We analyze the nonlinear dynamics of an isothermal system involving complex enzymatic reactions for L-DOPA (L-3,4-dihydroxyphenylalanine) production by numerical simulation. The mass action kinetics of the system forms a family of 9ordinary differential equations with 22 parameters. The multiple steady states are calculated by the chemical reaction network toolbox. Starting from one of the steady states, a limit point is guaranteed to be detected due to a change in the system parameters using the numerical continuation software MatCont. Other bifurcations are also obtained via the bifurcation continuations of the limit point, such as cusp bifurcations, Hopf bifurcations, limit cycles, zero Hopf bifurcations, generalized Hopf bifurcations, period-doubling, and so on. A transition of a period-doubling bifurcation to chaos occurs by numerical simulations. Positive values, 0.25~0.71, of Lyapunov exponents are obtained for the chaotic dynamics. Poincare maps and power spectrum densities are also plot. The Feigenbaum constant is computed to be 4.681~4.705.
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We are grateful to the National Center for High-Performance Computing (NCHC) of Taiwan for computer time and facilities. We also would like to thank Dr. Hil Meijer for kind help with the MatCont in bifurcation analysis.
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Yuan-Hong Luo, Hsing-Ya Li Numerical Analysis of Multiple Steady States, Limit Cycles, Period-Doubling, and Chaos in Enzymatic Reactions Involving Oxidation of L-tyrosine to Produce L-DOPA. Theor Found Chem Eng 54, 1340–1352 (2020). https://doi.org/10.1134/S004057952006007X
- multiple steady states
- Hopf bifurcation
- limit cycle
- period-doubling bifurcation