page_title: Logistic map
section_title: Chaos and the logistic map
hierarchical_section_title: Logistic map / Characteristics of the map / Chaos and the logistic map
caption_attribution_description: English: An animation of the logistic map, via construction with cobweb diagram. The parameter r ranges from 1 to 4, and the starting point is x0=0.2 This shows the chaotic nature of the logistic map for values of r near 4. Made by myself in MuPad.
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: The relative simplicity of the logistic map makes it a widely used point of entry into a consideration of the concept of chaos.  A rough description of chaos is that chaotic systems exhibit a great sensitivity to initial conditions—a property of the logistic map for most values of r between about 3.57 and 4 (as noted above).  A common source of such sensitivity to initial conditions is that the map represents a repeated folding and stretching of the space on which it is defined.  In the case of the logistic map, the quadratic difference equation describing it may be thought of as a stretching-and-folding operation on the interval (0,1).
  The following figure illustrates the stretching and folding over a sequence of iterates of the map.  Figure (a), left, shows a two-dimensional Poincaré plot of the logistic map's state space for r = 4, and clearly shows the quadratic curve of the difference equation (1).  However, we can embed the same sequence in a three-dimensional state space, in order to investigate the deeper structure of the map.  Figure (b), right, demonstrates this, showing how initially nearby points begin to diverge, particularly in those regions of xₜ corresponding to the steeper sections of the plot.
  This stretching-and-folding does not just produce a gradual divergence of the sequences of iterates, but an exponential divergence (see Lyapunov exponents), evidenced also by the complexity and unpredictability of the chaotic logistic map.  In fact, exponential divergence of sequences of iterates explains the connection between chaos and unpredictability: a small error in the supposed initial state of the system will tend to correspond to a large error later in its evolution.  Hence, predictions about future states become progressively (indeed, exponentially) worse when there are even very small errors in our knowledge of the initial state. This quality of unpredictability and apparent randomness led the logistic map equation to be used as a pseudo-random number generator in early computers.
  Since the map is confined to an interval on the real number line, its dimension is less than or equal to unity. Numerical estimates yield a correlation dimension of 0.500±0.005 (Grassberger, 1983), a Hausdorff dimension of about 0.538  (Grassberger 1981), and an information dimension of approximately 0.5170976 (Grassberger 1983) for r ≈ 3.5699456 (onset of chaos). Note: It can be shown that the correlation dimension is certainly between 0.4926 and 0.5024.
  It is often possible, however, to make precise and accurate statements about the likelihood of a future state in a chaotic system.  If a (possibly chaotic) dynamical system has an attractor, then there exists a probability measure that gives the long-run proportion of time spent by the system in the various regions of the attractor.  In the case of the logistic map with parameter r = 4 and an initial state in (0,1), the attractor is also the interval (0,1) and the probability measure corresponds to the beta distribution with parameters a = 0.5 and b = 0.5. Specifically, the invariant measure is
  Unpredictability is not randomness, but in some circumstances looks very much like it.  Hence, and fortunately, even if we know very little about the initial state of the logistic map (or some other chaotic system), we can still say something about the distribution of states arbitrarily far into the future, and use this knowledge to inform decisions based on the state of the system.
caption: A cobweb diagram of the logistic map, showing chaotic behaviour for most values of r > 3.57