Oscillations spatiotemporal waves

The 1970s saw an explosion of theoretical and experimental studies devoted to oscillating reactions. This domain continues to expand as more and more complex phenomena are observed in the experiments or predicted theoretically. The initial impetus for the smdy of oscillations owes much to the concomitance of several factors. The discovery of temporal and spatiotemporal organization in the Belousov-Zhabotinsky reaction [22], which has remained the most important example of a chemical reaction giving rise to oscillations and waves. 

In conclusion, the study of models for Ca signalling indicates that the phenomena of Ca oscillations and waves are closely intertwined. The spatiotemporal patterns correspond to the propagation of a Ca front in a biochemically excitable or oscillatory medium, at a rate much higher than that associated with simple diffusion. Such a property could also underlie a possible role of Ca waves in intercellular communication (Charles et al., 1991). The results presented in figs. 9.30 and 9.31 show that a unique mechanism, based on CICR, can account for the... 

The other case studies are all based on the exchange of energy between the two subvarieties, according to the three ways of considering the space-time purely temporal, purely spatial, and both in one- and three-dimensional space. The systems studied are temporal and spatial oscillators and wave propagations in the spatiotemporal case. 

This relationship is quite general, although restricted to the linear case, and works for any wave function. In the case of a conservative spatiotemporal oscillator, or wave, the wave function modulus is a constant ... 

The examples shown in this chapter are only a small part of the rich variety of behavior encountered in far-from-equilibrium chemical systems. Here our objective is only to show a few examples an extensive description would form a book in itself At the end of the chapter there is a list of monographs and conference proceedings that give a detailed descriptions of oscillations, propagating waves, Turing structures, pattern formation on catalytic surfaces, multistability and chaos (both temporal and spatiotemporal). Dissipative structures have also been found in other fields such as hydrodynamics and optics. 

We carefully dissected rat tail arteries and loaded them with a Ca2+ indicator, Fluo-3. After a rectangular glass capillary was inserted into the lumen of the excised arteries, [Ca2+] in smooth muscle cells within the arterial wall was visualized using a confocal microscope. Brief electrical shocks were delivered at 5 Hz to the preparations to stimulate the sympathetic nerve network present in the adventitia. We found Ca2+ signals with diverse spatiotemporal patterns, Ca2+ waves and oscillations in individual smooth muscle cells during the sympathetic nerve stimulation (lino et al 1994). 

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. 

Temporal patterns and oscillations have been observed in a very broad spectrum across various spatial scales. For example, periodic dynamics have been studied in gene expression patterns in the cell division cycle and the cellular redox state alterations, while the genome-wide oscillations at the transcript and protein levels indicate the cell cycle as a developmental process [33]. Spatiotemporal oscillations have been observed in mitochondria, transmembrane potentials, heart excitation waves, neural activities and brain dynamics, cognition and verbal working memory, and even bacteria [16, 34-37]. 

The most clear-cut examples of the influence of an insulating boundary on spatiotemporal dynamics come from experiments on the dissolution of iron. In 1969, Pigeau and Kirkpatrick presented a sequence of images in which a wave could be seen that emerged at the rim of a disk-shaped electrode and propagated toward the center. These radially symmetric waves constituted passivation waves accompanying the decrease in current density during typical relaxation oscillations. 

The discussion of the experimental results in Section in.2 shows that a fundamental understanding of pattern formation in electrochemical systems has been achieved. However, it also demonstrates that the present state represents just a first step toward a complete picture of possible dynamic behaviors. There are many observations that cannot yet be explained, for example, the spatiotemporal period-doubling bifurcation detected during the electrodissolution of iron, the occurrence of antiphase oscillations during Ni electrodissolution, or the emergence of modulated waves during the electrodissolution of Co. Nevertheless, these phenomena seem to be understandable through an extension of the models introduced in Section III.l. 

All the complex behavior described so far in this Chapter arises from the diffusive coupling of the local dynamics which in the homogeneous case have simple fixed points as asymptotic states. If the local dynamics becomes more complex, the range of possible dynamic behavior in the presence of diffusion becomes practically unlimited. It is clear that coupling chaotic subsystems could produce an extremely rich dynamics. But even the case of periodic local dynamics does so. Diffusively coupled chemical or biological oscillators may become synchronized (Pikovsky et ah, 2003), or rather additional instabilities may arise from the spatial coupling. This may produce target waves, spiral patterns, front instabilities and several different types of spatiotemporal chaos or phase turbulence (Kuramoto, 1984). 

The experimental study of Ca waves in fertilized eggs (Gilkey et al., 1978 Jaffe, 1983 Busa Nuccitelli, 1985) preceded the observation of temporal oscillations of Ca " in these and other cells. Likewise, the theoretical study of spatiotemporal Ca patterns was initially uncoupled from the study of models for Ca oscillations. Thus, empirical models were proposed to account for the propagation of Ca waves in amphib-... 

Basin of attraction, see Attraction basin Bell-shaped dependence, of Ca release on Ca, 358,359,379,499 Belousov-Zhabotinsky reaction chaos, 12,283,511 chemical waves, 169,513 excitability, 102,213 oscillations, 1,508 temporal and spatiotemporal organization, 7,169 Bifurcation... 

If the eigenvalues of Eqn (13.10) are real and at least one of them is positive due to the controlled parameters A, B, and the diffusion coefficients, only the spatial patterns of the Turing structure of sin Xr will arise without temporal oscillations. If, on the other hand, the eigenvalues are a complex-conjugate pair, then the solutions represent spatiotemporal instabilities and lead to propagating waves. If the real part is positive, the perturbation grows (Kondepudi and Prigogine, 1999). 

A system worth studying is the oscillator. The temporal oscillator uses only time and no space the converse is true for the spatial oscillator. The spatiotemporal oscillator combines both space-time properties for modeling the propagation of waves sound, light, or mechanical vibrations. 

This equation is the differential definition of the exponential function as already seen, thus modeling the wave function of the spatiotemporal oscillator with exactly the same expression as for the temporal and spatial oscillators... 

It seems feasible that the spatiotemporal oscillator, which is the generator of wave propagation, can be described as a combination of two oscillators, a temporal one and a spatial one. However, this has to be proven. 

Each oscillator possesses a pair of constitutive properties, an inductive one and a capacitive one they determine the pulsation co (hence the frequency) and the wave vector k (hence the wavelength) of the propagating wave. To establish this relationship between the spatiotemporal oscillator and its constituents, a relationship between these pairs of constitntive properties has to be found. 

This is done by applying separately to the wave function the operators that were defined previously for each oscillator. By applying first the pulsation operator co defined by Equation 9.116 that provided, through Equation 9.117, the pulsation eigen-value (O in the case of the temporal oscillator, one concludes that the pulsation is also an eigen-function of the same operator applied to the wave function of the spatiotemporal oscillator. The same conclusion is reached, by using Equation 9.150, for the wave vector k that appears also as an eigen-function of the wave vector operator k defined by Equation 9.149. 

In Chapter 9, the wave fnnction of a spatiotemporal oscillator has been demonstrated as being the product of two individual wave functions, one for the temporal oscillator and the other for the spatial one. From this property, the phase angle (here an apparent one for taking into acconnt the damping) was expressed as a double contribution of the individual phase angles, according to the relationship... 

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]. 

We employ a method of numerical continuation which has been earlier developed into a software tool for analysis of spatiotemporal patterns emerging in systems with simultaneous reaction, diffusion and convection. As an example, we take a catalytic cross-flow tubular reactor with first order exothermic reaction kinetics. The analysis begins with determining stability and bifurcations of steady states and periodic oscillations in the corresponding homogeneous system. This information is then used to infer the existence of travelling waves which occur due to reaction and diffusion. We focus on waves with constant velocity and examine in some detail the effects of convection on the fiiont waves which are associated with bistability in the reaction-diffusion system. A numerical method for accurate location and continuation of front and pulse waves via a boundary value problem for homo/heteroclinic orbits is used to determine variation of the front waves with convection velocity and some other system parameters. We find that two different front waves can coexist and move in opposite directions in the reactor. Also, the waves can be reflected and switched on the boundaries which leads to zig-zag spatiotemporal patterns. 

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