Logistic map

The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple nonlinear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation written down by Pierre François Verhulst. Mathematically, the logistic map is written where xn is a number between zero and one, which represents the ratio of existing population to the maximum possible population. This nonlinear difference equation is intended to capture two effects:

reproduction, where the population will increase at a rate proportional to the current population when the population size is small,
starvation (density-dependent mortality), where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of

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How to repair CSK using small perturbation control - Case study and performance analysis

Slobodan Kozic, Kumiko Oshima, Thomas Schimming

In this paper we study coded modulation schemes based on iterated chaotic maps. In particular, we consider

CSK

-like systems in which a chaotic map is switched depending on the payload. We show how small perturbation control can substantially improve the performance and we demonstrate using a minimum distance approach how absence/wrongly designed control can lead to bad performance.

2003

Minimum distance properties of coded modulations based on iterated chaotic maps

Slobodan Kozic, Kumiko Oshima, Thomas Schimming

In this paper we introduce a method for analyzing the performance of coded modulation schemes based on iterated chaotic maps. In particular, it is known from coding theory that the minimum distance plays an important role, defining the behavior of the error probability performance at sufficiently high signal to noise ratios. We introduce the method and illustrate it for two examples of chaotic maps, the Bernoulli shift map and the tent map. We emphasize that the performance of the map can not be understood by a reasoning in terms of ergodic behavior of the map. The given simulation results confirm the relevance of the minimum distance for describing the performance of such a system.

2003

Flow control of weakly non-parallel flows: application to trailing vortices

François Gallaire, Giacomo Valerio Iungo, Erica Pezzica, Francesco Viola

A general formulation is proposed to control the integral amplification factor of harmonic disturbances in weakly non-parallel amplifier flows. The sensitivity of the local spatial stability spectrum to a base-flow modification is first determined, generalizing the results of Bottaro et al. (J. Fluid Mech., vol. 476, 2003, pp. 293-302). This result is then used to evaluate the sensitivity of the overall spatial growth to a modification of the inlet flow condition. This formalism is applied to a non-parallel Batchelor vortex, which is a well-known model for trailing vortices generated by a lifting wing. The resulting sensitivity map indicates the optimal modification of the inlet flow condition enabling the stabilization of the helical modes. It is shown that the control, formulated using a single linearization of the flow dynamics carried out on the uncontrolled configuration, successfully reduces the total spatial amplification of all convectively unstable disturbances.

Cambridge Univ Press2017

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