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  1. 2 Open Physics

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  1. 1 2010

  2. 1 2008

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  1. 1 Dzwinel W.

  2. 1 Li F.-g.

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Spatially extended populations reproducing logistic map

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Dzwinel W.
Open Physics
|
2010
|
vol. 8
|
issue 1
33-41
EN
We discuss here the conditions that the spatially extended systems (SES) must satisfy to reproduce the logistic map. To address this dilemma we define a 2-D coupled map lattice with a local rule mimicking the logistic formula. We show that for growth rates of k⩽k ∞ (k ∞ is the accumulation point) the global evolution of the system exactly reproduces the cascade of period doubling bifurcations. However, for k > k ∞, instead of chaotic modes, the cascade of period halving bifurcations is observed. Consequently, the microscopic states at the lattice nodes resynchronize producing dynamically changing spatial patterns. By downscaling the system and assuming intense mobility of individuals over the lattice, the spatial correlations can be destroyed and the local rule remains the only factor deciding the evolution of the whole colony. We found the class of “atomistic” rules for which uncorrelated spatially extended population matches the logistic map both for pre-chaotic and chaotic modes. We concluded that the global logistic behavior can be expected for a spatially extended colony with high mobility of individuals whose microscopic behavior is governed by a specific semi-logistic rule in the closest neighborhood. Conversely, the populations forming dynamically changing spatial clusters behave in a different way than the logistic model and reproduce at least the steady-state fragment of the logistic map.
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Effects of noise on periodic orbits of the logistic map

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Li F.-g.
Open Physics
|
2008
|
vol. 6
|
issue 3
539-545
EN
Noise can induce an inverse period-doubling transition and chaos. The effects of noise on each periodic orbit of three different period sequences are investigated for the logistic map. It is found that the dynamical behavior of each orbit, induced by an uncorrelated Gaussian white noise, is different in the mergence transition. For an orbit of the period-six sequence, the maximum of the probability density in the presence of noise is greater than that in the absence of noise. It is also found that, under the same intensity of noise, the effects of uncorrelated Gaussian white noise and exponentially correlated colored (Gaussian) noise on the period-four sequence are different.
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