Oscillatory system, chemical wave

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

Such oscillations are observed in the form of chemical waves or stationary structures (Turing type) in an oscillatory system (Chapter 10). The relative magnitudes of diffusion... 

Smoes, M. L. (1980). Chemical waves in the oscillatory Zhabotinsky system. A transition from temporal to spatio-temporal organization. In Dynamics of synergetic systems, ed. H. Haken (Springer Series in Synergetics, Vol. 6), pp 80-95. Springer Verlag, Berlin. 

Various types of oscillating behaviors such as emergence of chemical waves, chaotic patterns, and a rich variety of spatiotemporal structures are investigated in oscillatory chemical reactions in association with nonlinear chemical dynamics [1-3]. In non-equilibrium condition, the characteristic dynamics of such chemically reacting systems are capable to self-organize into diverse kinds of assembly patterns. With the help of nonlinear chemical dynamics, the complexity and orderliness of those chemical processes can be explained properly. Various biological processes which exhibited very time-based flucmations especially when they are away from equilibrium have also been described by mechanistic considerations and theoretical techniques of nonlinear chemical dynamics [4-7]. 

The BZ reaction is carried out in two-dimensional, i.e., Petri dish filled with the reaction mixture followed by the monitoring of oscillatory phenomenon in the stirred system which can manifest itself formation of traveling chemical waves as could be seen in Fig. 1.6. In the oxidized state of the reaction, the autocatalyst... 

However, physico-chemical systems of small size as well as biological tissues may also self-organize into coherent spatio-temporal activity. For example, it has been shown that calcium waves propagate into oocytes [2J, and spiral activity has been demonstrated in small pieces of heart tissue [3]. Therefore, the boundaries of small systems may influence wave or spiral propagation. Our aim in this chapter is to investigate the role of boundaries on spiral activity and wave propagation in oscillatory systems of small size. 

The bottom line of Table IV gives the calculated recurrence times for critical fluctuations in Br. The smallest recurrence time seen in Table IV is the amount of time necessary to wait for [Br ] to fluctuate to the critical bromide concentration within at least one small sphere of radius 0.5 /rm anywhere within the total volume of the excitable system studied. This minimum recurrence time of 10 seconds, or one billion years, is in fact the smallest recurrence time obtained from all our calculations considering fluctuations in any species for either the excitable or oscillatory medium. Clearly, such critical perturbations predicted by the deterministic equations necessary for the initiation of a chemical wave have vanishingly small probabilities of occurring spontaneously in solution. 

The harmonic oscillator is an important system in the study of physical phenomena in both classical and quantum mechanics. Classically, the harmonic oscillator describes the mechanical behavior of a spring and, by analogy, other phenomena such as the oscillations of charge flow in an electric circuit, the vibrations of sound-wave and light-wave generators, and oscillatory chemical reactions. The quantum-mechanical treatment of the harmonic oscillator may be applied to the vibrations of molecular bonds and has many other applications in quantum physics and held theory. 

The Hopf bifurcation approach is a mathematically rigorous technique for locating and analysing the onset of oscillatory behaviour in general dynamical systems. Another approach which has been particularly well exploited for chemical systems is that of looking for relaxation oscillations. Typically, the wave form for such a response can be broken down into distinct periods,... 

Another form of behaviour exhibited by a number of chemical reactions, including the Belousov-Zhabotinskii system, is that of excitability. This concerns a mixture which is prepared under conditions outside the oscillatory range. The system sits at the stationary state, which is stable. Infinitesimal perturbations decay back to the stationary state, perhaps in- a damped oscillatory manner. The effect of finite, but possibly still quite small, perturbations can, however, be markedly different. The system ultimately returns to the same state, but only after a large excursion, resembling a single oscillatory pulse. Excitable B-Z systems are well known for this propensity for supporting spiral waves (see chapter 1). 

M. Stich and A. S. Mikhailov. Complex pacemakers and wave sinks in heterogeneous oscillatory chemical systems. Z. Phys. Chem., 216 521-533, 2002. 

The spiral or concentric waves observed for the spatial distribution of cAMP (fig. 5.6) present a striking analogy with similar wavelike phenomena found in oscillatory chemical systems, of which the Belousov-Zhabotinsky reaction (fig. 5.7) provides the best-known example (Winfree, 1972a). 

A typical feature of a non-potential systems is the non-stationary oscillatory behavior that usually manifests itself in the propagation of waves. We have shown that the nonlinear evolution of waves near the instability threshold is described by the complex Ginzburg-Landau (CGL) equation. This equation is capable of describing various kinds of instabilities of wave patterns, like the Benjamin-Feir instability. In two dimensions, the CGL equation describes the formation of spiral waves that are observed in many biological and chemical systems characterized by the interplay of diffusion and chemical reactions at nano-scales. 

---