The Logistic Map

 Historical account

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

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,

It was investigated in its discrete form

by Pekka Juhana Myrberg (1892–1976) in the early 1960s [2] : 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 [3].

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 [4]. 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 [5]. May then used the logistic map to show that a simple map can induce very complicated behaviors [6]. 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 [7]. In 1977, John Guckenheimer provided a formal introduction to the `universal’’ sequence of bifurcations which take place with respect to the bifurcation parameter \(\mu\) [8]. 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 [9] and Mitchell Feigenbaum [10]
independently investigated the renormalization properties of continuous unimodal maps and found that there is a scaling law on the parameters

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 \(\mu = 3.828 429...\).

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