Takens-Bogdanov Point Research Articles

Micromixing effects in the nicolis—puhl reaction: numerical bifurcation and stability analysis of the IEM model

The Nicolis—Puhl reaction, an isothermal, two-component, autocatalytic reaction exhibiting bistability, oscillatory solutons, and Takens-Bogdanov points in a perfectly micromixed CSTR has been shown to be sensitive to the degree of micromixing under non-premixed feed conditions. These properties make it an interesting theoretical model for exploring the effect of micromixing on the dynamics of nonlinear chemical reactions; in particular, for examining whether micromixing effects suffice to produce complex dynamics in an otherwise two-dimensional dynamical system. Here, the reaction is studied in conjunction with the interaction-by-exchange-with-the mean (IEM) model of micromixing. Numerical bifurcation and stability analysis techniques applicable to the specific class of integro-differential equations to which this model belongs are employed to carry out the analysis. For various values of the reaction rates, the micromixing rate and the inverse residence time have been used as bifurcation parameters to construct two-dimensional bifurcation sets containing saddle-node and Hopf bifurcation curves whose locations are shown to be strongly dependent on the degree of micromixing. As is observed experimentally by lowering the stirring rate for some real chemical reactions in flow reactors, regions of bistability and oscillatory behavior are shown to be accessible, by decreasing the micromixing rate, from regions exhibiting a single steady state. Manifolds of Takens-Bogdanov points for finite micromixing rates have also been computed; however, it is shown that the dynamical dimension of the IEM model remains the same for all values of the micromixing parameter.

Time-modulated oscillatory convection.

We investigate the effect of temporal modulation on the spatial patterns produced by a spatially extended system undergoing a Hopf bifurcation. It is shown that physically this modulation can stabilize standing waves, which would otherwise be stable to traveling waves. Mathematically a codimension-3 point is naturally introduced which encompasses a Takens-Bogdanov codimension-2 point with vanishing Hopf frequency. Experimentally this setup is attractive since the three relevant parameters are easily accessible and the steady bifurcation can be made forward as well as backward near a codimension-2 point.

Normal form analysis of multiple bifurcations in incompletely mixed chemical reactors

Using the theory of normal forms, we investigate the effects of mixing in a continuous flow stirred tank reactor (CSTR) for a reaction model exhibiting oscillatory behavior in the vicinity of a degenerated bifurcation point (here, a Takens–Bogdanov point). In addition we show without specification of a particular reaction system that, as long as reaction rates remain much slower than the inverse mixing time, incomplete mixing introduces a new bifurcation parameter for nonpremixed feeding conditions, whereas premixed feeding conditions merely lead to a renormalization of flow rate.