Logistic map

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The logistic map is a polynomial mapping (equivalently, recurrence relation) 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 seminal 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.[1] Mathematically, the logistic map is written

where:

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 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. This problem does not appear in the older Ricker model, which also exhibits chaotic dynamics.

The r=4 case of the logistic map is a nonlinear transformation of both the bit shift map and the μ = 2 case of the tent map.

Contents

Behavior dependent on r

By varying the parameter r, the following behavior is observed:

  • With r between 0 and 1, the population will eventually die, independent of the initial population.
  • With r between 1 and 2, the population will quickly approach the value
  • With r between 2 and 3, the population will also eventually approach the same value
  • With r between 3 and 1+\sqrt{6} (approximately 3.45), from almost all initial conditions the population will approach permanent oscillations between two values. These two values are dependent on r.
  • With r between 3.45 and 3.54 (approximately), from almost all initial conditions the population will approach permanent oscillations among four values.
  • With r increasing beyond 3.54, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc. The lengths of the parameter intervals which yield oscillations of a given length decrease rapidly; the ratio between the lengths of two successive such bifurcation intervals approaches the Feigenbaum constant δ = 4.669\dots. This behavior is an example of a period-doubling cascade.
  • At r approximately 3.57 is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions we can no longer see any oscillations of finite period. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
  • Most values beyond 3.57 exhibit chaotic behaviour, but there are still certain isolated ranges of r that show non-chaotic behavior; these are sometimes called islands of stability. For instance, beginning at 1+\sqrt{8} (approximately 3.83) there is a range of parameters r which show oscillation among three values, and for slightly higher values of r oscillation among 6 values, then 12 etc.
  • The development of the chaotic behavior of the logistic sequence as the parameter r varies from approximately 3.5699 to approximately 3.8284 is sometimes called the Pomeau–Manneville scenario, which is characterized by a periodic (laminar) phase interrupted by bursts of aperiodic behavior. Such a scenario has an application in semiconductor devices [2]. There are other ranges which yield oscillation among 5 values etc.; all oscillation periods occur for some values of r. A period-doubling window with parameter c is a range of r-values consisting of a succession of sub-ranges. The kth sub-range contains the values of r for which there is a stable cycle (a cycle which attracts a set of initial points of unit measure) of period c2k. This sequence of sub-ranges is called a cascade of harmonics.[3] In a sub-range with a stable cycle of period c2^{k^{*}}, there are unstable cycles of period c2k for all k < k * . The r value at the end of the infinite sequence of sub-ranges is called the point of accumulation of the cascade of harmonics. As r rises there is a succession of new windows with different c values. The first one is for c = 1; all subsequent windows involving odd c occur in decreasing order of c starting with arbitrarily large c.[3][4]
  • Beyond r = 4, the values eventually leave the interval [0,1] and diverge for almost all initial values.

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