Bifurcation continuation methods

Procedures enabling the calculation of bifurcation and limit points for systems of nonlinear equations have been discussed, for example, by Keller (13) Heinemann et al. (14-15) and Chan (16). In particular, in the work of Heineman et al., a version of Keller s pseudo-arclength continuation method was used to calculate the multiple steady-states of a model one-step, nonadiabatic, premixed laminar flame (Heinemann et al., (14)) a premixed, nonadiabatic, hydrogen-air system (Heinemann et al., (15)). 

We know of no numerical procedures that will guarantee finding all the solutions to an arbitrary set of nonlinear equations. Continuation methods are often capable of finding more than one solution if several exist. Fidkowski et al. (1993) propose using such a method along with discovering bifurcation points to compute all the azeotropic compositions for a mixture. Their homotopy function... 

The continuation of Hopf bifurcation (HB) points. The continuation method described by KublCek Holodniok [6] has been used. 

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

The power load Pl increases by a factor until the voltage collapse is reaehed. The analysis is tested on the distribution system. Continuation method is used to identify the location of bifurcation point (2 = 2.45), as presented in Fig. 30. 

Abstract. Numerical bifurcation analysis and continuation methods are proposed for the investigation of chemical reactors. An algorithm for the continuation of periodic solutions of large systems is presented. Steady state and periodic continuation are applied to the circulation loop reactor, which makes use of travelling reaction zones for autothermal operation and serves as an example for unsteady state operation of a chemical process. 

When investigating a model of a chemical plant or process, one of the most important tasks is to determine the influence of model parameters like operation conditions or geometric dimensions on performance and dynamics. Because in most cases a large number of parameters has to be examined, an efficient tool for the determination of parameter dependencies is required. Continuation methods in conjunction with the concepts of bifurcation theory have proved to be useful for the analysis of nonlinear systems and are increasingly used in chemical engineering science. They offer the possibility to compute steady states or periodic solutions directly as a function of one or several parameters and to detect changes in the qualitative behaviour of a system like the appearance or disappearance of multiple steady states. In this paper, numerical methods for the continuation of steady states and periodic solutions for large sparse systems with arbitrary structural properties are presented. The application of this methods to models of chemical processes and the problems which arise in this context are discussed for the example of a special type of catalytic fixed bed reactor, the so-called circulation loop reactor. 

The predictor/corrector algorithm in Diva includes a stepsize control in order to minimize the number of predictor and corrector steps. Finally, the continuation package contains methods for the computation of the dominating eigenvalues of DAEs. This allows a stability analysis of the steady state solutions and a detection of local bifurcations for large sparse systems. As the continuation method is embedded into a dynamic simulator, the user has the opportunity to switch interactively from continuation to time integration. This allows additional investigations of transient behaviour or domains of attraction with the same simulation tool[2]. 

The continuation method for periodic solutions allows the completion of the bifurcation diagram in Fig.3.1. In Fig.4.1 the periodic solutions are represented by the maximum value of the corresponding characteristic temperature. As it can be seen, each pair of Hopf bifurcation points is connected by... 

State bifurcations, arise. Two widely used packages that implement numerical continuation methods are AUTO (Doedel et al., 1991) and CONT (Marek and Schreiber, 1991). These programs trace out curves of steady states and follow the properties of limit cycles by using techniques similar to the predictor-corrector approach described above to improve their computational efficiency. An example is shown in Figure 7.8. 

In addition to multiple steady states also existence of oscillations of various types has been observed in the model. Continuation methods (Kubicek and Marek, 1983) can be used to locate positions of limit points (multiple solutions), Hopf bifurcation points (origin of oscillations) and period doubling bifurcation points. Fig. 2 shows an example of the results of such computations, using the continuation software CONT (Kohout et al.. 

The method sketched so far is based on the assumption that there exists a smooth solution path x(s) without bifurcation and turning points. Before describing continuation methods in more algorithmic details we look for criteria for an existence of such a solution path. 

In their subsequent works, the authors treated directly the nonlinear equations of evolution (e.g., the equations of chemical kinetics). Even though these equations cannot be solved explicitly, some powerful mathematical methods can be used to determine the nature of their solutions (rather than their analytical form). In these equations, one can generally identify a certain parameter k, which measures the strength of the external constraints that prevent the system from reaching thermodynamic equilibrium. The system then tends to a nonequilibrium stationary state. Near equilibrium, the latter state is unique and close to the former its characteristics, plotted against k, lie on a continuous curve (the thermodynamic branch). It may happen, however, that on increasing k, one reaches a critical bifurcation value k, beyond which the appearance of the... 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

The continuation of the strategy presents at this point a bifurcation. The solute M may be described with a semiclassical procedure similar to that used for solvent molecules, or with a QM approach. The first method is often called classical (or semiclassical) MM description [3], the second a combined QM/MM approach [4], The physics of the first method is rather elementary, but notwithstanding this it opened the doors to our present understanding of the solvation of molecules. The second method is markedly more accurate, because the QM description of the solute has the potential of taking into account subtler solvent effects, such as the solvent polarization of the solute electronic polarization and the changes in geometry within M. 

Hie continuation of points of fiogdanov-Takens bifurcation (origins of Hopf point branches Roose, [10]). The method has been developed with the adoption of shooting method. 

In this chapter, we describe an algorithm for predicting feasible splits for continuous single-feed RD that is not limited by the number of reactions or components. The method described here uses minimal information to determine the feasibility of reactive columns phase equilibrium between the components in the mixture, a reaction rate model, and feed state specification. This is based on a bifurcation analysis of the fixed points for a co-current flash cascade model. Unstable nodes ( light species ) and stable nodes ( heavy species ) in the flash cascade model are candidate distillate and bottom products, respectively, from a RD column. Therefore, we focus our attention on those splits that are equivalent to the direct and indirect sharp splits in non-RD. One of the products in these sharp splits will be a pure component, an azeotrope, or a kinetic pinch point the other product will be in material balance with the first. 

We employ a method of numerical continuation which has been earlier developed into a software tool for analysis of spatiotemporal patterns emerging in systems with simultaneous reaction, diffusion and convection. As an example, we take a catalytic cross-flow tubular reactor with first order exothermic reaction kinetics. The analysis begins with determining stability and bifurcations of steady states and periodic oscillations in the corresponding homogeneous system. This information is then used to infer the existence of travelling waves which occur due to reaction and diffusion. We focus on waves with constant velocity and examine in some detail the effects of convection on the fiiont waves which are associated with bistability in the reaction-diffusion system. A numerical method for accurate location and continuation of front and pulse waves via a boundary value problem for homo/heteroclinic orbits is used to determine variation of the front waves with convection velocity and some other system parameters. We find that two different front waves can coexist and move in opposite directions in the reactor. Also, the waves can be reflected and switched on the boundaries which leads to zig-zag spatiotemporal patterns. 

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