Spatiotemporal chaos

Turbulence is generally understood to refer to a state of spatiotemporal chaos that is to say, a state in which chaos exists on all spatial and temporal scales. If the reader is unsatisfied with this description, it is perhaps because one of the many important open questions is how to rigorously define such a state. Much of our current understanding actually comes from hints obtained through the study of simpler dynamical systems, such as ordinary differential equations and discrete mappings (see chapter 4), which exhibit only temporal chaosJ The assumption has been that, at least for scenarios in which the velocity field fluctuates chaotically in time but remains relatively smooth in space, the underlying mechanisms for the onset of chaos in the simpler systems and the onset of the temporal turbulence in fluids are fundamentally the same. 

An altogether different behavior emerges for slightly larger values of 7 ( 40 -tSee also our discus.sion of spatiotemporal chaos in coupled-map lattices in section 8.2. 

In the modern theory of fluid dynamic systems the term turbulence is accepted to mean a state of spatiotemporal chaos (e.g., [155], chap 5). That is, the fluid exhibits chaos on all scales in both space and time. Chaos theory involves the behavior of non-linear dynamical systems and their response to initial and boundary conditions. Using such methods it can be shown that although the solution of the Navier-Stokes is apparently random for turbulent flows, its behavior presents some orderly structures. In addition, the numerical solution of the Navier-Stokes equations is sometimes strongly dependent on the initial conditions, thus even very small inaccuracies in the initial conditions may be fatal providing completely erroneous results. ... 

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). 

B.I. Shraiman, A. Pumir, W. van Saarloos, P.C. Hohenberg, H. Chate, and M. Holen. Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation. Physica D, 57 241-248, 1992. 

H. G. Winful and L. Rahman. Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers. Phys. Rev. Lett, 65 1575, 1990. 

Petrov, V. Metens, S. Borckmans, P. Dewel, G. Showalter, K. Tracking unstable Turing patterns through mixed-mode spatiotemporal chaos. Phys. Rev. Lett. 1995, 75, 2895-2898. 

Over the past several years there have been many experimental and theoretical studies aimed at developing a better understanding of pattern formation in reaction-diffusion systems. The focus of recent studies has been on more complex behavior away from the onset of instability. For some parameter values, spatiotemporal chaos may occur near the boundary between the Turing region and the region of homogeneous oscillations (Figure 12). 

W. Zhang and J. Vinals, Secondary instabilities and spatiotemporal chaos in parametric surface waves, Phys. Rev. Lett, 74, 690, 1995. 

The spontaneous switching between the stabilizing and destabilizing modes (i.e., the transition among solutions) was observed as nonperiodic and stochastic. As one of its possible origins, we suppose the combination of intrinsic microscopic fluctuations and spatiotemporal chaos characterized as nonperiodic but nonrandom behavior. Indeed, the system s behavior is chaotic as its time evolution is unstable and unreproducible. Its capability of successively searching for multiple solutions, however, is robustly maintained and qualitatively reproducible. This resembles the robustness of strange attractors of chaotic systems. 

S.J. Moon, M.D. Shattuck, C. Bizon, D.I. Goldman, J.B. Swift, and H.L. Swinney. Phase bubbles and spatiotemporal chaos in granular patterns. 

A. Kudrolli and J.P. Gollub. Localized spatiotemporal chaos in surface waves. 

E. Scholl, Nonlinear Spatiotemporal Dynamics and Chaos in Semiconductors, Cambridge University Press, Cambridge (2001). 

Rate oscillations, spatiotemporal patterns and chaos, e.g. dissipative structures were also observed in heterogeneous catalytic reactions. If compared with pattern formation in homogeneous systems, the surface studies introduced new aspects, like anisotropic diffusion, and the possibility of global synchronization via the gas phase. Application of field electron and field ion microscopy to the study of oscillatory surface reactions provided the capability of obtaining images with near-atomic resolution. The most extensively studied reaction is CO oxidation, which is catalyzed by group VIII noble metals. 

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... 

Differentiation of Dictyostelium, 163-5 effect of cAMP pulses on, 304,305 Diffusion of Ca" 399,401 of IP3,399,405 Dissipative structures, 5,491 rhythms as temporal, 4-7 spatial and spatiotemporal, 82,491 Domain in parameter space of chaos, 131,245,267,282,509,510 of complex oscillatory phenomena, 507, 512... 

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. 

If this were the only context in which CML models were used, their utility would be severely limited. For values y beyond the stability limit, the Euler method fails and one obtains solutions that fail to represent the solutions of the reaction-diffusion equation. However, it is precisely the rich pattern formation observed in CML models beyond the stability limit that has attracted researchers to study these models in great detail. Coupled map models show spatiotemporal intermittency, chaos, clustering, and a wide range of pattern formation processes." Many of these complicated phenomena can be studied in detail using CML models because of their simplicity and, if there are generic aspects to the phenomena, for example, certain scaling properties, then these could be carried over to real systems in other parameter regimes. The CML models have been used to study chemical pattern formation in bistable, excitable, and oscillatory media." ... 

Chaos and pattern formation spatiotemporal aspects of surface reactivity... 

We begin this chapter with a discussion of the automaton and present the details of the model construction in Section 2. A number of different systems has been studied using this method in order to investigate fluctuation effects on chemical wave propagation and domain growth in bistable chemical systems [6], excitable media and Turing pattern formation [3,4,7], surface catalytic oxidation processes [8], as well as oscillations and chaos [9]. Our discussions will be confined to the Willamowski-Rossler [10] reaction which displays chemical oscillations and chaos as well as a variety of spatiotemporal patterns. This reaction scheme is sufficiently rich to illustrate many of the internal noise effects we wish to present the references quoted above can be consulted for additional examples. Section 3 applies the general considerations of Section 2 to the Willamowski-Rossler reaction. Sections 4 and 5 describe a variety of aspects of the effects of fluctuations on pattern formation and reaction processes. Section 6 contains the conclusions of the study. 

The Willamowski-Rossler [10] chemical mechanism gives rise to a rich bifurcation structure including oscillations, chaos and bistability thus, we can use it to explore the effects of internal noise on different spatiotemporal states. Since it is based on a scheme with mass action kinetics it is especially suitable for the investigation of the microscopic basis of macroscopic self-organized states. 

Oscillations and chaos are observed frequently in chemical systems. Most of the experimental investigations of these phenomena have been carried out for well-stirred systems where spatial degrees of freedom are assumed to play no role. If this is the case the system may be described in terms of chemical rate equations for a small number of macroscopic chemical concentrations. The periodic or chaotic attractors typically have low dimensions and can be characterized using the tools of dynamical systems theory [20]. Chemical systems may also display spatiotemporal oscillations and chaos. If spatial degrees of freedom are important the appropriate macroscopic model is the reaction-diffusion equation. The attractors may have high dimension and the theoretical description of such spatiotemporal states is less well developed. 

This brief overview of the wave propagation processes that occur near chaos and period-doubling cascades has served to show that fluctuations can lead to unusual modifications of these spatiotemporal reaction-diffusion system states. 

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