steady state, oscillations, and chaos. Currently, we are working on three different projects with the central theme of nonlinear dynamics.
In chemical engineering, chemical reactors have long been known to exhibit such nonlinear phenomenon. The dynamics and stability of these nonlinear reactors have been studied extensively in our lab [2]-[6]. The classic reactor models can be used to represent various other processes in nature, apart from industrial reactors. Here, we study one such class of reactions, viz. autocatalytic reactions, which can capture the self-replication observed in diverse disciplines. Specifically, we are working on modeling the inter-conversion of species (like isomers, social groups) by coupled autocatalytic reactions1 (Fig.1). The conditions for complete conversion from one species to another are determined, and the effect of the stoichiometry of these reactions and the type of reactors (Batch or CSTR) on the conversion characteristics is analyzed. In another research project, we are modeling the population interaction of farmers and industrialists. The population model for this kind of occupation interaction can be captured with chemical reactions as well. The focus of the project is to predict the population dynamics and accordingly develop strategies to optimize the growth of each sector for the maximum productivity of the society.
Fig. 1: Bifurcation Diagrams of b (dimensionless concentration of species B) versus the pa (dimensionless production rate of species A). The values of other parameters are: (a) d_a=10, d_b=1.4, p_b=12, (b) d_a=10, d_b=0.5, p_b=4.5, (c) d_a=10, d_b=1.3, p_b=12.5. The system has a stable single solution in (a) and exhibits complexities like multiple steady states in (b) and oscillations in (c). The dotted line indicates unstable branch and LP and H denote Saddle-node and Hopf bifurcation respectively [1].
Computational neuroscience is another fascinating application of nonlinear dynamics due to the complexity and highly nonlinear nature of the brain. We are working on modeling the effect of general anesthesia on the brain. General anesthesia is used during surgery to induce unconsciousness, analgesia, and amnesia. A clear understanding of mechanisms of these drugs is essential for effective drug delivery during surgery. EEG (Electroencephalogram) shows distinct patterns under general anesthesia. Here, we develop theoretical models which can reproduce these EEG patterns, and hence, provide a better understanding of the cellular mechanism of general anesthesia. Currently, we are focusing on modeling the burst suppression patterns in EEG observed at a high dose of anesthesia.
References:
[1] Khot, Aditi, and S. Pushpavanam. "Coupled autocatalytic reactions: Interconversion and extinction of species." Chemical Engineering Science 160 (2017): 254-268.
[2] Kumar M. V., Zeyer K. P., Kienle A., Pushpavanam S., Conceptual Analysis of the Effect of Kinetics on the Stability and Multiplicity of a Coupled Bioreactor-Separator System Using a Cybernetic Modeling Approach, INDUSTRIAL & ENGINEERING CHEMISTRY RESEARCH Volume: 48, Issue: 24, Pages: 10962-10975, 2009.
[3] Balasubramanian P., Pushpavanam S., Kienle A., Balaraman K. S., Effect of delay on the stability of a coupled reactor-flash system sustaining an elementary non-isothermal reaction, INDUSTRIAL & ENGINEERING CHEMISTRY RESEARCH, Volume: 44, Issue: 10, Pages: 3619-3625, MAY 2005.
[4] Painuly A., Pushpavanam S., Kienle A., Steady state behavior of coupled nonlinear reactor-separator systems: Effect of different separators, INDUSTRIAL & ENGINEERING CHEMISTRY RESEARCH, Volume: 44, Issue: 7, Pages: 2165-2173, MAR 2005.
[5] Balasubramanian P., Kosuri M. R., Pushpavanam S., Kienle A., Effect of delay on the stability of a coupled reactor-separator system, INDUSTRIAL & ENGINEERING CHEMISTRY RESEARCH, Volume: 42, Issue: 16, Pages: 3758-3764, AUG 2003.
[6] Waschler R., Pushpavanam S., Kienle A., Multiple steady states in two-phase reactors under boiling conditions, CHEMICAL ENGINEERING SCIENCE, Volume: 58, Issue: 11, Pages: 2203-2214, JUN 2003.