Christophe LETELLIER
11/01/2008

**Historical account**

**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

where

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 on the interval

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 [8]. The family of maps considered have the following properties : 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 : *x*_{0}=0 and . The point *x*_{0} is stable on the range . The second period-1 point is stable on the range . These two period-1 points exchange their stability through a transcritical bifurcation. Then the point *x*_{0} 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 . 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 .

**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 is increased can be predicted by the so-called unimodal order, built from the symbolic dynamics based on the partition

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

[4] **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.

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

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

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

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

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

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