The Logistic Map

Christophe LETELLIER

- Historical account

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Pierre François de Verhulst

In 1845, the discrete map was introduced in its continuous form by Pierre-François de Verhulst (1804-1849) [1]. It is the first order differential equation

 \dot{N} = \mu N \left( \displaystyle 1 - \frac{N}{\kappa} 

where N is the number of individuals, \mu is the maximum growth rate, and \kappa is the biotic capacity. Dividing by \kappa and using x = \frac{N}{\kappa}, the most often form of the logistic equation is obtained, that is,  \dot{x} = \mu x \left( 1 - x \right) .

It was investigated in its discrete form

x_{n+1}= \mu x_n \left( 1 - x_n \right)

by Pekka Juhana Myrberg (1892–1976) in the early 1960s [2] [3] : in particular, Myrberg identified the period-doubling cascade and the corresponding accumulation point. In fact, Myrberg did not investigate one map, but a family of quadratic one-dimensional continuous maps f:I \mapsto I on the interval I. For all maps, the accumulation point is provided by \lim_{n \mapsto \infty} \mu_n - \mu_0 = 1.401155... where \mu_\infty is the parameter value at which the orbit of period 20=1 occurs. In 1964, Andrei Sharkovskii obtained a theorem providing the order with which period-p orbits appear for the first time when the bifurcation parameter is varied [4].

In 1973, Metropolis, Stein & Stein investigated quadratic maps - the logistic map being a simple particular example - and found that the order in which all limit cycles were observed was ``universal’’, that is, was not dependent on the specific algebraic form of the map considered [5]. An important aspect of their work is that they used symbolic dynamics. In 1974, using numerical simulations, Robert May rediscovered the period-doubling cascades and showed that chaotic regimes occur after the accumulation point [6]. May then used the logistic map to show that a simple map can induce very complicated behaviors [7]. In 1975, Igor Gumowski and Christian Mira showed that the sequences of period-doubling cascades was fractal in the sense that the whole bifurcation diagram can be recovered in a part of it : they called this the box-within-the box structure [8]. In 1977, John Guckenheimer provided a formal introduction to the `universal’’ sequence of bifurcations which take place with respect to the bifurcation parameter \mu [9]. The family of maps considered have the following properties : f_\mu is a smooth function and has a single critical point (quadratic maps are thus included, but unimodal maps is a more general designation). Guckenheimer provided an algorithm for computing the order of the period-p of the stable limit cycle observed while varying the bifurcation parameter.

In 1978, Pierre Coullet and Charles Tresser [10] and Mitchell Feigenbaum [11] independently investigated the renormalization properties of continuous unimodal maps and found that there is a scaling law on the parameters

\lim_{n\mapsto \infty} \frac{\mu_n - \mu_{n-1}}{\mu_{n+1}-\mu_n} = 4.669~201... .

The logistic map is thus a very convenient example to start to investigate the period-doubling cascade as a route to chaos. It should be keep in mind, that it also serve as a model for the first-return map to a Poincaré section of any set of differential equations which produces a period-doubling cascade as a route to chaos.

- Dynamical analysis

The logistic map presents a period-doubling cascade as a route to chaos as shown by a bifurcation diagram (Fig. 1). The Logistic map has two period-1 points : x0=0 and x_1 = \frac{\mu-1}{\mu}. The point x0 is stable on the range \mu \in [-1;+1]. The second period-1 point is stable on the range \mu \in [+1;+3]. These two period-1 points exchange their stability through a transcritical bifurcation. Then the point x0 becomes unstable through a period-doubling birfucation. This first period-doubling bifurcation is followed by a cascade of period-doubling bifurcations up to the accumulation point \mu_\infty = 3.569~945.... Then a first chaotic regime is observed. Many other period-doubling cascades can be found in this bifurcation diagram. Let us note the cascade emerging from the stable period-3 limit cycle appearing at \mu = 3.828~429....

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Fig. 1. Bifurcation diagram of the logistic map.

When a blow-up is made within the period-3 window, a picture equivalent to the full bifurcation diagram is recovered (Fig. 2). This is a simple illustration of the nature (the box-within-the box structure) of the bifurcation diagram. The order with which all the limit cycles when parameter \mu is increased can be predicted by the so-called unimodal order, built from the symbolic dynamics based on the partition

\sigma_n = 
  \left| \begin{array}{l} 0 \mbox{ if } x_n > 0.5 \\ 1 \mbox{ if } x_n > 0.5 \end{array} \right.
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Fig. 2. Period-3 window. A period-doubling cascade is observed from the stable period-3 limit cycle.

[1] P. F. de Verhulst, Recherches mathématiques sur la loi d’accroissement de la population, Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Bruxelles, 18, 1-42, 1845.

[2] P. J. Myrberg, Sur l’itération des polynômes réels quadratiques, Journal de Mathématiques pures et appliquées 9 (41), 339-351, 1962.

[3] P. J. Myrberg, Iteration der reellen Polynome zweiten Grades III, Annales Academiae Scientiarum Fennicae A, 336, 3-16, 1963.

[4] A. N. Sharkovskii, Coexistence of cycles of a continuous map of the line into itself, Ukrain. Mat. Zh. , 16, 61–71, 1964.

[5] N. Metropolis, M. L. Stein & P. R. Stein, On finite limit sets for transformations on the unit interval, Journal of Combinatorial Theory A , 15, 25–44, 1973.

[6] R. May, Biological populations with nonoverlapping Generations : stable points, stable cycles, and chaos, Science, 186, 645-647, 1974.

[7] R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261, 459 - 467, 1976.

[8] I. Gumowski & C. Mira, Accumulations de bifurcations dans une récurrence, Comptes-Rendus de l’Académie des Sciences, 281, 45-48, 1975.

[9] J. Guckenheimer, On the bifurcation of maps of the interval, Inventiones Mathematicae, 39, 165-178, 1977.

[10] P. Coullet & C. Tresser, Itérations d’endomorphismes et groupe de rénormalisation, Journal de Physique, C5, 25–28, 1978.

[11] M. Feigenbaum, Quantitative universality for a class of non-linear transformations, Journal of Statistical Physics, 19, 25–52, 1978.

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