A problem of frequent interest in the engineering and science is the analysis of nonlinear ordinary differential equation (ODE) systems. These systems often display rich mathematical behavior, including varying numbers of steady states (equilibria). The number and/or stability of equilibria in a model may change in a bifurcation as one or more model parameters change. Bifurcations of equilibria are generally found by solving a nonlinear algebraic equation system consisting of the equilibrium (steady-state) conditions, plus one (in the case of codimension-1 bifurcations) or two (codimension-2 bifurcations) additional conditions. Typically this equation system is solved using some continuation-based strategy (Kuznetsov, 1991). However, in general, these methods are initialization dependent and do not provide any guarantee that all bifurcations will be found. Thus, without some a priori knowledge of system behavior, one may not know with complete certainty if all bifurcation curves have been identified and explored.

Interval mathematics (e.g., Kearfott, 1996) provides tools with which one can resolve this issue with computational certainty. Specifically, in solving a nonlinear algebraic equation system, use of an interval-Newton approach provides a mathematical and computational guarantee that all solutions will be found (or more precisely enclosed within a very narrow interval). Using this approach, all equilibrium states and bifurcations within parameter intervals of interest can be located with certainty and without need for initialization. In previous work, this method was applied to the problem of locating equilibrium states and bifurcations in simple ecosystem models, specifically a tritrophic Rosenzweig-MacArthur food-chain model (Gwaltney, et al., 2004) and a tritrophic food chain in a chemostat (Canale's model) (Gwaltney and Stadtherr, 2006). In the context of dynamical systems in chemical engineering, interval methods have also been used to locate equilibrium states and singularities in some reaction and separation operations (Schnepper and Stadtherr, 1996; Gehrke and Marquardt, 1997; Bischof, et al., 2000; Mönnigmann and Marquardt, 2002).

In this presentation, we consider a novel technique for locating equilibrium states, codimension-1 bifurcations (including fold, transcritical, and Hopf bifurcations), and codimension-2 bifurcations (including double-fold, fold-Hopf, and cusp bifurcations) in chemical reactor models. Cusp bifurcations are often found in systems exhibiting multiple steady-state solutions, and can be indicative of the presence of isolated solution branch curves (isola). When local methods or continuation-based methods are used, the presence of isola may be, in many cases, difficult to detect without a priori knowledge of their existence. However, the method presented here, which is based on an interval-Newton strategy, provides both a mathematical and computational guarantee of locating all equilibrium states, including those on isola, as well as all the codimension-1 and codimension-2 bifurcations of interest.

The solution method will be demonstrated using a variety of reactor modeling problems. One such problem is the first-order, exothermic, jacketed CSTR system studied by Russo and Bequette (1996). This system exhibits both cusp bifurcations and isola. In this presentation we will show that on such problems the interval-Newton approach can be used to reliably locate all nonlinear dynamical behavior of interest, including multiple equilibrium states and bifurcations of equilibria.

References

Bischof, C.H., B. Lang, W. Marquardt and M. Mönnigmann, “Verified determination of singularities in chemical processes”, In SCAN 2000, 9th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, Karlsruhe, Germany (September 18-22, 2000).

Gehrke, V. and W. Marquardt, “A singularity theory approach to the study of reactive distillation”, Computers and Chemical Engineering, 21 (Supplement), S1001–S1006 (1997).

Gwaltney, C.R., and M.A. Stadtherr, “Reliable computation of equilibrium states and bifurcations in nonlinear dynamics”, Lecture Notes in Computer Science, 3732, 121 (2006).

Gwaltney, C.R., M.P. Styczynski, and M.A. Stadtherr, “Reliable computation of equilibrium states and bifurcations in food chain models”, Computers and Chemical Engineering, 28, 1981 (2004).

Kearfott, R.B., Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, 1996.

Kuznetsov, Y.A., Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1991.

Mönnigmann, M. and W. Marquardt, “Normal vectors on manifolds of critical points for parametric robustness of equilibrium solutions of ODE systems”, Journal of Nonlinear Science, 12, 85 (2002).

Russo, L.P. and B.W. Bequette, “Effect of process design on the open-loop behavior of a jacketed exothermic CSTR”, Computers and Chemical Engineering, 20(4), 417 (1996).

Schnepper, C.A. and M. A. Stadtherr, “Robust process simulation using interval methods”, Computers and Chemical Engineering, 20(2), 187 (1996).