Linear oscillating field, relationship

Equation (1) is, strictly speaking, not suitable for optical fields, which are rapidly varying in time. Even for linear polarization, the oscillation of the induced dipole moment may be damped (by material resonances) and thereby phase-shifted with respect to the oscillation of the external electric field. The usual way of expressing this phase shift is by considering the relationship between the Fourier components of the induced effect (oscillation of the induced dipole) and the stimulus (the electric field), with the damping and phase shift conveniently expressed by treating the terms involved as complex. Thus, the linear polarizability can be written as... 

We have seen that chemical and biological interactions lead to mathematical models displaying a variety of linear and nonlinear behavior relaxation to fixed points, multistability, excitability, oscillations, chaos, etc. Despite the different origin of the models, and the diverse nature of the variables they represent (chemical concentrations, population numbers, or even membrane electric potentials) the mathematical structures are quite similar, and it is possible to understand some aspects of the dynamics in one field (e.g. the chemical oscillations in the BZ reaction) with the help of models from other fields (for example the FN model of neurophysiology, or a phytoplankton-zooplankton model). This possibility of common mathematical description will be used in the rest of the book to highlight the similarities and relationships between chemical and biological dynamics when occurring in fluid flows. 

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