Chaos interacting oscillators

Before we can start to develop a model we also have to decide how to interpret the behavior observed in Fig. 2.1. The variations in insulin and glucose concentrations could be generated by a damped oscillatory system that was continuously excited by external perturbations (e.g. through interaction with the pulsatile release of other hormones). However, the variations could also represent a disturbed self-sustained oscillation, or they could be an example of deterministic chaos. Here, it is important to realize that, with a sampling period of 10 min over the considered periods of 20-24 h, the number of data points are insufficient for any statistical analysis to distinguish between the possible modes. We need to make a choice and, in the present case, our choice is to consider the insulin-glucose regulation to operate... 

In a review article on oscillatory reactions (294), Sheintuch discusses the effect of introducing a heat balance for the catalyst rather than a mass balance for the reactor into the differential equation system for a surface reaction with oxidation/reduction cycles. Although the coverage equations alone can yield oscillatory behavior, as was the case for the models discussed in the previous section, Sheintuch s model is discussed in this section because introduction of the heat balance adds qualitatively new features. In this extended system complex, multiple peak behavior and quasiperiodicity was observed as shown in Fig. 8. Sheintuch also investigated the interaction of two oscillators. This work, however, will be treated in detail in Section V, were synchronization and chaos are discussed. 

We have seen that chemical and biological interactions lead to mathematical models displaying a variety of linear and nonlinear behavior relaxation to fixed points, multistability, excitability, oscillations, chaos, etc. Despite the different origin of the models, and the diverse nature of the variables they represent (chemical concentrations, population numbers, or even membrane electric potentials) the mathematical structures are quite similar, and it is possible to understand some aspects of the dynamics in one field (e.g. the chemical oscillations in the BZ reaction) with the help of models from other fields (for example the FN model of neurophysiology, or a phytoplankton-zooplankton model). This possibility of common mathematical description will be used in the rest of the book to highlight the similarities and relationships between chemical and biological dynamics when occurring in fluid flows. 

Theoretical investigations of this model (A. G. Makeev, B. E. Nieuwenhuys, Mathematical modeling of the NO + H2/Pt(100) reaction "Surface explosion," kinetic oscillations, and chaos, Journal of Chemical Physics, 108 (1998) 3740-3749) with 11 reversible and irreversible elementary steps included lateral interactions for only two steps in the forward direction and two steps in the reverse direction, leading to the following rate expressions... 

The complex oscillations predicted by the model can be related to those sometimes observed, at low values of the substrate injection rate, in glycolysing yeast extracts. Such complex glycolytic oscillations (fig. 4.31) could represent chaos resulting from the interaction between oscillating phosphofructokinase and a second instability-generating reaction, catalysed by another glycolytic enzyme, in a small range of values of the substrate injection rate. 

Fig. 6.25. Origin of complex oscillations in the cAMP signalling system of D. discoideum. Complex behaviour (birhythmidty, bursting and chaos) originates from the interaction of two endogenous oscillatory mechanisms that are coupled in parallel. The two mechanisms share the same feedback loop of selfamplification by extracellular cAMP, via the binding of the latter to the receptor, and differ by the process responsible for limiting autocatalysis (dotted area) the first limiting process is based on receptor desensitization, and the second on substrate availability at the adenylate cyclase reaction site (Goldbeter Martiel, 1987).

While the coexistence between two limit cycles or between a limit cycle and a stable steady state is also shared by the two-variable models of fig. 12.1b and c, new modes of complex dynamic behaviour arise because of the presence of a third variable in the multiply regulated system. The coexistence between three simultaneously stable limit cycles, i.e. trirhythmicity, is the first of these. Moreover, the interaction between two instability-generating mechanisms allows the appearance of complex periodic oscillations, of the bursting type, as well as chaos. The system also displays the property of final state sensitivity (Grebogi et ai, 1983a) when two stable limit cycles are separated by a regime of unstable chaos. 

Whereas chaos and complex oscillations in the models studied here result from the interaction between two endogenous oscillators, in these experiments and in the associated models they result from the coupling between an endogenous oscillator and a periodic extem d source, as indicated by the analysis of normal forms for such a situation (Baesens Nicolis, 1983). Evidence for autonomous chaos has nevertheless been obtained in molluscan neurons (Holden et al, 1982 Hayashi Ishizuka, 1992). 

Similar to energy transfer between different shapes of a liquid droplet, coupling between the volume oscillation and different shape oscillations occur for bubbles in acoustic fields [43]. Interaction between modes can lead to chaotic response of the bubble to the external forcing. For a large enough bubble, the spectrum of distortion modes is dense, and several distortion modes attribute to the shape. Development of chaos depends on the number of excited shape modes. 

When we think of diffusion acting on a system in which there are concentration inhomogeneities, our intuition suggests that diffusion should act to lessen, and eventually eliminate, the inhomogeneities, leaving a stable pattern with concentrations that are equal everywhere in space. As in the case of temporal oscillation, for a closed system the laws of thermodynamics require that this intuition be valid and that the eventual concentration distribution of the system be constant, both in time and in space. In an open system, however, just as appropriate nonlinear rate laws can lead to temporal structure, like oscillations and chaos, the interaction of nonlinear chemical kinetics and diffusion can produce nonuniform spatial structure, as suggested schematically in Figure 14.1. 

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