Chaos origins

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).

Rt is ironic that such an intensely computational mathematical science as chaos theory owes much of its modern origin to calculations that were performed not on a large mainframe computer but rather a simple programmable pocket calculator, a Hewlett-Packard HP-65 Feigenbaum would explain years later that had the series of intermediate calculations not been carried out sufficiently slowly, it is likely that most of the key observations would have been missed [feigSOa]. 

In their correlation, Chao and Seader use the original Redlich-Kwong equation of state for vapor-phase fugacities. For the liquid phase, they use the symmetric convention of normalization for y and partial molar volumes which are independent of composition, depending only on temperature. For the variation of y with temperature and composition, Chao and Seader use the equation of Scatchard and Hildebrand for a multicomponent solution ... 

G. Houart, G. Dupont, and A. Goldbeter, Bursting, chaos and birhythmicity originating from self-modulation of the inositol 1,4,5-triphosphate signal in a model for intracellular Ca + oscillations. Bull. Math. Biol. 61, 507-530 (1999). 

Following the collapse of the Third Reich, for German science, as for many sectors of public life, the need for a new start was essential. The state of Germany s institutions at the end of the war corresponded to the general chaos accompanying the defeat. The various institutes of the Kaiser Wilhelm Society (KWS) originally founded in 1911, the predecessor of the Max Planck Society, were damaged or housed provisionally at different evacuation sites. The years of... 

The stability matrix carries the necessary information related to the vicinity of the trajectory and provides an efficient numerical procedure for computing the response function. It plays an important role in the field of classical chaos the sign of its eigenvalues (related to the Lyapunov exponents) controls the chaotic nature of the system. Interference effects in classical response functions have a different origin than their quantum counterparts. For each initial phase-space point we need to launch two trajectories with very close initial conditions. [For 5(n) we need n trajectories.] The nonlinear response is obtained by adding the contributions of these trajectories and letting them interfere. 

Intramolecular dynamics and chemical reactions have been studied for a long time in terms of classical models. However, many of the early studies were restricted by the complexities resulting from classical chaos, Tlie application of the new dynamical systems theory to classical models of reactions has very recently revealed the existence of general bifurcation scenarios at the origin of chaos. Moreover, it can be shown that the infinite number of classical periodic orbits characteristic of chaos are topological combinations of a finite number of fundamental periodic orbits as determined by a symbolic dynamics. These properties appear to be very general and characteristic of typical classical reaction dynamics. 

Results demonstrate that when agitators are switched the slope of the pathline becomes discontinuous. We will see later in this chapter how this mechanism may produce an essentially stochastic response in the Lagrangian sense. Aref termed this chaotic advection, which he suggested to be a new intermediate regime between turbulent and laminar advection. The chaos has a kinematic origin, it is temporal—that is, along trajectories associated with the motion of individual fluid particles. Chaos is used in the sense of sensitivity of the motion to the initial position of the particle, and exponential divergence of adjacent trajectories. 

Krimsky, Sheldon. 2000. Hormonal Chaos The Scientific and Social Origins of the Environmental Endocrine Hypothesis. Baltimore Johns Hopkins University Press. 

Cavalier-Smith T (2006b) Origin of mitochondria by intracellular enslavement of a photosynthetic purple bacterium. Proc R Soc Lond B Biol Sci 273 1943-1952 Cavalier-Smith T (2006c) Rooting the tree of life by transition analyses. Biol Direct 1 19 Cavalier-Smith T, Chao EE (2003a) Phylogeny and classification of phylum Cercozoa (Protozoa). Protist 154 341-358... 

Although the formulation of the hypothesis of molecular chaos still contains many gaps, it certainly shows clearly that the Umkehreinwand and the Wiederkehrein-wand affect only the original formulation of the Stosszahl-amatz in fact, they prove its untenability. The improved statistical version of the Stosszahlamaiz, however, takes into account all those requirements arising from these objections.168... 

The stochastic formulation would be the most appropriate choice to capture the structural and functional heterogeneity in these biological media. When the process is heterogeneous, one frequently observes chaos-like behaviors. Heterogeneity is at the origin of fluctuations, and fluctuations are the prelude of instability and chaotic behavior. Stochastic modeling is able to ... 

In order to have a metabolic and enzymatic system, it would be necessary to have proteinaceous enzymes which can catalyse the synthesis of other enzymes, and for this they should be capable of making peptide bonds. Such systems have not been found in nature, so far, but according to Stuart Kauffman (1986) they could have existed in the past, and in primitive compartments could have produced autocatalytic networks which had the potential to jump from chaos to order. Even if we admit that those enzymes existed, however, we still have the problem of accounting for the origin of the complex autocatalytic networks that housed them. 

The most plausible current models for the origin and evolution of life revolve around self-assembling molecular systems Order begot order from chaos [19, 20], These models have the following features ... 

F. Diacu and P. Holmes, Celestial Encounter The Origin of Chaos and Stability, Princeton University Press, Princeton, 1996. 

At the microscopic level, chemical reactions are dynamical phenomena in which nonlinear vibrational motions are strongly coupled with each other. Therefore, deterministic chaos in dynamical systems plays a crucial role in understanding chemical reactions. In particular, the dynamical origin of statistical behavior and the possibility of controlling reactions require analyses of chaotic behavior in multidimensional phase space. 

---