page_title: Logistic map
section_title: Feigenbaum universality of 1-D maps
hierarchical_section_title: Logistic map / Related concepts / Feigenbaum universality of 1-D maps
caption_attribution_description: English: Bifurcation diagram tanh map chaos by Okulov
context_page_description: The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear 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 first created by Pierre François Verhulst.
  Mathematically, the logistic map is written
  where xₙ is a number between zero and one that represents the ratio of existing population to the maximum possible population. The values of interest for the parameter r are those in the interval [0,4].
  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 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.
  However, as a demographic model the logistic map has the pathological problem that some initial conditions and parameter values lead to negative population sizes.
context_section_description: Universality of one-dimensional maps with parabolic maxima  and Feigenbaum constants ,   is well visible with map proposed as a toy 
  model for discrete laser dynamics: 
  ,
  where  stands for electric field amplitude,  is laser gain as bifurcation parameter. 
  The gradual increase of  at interval  changes dynamics from regular to chaotic one  with qualitatively the same bifurcation diagram as those for logistic map.
caption: Bifurcation diagram for the hyperbolic tangent map. It is a self-similar in a wider range of bifurcation parameter G. This is the else ubiquitous connection between chaos and fractals.
exif: {"Image SubfileType": "Full-resolution Image", "Image ImageWidth": "753", "Image ImageLength": "610", "Image BitsPerSample": "[8, 8, 8]", "Image Compression": "LZW", "Image PhotometricInterpretation": "2", "Image StripOffsets": "[8, 179, 407, 1377, 2702, 3891, 4155, 4384, 4810, 5219, 5554, 5772, 5990, 6208, 6483, 6824, 7234, 7766, 8442, 9159, ... ]", "Image SamplesPerPixel": "3", "Image RowsPerStrip": "5", "Image StripByteCounts": "[171, 228, 970, 1325, 1189, 264, 229, 426, 409, 335, 218, 218, 218, 275, 341, 410, 532, 676, 717, 660, ... ]", "Image XResolution": "120", "Image YResolution": "120", "Image PlanarConfiguration": "1", "Image ResolutionUnit": "Pixels/Inch", "Image Predictor": "None"}