LINEAR VERSUS NONLINEAR AMPLIFICATION - A GENERALIZATION OF THE NOVIKOV-MONTROLL-SHLESINGER-WEST CASCADE

The framework of this paper is the theory of statistical fractals. A general approach for the scale-invariant stochastic amplification is derived. The model is based on the assumption that the amplification coefficients as well as the probability of an amplification event are random variables selected from a certain probability law. Both linear and nonlinear cases are considered. For linear processes a general solution is available. Like in the case of constant amplification factors the probability density of an amplified variable has a long tail. However, the analytical form of the tail is different: it can be expressed as a sum of inverse power laws modulated by oscillatory functions of the logarithm of the amplified variable, having different characteristic periods. In the nonlinear case no general analytic solutions are available. For a mapping with saturation and a single stable fixed point the probability density of the amplified varibale may have a singularity at a fixed point. A situation with amplification and diminution is also considered. In this case the solution has the features of a double statistical fractal. A generalized version of the linear model is applied to the problem of overdispersed molecular clocks.