Chaotic transitions connections

In the presence of weak noise there is a finite probability of noise-induced transitions between the chaotic attractor and the stable limit cycle. In Fig. 14 the filled circles show the intersections of one of the real escape trajectories with the given Poincare section. The following intuitive escape scenario can be expected in the Hamiltonian formalism. Let us consider first the escape of the system from the basin of attraction of a stable limit cycle that is bounded by an saddle cycle. In general, escape occurs along a single optimal trajectory qovt(t) connecting the two limit cycles. 

To connect the two markedly different scenarios observed in the static and the well-mixed environments, it is natural to analyze the role of increasing mobility (Reichenbach et al., 2007). Karolyi et al. (2005) studied the above competition model combined with dispersion by a chaotic map that represents advection of fluid elements in the alternating sine-flow. By continuously changing the frequency of the chaotic dispersion as a control parameter, it is possible to follow the transitions between the two limiting situations. When the chaotic mixing is much faster than the local population dynamics, the killer and resistant cells gradually disappear from the population and only the sensitive cells survive. This is because the killer cells... 

As a rule, the hydration layer around a dissolved ion extends over several molecular H2O layers, the extension being primarily determined by the electric field strength around the ion. This field strength, in turn, is a function of the valency and the radius of the ion. The preferred direction of the H2O dipoles in the hydration layer leads to a reduction of the entropy. The structured hydration layer is connected to the differently structured bulk water via a transition layer in which the H2O molecules have to switch over from one structure to the other. As a result, the transition layer is relatively chaotic having a higher entropy, as shown in Figure 4.4. 

The generation of self-sustained oseillations is a particular case of bifurcation. The term bifurcation is often used in connection with the mathematical study of dynamical systems. It denotes a sudden qualitative ehange in the behavior of a system upon the smooth variation of a parameter, the so-eaUed bifureation parameter, and is applied to the point of the fundamental reeonstmetion of the phase portrait where the bifurcation parameter attains its critical value. The simplest examples of bifurcation are the appearance of a new rest point in the phase space, the loss of the rest-point stability, and the appearance of a new limit cycle. Bifurcation relates to physicochemical phenomena such as ignition and extinction, that is, a jump-like transition from one steady state to another one, the appearance of oscillations, or a chaotic regime, and so on. 

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