Abstract
The logistic map is one of the most important but common examples of chaotic dynamics. The object shows the crucial belief of the deterministic chaos theory that brings a new procedural structure and apparatus for exploring and understanding complex behavior in dynamical systems. We put an importance on report of the Verhulst logistic map which is one of the potential models and methods for researching dynamical systems that could develop to chaotic. Chaotic signals present a special difficulty in parameter estimation. The difficulty arises from the definition of a chaotic system because of sensitive dependence on initial conditions. It is seen that very slight changes in the initial conditions cause significant effects in the evolution. In general the chaotic systems are nonlinear and apparently random but they are deterministic. The main objective of this paper is how can find the logistic map equation and investigated the chaotic behavior for the logistic equation by varying the control parameters and finally discover Lyaponov exponent, Bifurcation diagrams etc.
Highlights
The logistic map is a one dimensional discrete time demographic model how very complex, chaotic behavior can arise from very simple non-linear dynamical equations
Xn ∈[0,1] and represents the ratio of existing population to the maximum possible population at year n and x0 represents the initial ratio of population to maximum, r ∈ [0,4] and represents a combined rate for reproduction and starvation. It was originally made as a very simple model for the population numbers of species in the presence of limiting factors such as food supply or disease containing two causal loops: i Due to reproduction the population will increase at a rate proportional to the current population when the population size is small. ii Due to starvation where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical carrying capacity of the environment less the current population
Paper has shown that some nonlinear systems that have their sub-domains on the logistic map are often characterized with unique chaotic attractors [3]
Summary
The logistic map is a one dimensional discrete time demographic model how very complex, chaotic behavior can arise from very simple non-linear dynamical equations. Xn ∈[0,1] and represents the ratio of existing population to the maximum possible population at year n and x0 represents the initial ratio of population to maximum (at year 0), r ∈ [0,4] and represents a combined rate for reproduction and starvation It was originally made as a very simple model for the population numbers of species in the presence of limiting factors such as food supply or disease containing two causal loops: i Due to reproduction the population will increase at a rate proportional to the current population when the population size is small. Ii Due to starvation where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical carrying capacity of the environment less the current population This type of simple equation already exhibits wonderful dynamics, quickly summarized. The present paper which is strongly motivated by the quest to introduce the beginners to the theories of chaos and nonlinear dynamics focus on the development of time series and cobweb plot and bifurcation and Lyapunov exponent based-chaos diagram in one dimensional logistic map
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