Multiple limit cycle

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

Although most of the results can be established with mathematical rigor, there are some elusive problems. These center around the possibility of multiple limit cycles and the difficulty of determining the stability of such limit cycles. At this point one must simply make a hypothesis and resort to numerical evidence in any specific case. Determining the number of limit cycles is a deep mathematical problem, and even in very simple cases the solution is not known. Hilbert s famous sixteenth problem, concerning the number of limit cycles of a second-order system with polynomial right-hand sides, remains basically unresolved. In principle, the stability of a limit cycle can be determined from the Floquet exponents (see Section 4), but this is a notoriously difficult computation - indeed, generally an impossible one. 

If the inequality is reversed then the rest point E. is unstable - a repeller. The Poincar -Bendixson theorem then allows one to conclude that there exists a limit cycle. Unfortunately, there may (theoretically) be several limit cycles. If all limit cycles are hyperbolic then there is at least one asymptotically stable one, for if there are multiple limit cycles the innermost one must be asymptotically stable. Moreover, since all trajectories eventually lie in a compact set, there are only a finite number of limit cycles and the outermost one must be asymptotically stable. Since the system is (real) analytic, one could also appeal to results for such systems. For example, Erie, Mayer, and Plesser [EMP] and Zhu and Smith [ZSJ show that if E is unstable then there exists at least one limit cycle that is asymptotically stable. Stability of limit cycles will be discussed in the next section. We make a brief digression to outline the principal parts of this theory, and then return to the food-chain problem. 

Theorem 7.1 guarantees the coexistence of both the Xi and X2 populations when Ec exists. However, it does not give the global asymptotic behavior. The further analysis of the system is complicated by the possibility of multiple limit cycles. Since this is a common difficulty in general two-dimensional systems, it is not surprising that such difficulties occur in the analysis of three-dimensional competitive systems. 

Model 4 Multiple Singular Points and Multiple Limit Cycles... 

Solutions obtained are multiple limit cycles, and multiple singular points as shown in Fig. III.20. 

Fig. III.20. Multiple limit cycles, one stable and two unstable, and multiple singular points, two stable focus and one saddle point. (From Kaimachnikov and Sel kov (1975))...

In this table 1 LC stands for a single limit cycle, and M LC indicates multiple limit cycles. 

Multiple solutions more complex than this are also present in the literature. For example for two-dimensional systems Aris and Amundson (1958) discussed multiple singular solutions as well as multiple limit cycles appearing simultaneously. 

Linear control theory will be of limited use for operational transitions from one batch regime to the next and for the control of batch plants. Too many of the processes are unstable and exhibit nonlinear behavior, such as multiple steady states or limit cycles. Such problems often arise in the batch production of polymers. The feasibility of precisely controlling many batch processes will depend on the development of an appropriate nonlinear control theory with a high level of robustness. 

In the field of responsive agents, enzyme targeting has specific advantages. A small concentration of the enzyme can convert a relatively high amount of the probe in multiple catalytic cycles which considerably decreases the detection limit for the enzyme as compared to other biomolecules. Moreover, enzymatic reactions are usually highly specific therefore, the observed change... 

Cutlip and Kenney (44) have observed isothermal limit cycles in the oxidation of CO over 0.5% Pt/Al203 in a gradientless reactor only in the presence of added 1-butene. Without butene there were no oscillations although regions of multiple steady states exist. Dwyer (22) has followed the surface CO infrared adsorption band and found that it was in phase with the gas-phase concentration. Kurtanjek et al. (45) have studied hydrogen oxidation over Ni and have also taken the logical step of following the surface concentration. Contact potential difference was used to follow the oxidation state of the nickel surface. Under some conditions, oscillations were observed on the surface when none were detected in the gas phase. Recently, Sheintuch (46) has made additional studies of CO oxidation over Pt foil. 

In Section 5.03.6.2, a stereoselective synthesis of L-homophenylalanine from the racemic AAacetylated amino acid is described. The authors, however, found that substrate solubility limited the utility of this procedure. Having found an L-N-carbamoylase in Bacillus kaustophilus, they introduced the gene for this enzyme together with that for the N-acyl amino acid racemase from D. radiodurans into E. coli for coexpression. These cells, permeabilized with 0.5% toluene, were able to deliver L-homophenylalanine in 99% yield and were able to be used for multiple reaction cycles. 

When the steady state becomes unstable, the system moves away from it and often undergoes sustained oscillations around the unstable steady state. In the phase space defined by the system s variables, sustained oscillations generally correspond to the evolution toward a limit cycle (Fig. 1). Evolution toward a limit cycle is not the only possible behavior when a steady state becomes unstable in a spatially homogeneous system. The system may evolve toward another stable steady state— when such a state exists. The most common case of multiple steady states, referred to as bistability, is of two stable steady states separated by an unstable one. This phenomenon is thought to play a role in differentiation [30]. When spatial inhomogeneities develop, instabilities may lead to the emergence of spatial or spatiotemporal dissipative stmctures [15]. These can take the form of propagating concentration waves, which are closely related to oscillations. 

Yet another type of complex oscillatory behavior involves the coexistence of multiple attractors. Hard excitation refers to the coexistence of a stable steady state and a stable limit cycle—a situation that might occur in the case of circadian rhythm suppression discussed in Section VI. Two stable limit... 

Experimentally, in chemical systems a variety of dynamic states can be observed resulting from thermodynamic and kinetic conditions as defined as a prerequisite for the evolution of dissipative structures. Today we distinguish the following states (1) the maintenance of multiple steady states with transitions from one to another, (2) rotation on a limit cycle... 

Another useful rule which can frequently guide us to situations where oscillatory solutions will be found is the Poincare-Bendixson theorem. This states that if we have a unique stationary state which is unstable, or multiple stationary states all of which are unstable, but we also know that the concentrations etc. cannot run away to infinity or become negative, then there must be some other non-stationary atractor to which the solutions will tend. Basically this theorem says that the concentrations cannot just wander around for an infinite time in the finite region to which they are restricted they must end up somewhere. For two-variable systems, the only other type of attractor is a stable limit cycle. In the present case, therefore, we can say that the system must approach a stable limit cycle and its corresponding stable oscillatory solution for any value of fi for which the stationary state is unstable. 

Escher, C. and Ross, J. (1983). Multiple ranges of flow rate with bistability and limit cycles for Schlogl s mechanism in a CSTR. J. Chem. Phys., 79, 3773-7. 

Fig. 8.12. The loci DH, and DH2 corresponding to degenerate Hopf bifurcation points at which the stability of the emerging limit cycle is changing. Again, these are shown relative to the loci for stationary-state multiplicity (broken curves).

In chapter 12 we discussed a model for a surface-catalysed reaction which displayed multiple stationary states. By adding an extra variable, in the form of a catalyst poison which simply takes place in a reversible but competitive adsorption process, oscillatory behaviour is induced. Hudson and Rossler have used similar principles to suggest a route to designer chaos which might be applicable to families of chemical systems. They took a two-variable scheme which displays a Hopf bifurcation and, thus, a periodic (limit cycle) response. To this is added a third variable whose role is to switch the system between oscillatory and non-oscillatory phases. 

FIGURE 29 The development of the stroboscopic phase-plane. Segments (a) and (d) show the trajectory settling down to a limit-cycle through a sequence of points at times that are multiples of t. This is drawn out in the time dimension in (b) and shown in its regularity in (c). If there is a unique periodic solution, the stroboscopic plane will show a sequence of states converging on (e). 

FIGURE I Bifurcation diagrams of the autonomous system for y, = 0.001, y2 = 0.002. (a) Multiple-steady states are found inside the finger-shaped region, and limit cycles are born upon crossing the Hopf curves, (b) Steady-state and limit-cycle branches for < 1 = 0.019. The location of the average value of a2 used for forced oscillations is 0.028, which is in the oscillatory region and a distance of Ao from the left Hopf bifurcation point labelled c. 

The complicated situations of simultaneous occurrence of multiplicity a limit cycles will not be considered in this text. 

Next we display 13 single, double, or multiple plots drawn by our MATLAB program neurocycle.m of (a) the acetylcholine concentration profile in compartment (II) above the phase plot of the acetylcholine concentration in compartment (I) versus that in compartment (II), or (b) the limit cycle plot, or (c) the plot of all 8 profiles. We include interpretative comments on the solution s behavior in each case. 

This result can be generalized for multi-dimensional systems in which a limit set for every motion is a fixed point or a limit cycle, linear approximation matrices at fixed points have no eigenvalues in the imaginary axis and limit cycles have no multiplicators on the unit circle. In this case, k should be treated for fixed points as the sums of those eigenvalues that have positive real parts (they are "unstable ), and for limit cycles as the sums of unstable characteristics indices. 

The available data from emulsion polymerization systems have been obtained almost exclusively through manual, off-line analysis of monomer conversion, emulsifier concentration, particle size, molecular weight, etc. For batch systems this results in a large expenditure of time in order to sample with sufficient frequency to accurately observe the system kinetics. In continuous systems a large number of samples are required to observe interesting system dynamics such as multiple steady states or limit cycles. In addition, feedback control of any process variable other than temperature or pressure is impossible without specialized on-line sensors. This note describes the initial stages of development of two such sensors, (one for the monitoring of reactor conversion and the other for the continuous measurement of surface tension), and their implementation as part of a computer data acquisition system for the emulsion polymerization of methyl methacrylate. 

Figure 4a shows Xq the stationary state value of X for the nonlinear system as a function of P. With this set of parameters no multiple steady states exist but the steady states marked by the broken line are unstable and evolve to stable limit cycles. The amplitude of the limit cycles is shown in Figure 4B. 

Fig. 16 are only rarely strictly periodic, because usually rather small fluctuations in the external parameters are sufficient to trigger abrupt changes. However, in principle, mixed-mode oscillations belong to the category of multiple-periodic limit cycles. If the behavior is governed by two incommensurate frequencies, i.e., the ratio of two periodicities is an irrational number. This situation is denoted by quasiperiodicity and has been realized experimentally with periodically forced oscillations, as will be described next. 

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