Spatio-temporal focusing

We shall describe some of tire common types of chemical patterns observed in such experiments and comment on tire mechanisms for tlieir appearance. In keeping witli tire tlieme of tliis chapter we focus on states of spatio-temporal chaos or on regular chemical patterns tliat lead to such turbulent states. We shall touch only upon tire main aspects of tliis topic since tliere is a large variety of chemical patterns and many mechanisms for tlieir onset [2,3, 5,31]. 

The distribution and fate of PAHs in the atmosphere has been the subject of numerous studies covering a wide range of spatio-temporal scales and relevant sources (references in Finlayson-Pitts and Pitts 2000 Prevedouros et al. 2005 Tsapakis and Stephanou 2005 Lammel et al. 2009a Balasubramanian and He 2010). Most studies focused on occurrence of the parent PAHs, while nitro-PAHs gained interest in the chemistry of nitrogen oxides rich atmospheres (Finlayson-Pitts and Pitts 2000) and alkylated and partly oxygenated PAHs had significantly less been addressed so far (e.g. Albinet et al. 2008). 

This chapter will focus on practicable methods to perform both the model specification and model estimation tasks for systems/models that are static or dynamic and linear or nonlinear. Only the stationary case win be detailed here, although the potential use of nonstationary methods will be also discussed briefly when appropriate. In aU cases, the models will take deterministic form, except for the presence of additive error terms (model residuals). Note that stochastic experimental inputs (and, consequently, outputs) may stiU be used in connection with deterministic models. The cases of multiple inputs and/or outputs (including multidimensional inputs/outputs, e.g., spatio-temporal) as well as lumped or distributed systems, will not be addressed in the interest of brevity. It will also be assumed that the data (single input and single output) are in the form of evenly sampled time-series, and the employed models are in discretetime form (e.g., difference equations instead of differential equations, discrete summations instead of integrals). 

People often speak of chemical turbulence whereby either of two distinct chaotic phenomena may be meant. One is the spatially uniform but temporally chaotic dynamics exhibited by the concentrations of chemical species, while the other involves spatial chaos too. For chemical turbulence in the latter sense, our attention is usually focused upon systems in which the local dynamics itself is non-chaotic, while such non-chaotic elements are coupled through diffussion to produce spatio-temporal chaos. In fact, if the local elements were already chaotic, the fields composed of them would trivially exhibit spatio-temporal chaos. Hence non-trivial chemical turbulence involving spatio-temporal chaos may be called diffusion-induced chemical turbulence. 

Moonen, C. T. W., Spatio-Temporal Control of Gene Expression and Cancer Treatment Using Magnetic Resonance Imaging-Guided Focused Ultrasound, Clin. Cancer Res., 13, 3482 (2007). 

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