Thelogistic map is a discretedynamical system defined by the quadraticdifference equation

${\displaystyle x_{n+1}=r{x}_n(1-x_n).}$

1

Equivalently, it is arecurrence relation and apolynomialmapping ofdegree 2. It is often referred to as an archetypal example of how complex,chaotic behaviour can arise from very simplenonlinear dynamical equations.

The map was initially utilized byEdward Lorenz in the 1960s to showcase properties of irregular solutions in climate systems.^[1] It was popularized in a 1976 paper by the biologistRobert May,^{[May, Robert M. (1976) 1]} in part as a discrete-time demographic model analogous to thelogistic equation written down byPierre François Verhulst.^[2]Other researchers who have contributed to the study of the logistic map includeStanisław Ulam,John von Neumann,Pekka Myrberg,Oleksandr Sharkovsky,Nicholas Metropolis, andMitchell Feigenbaum.^{[3][citation needed]}

In the logistic map,x is a variable, andr is a parameter. It is amap in the sense that it maps a configuration orphase space to itself (in this simple case the space is one dimensional in the variablex):

${\displaystyle f:x\mapsto rx(1-x).}$

1-2

It can be interpreted as a tool to get next position in the configuration space after one time step. The difference equation is a discrete version of thelogistic differential equation, which can be compared to a time evolution equation of the system.

Given an appropriate value for the parameterr and performing calculations starting from an initial condition${\displaystyle x_0}$, we obtain the sequence${\displaystyle x_0}$,${\displaystyle x_1}$,${\displaystyle x_2}$, ..., which can be interpreted as a sequence of time steps in the evolution of the system.

In the field ofdynamical systems, this sequence is called anorbit, and the orbit changes depending on the value given to the parameter. When the parameter is changed, the orbit of the logistic map can change in various ways, such as settling on a single value, repeating several values periodically, or showingnon-periodic fluctuations known aschaos.^{[Devaney 1989 1][4]}

Another way to understand thissequence is to iterate the logistic map (here represented by${\displaystyle f(x)}$) to the initial state${\displaystyle x_0}$:^{[Devaney 1989 2]}

This was the initial approach ofHenri Poincaré to studydynamical systems and ultimately chaos starting from the study offixed points or, in other words, states that do not change over time (i.e. when${\displaystyle x_n=...=x_1=x_0=f(x_0)}$). Many chaotic systems such as theMandelbrot set emerge from iteration of very simple quadratic nonlinear functions such as the logistic map.^[5]

Taking the biologicalpopulation model as an examplex_n is a number between zero and one, which represents the ratio of existingpopulation to themaximum possible population.^{[May, Robert M. (1976) 2]}This nonlinear difference equation is intended to capture two effects:

• reproduction, where the population will increase at a rateproportional to the current population when the population size is small,
• starvation (density-dependent mortality), where thegrowth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.

The usual values of interest for the parameterr are those in the interval[0, 4], so thatx_n remains bounded on[0, 1]. Ther = 4 case of the logistic map is a nonlinear transformation of both thebit-shift map and theμ = 2 case of thetent map. Ifr > 4, this leads to negative population sizes. (This problem does not appear in the olderRicker model, which also exhibits chaotic dynamics.) One can also consider values ofr in the interval[−2, 0], so thatx_n remains bounded on[−0.5, 1.5].^[6]

You can helpexpand this article with text translated fromthe corresponding article in Japanese.Click [show] for important translation instructions. Machine translation, likeDeepL orGoogle Translate, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English Wikipedia. Consideradding a topic to this template: there are already 1,540 articles in themain category, and specifying|topic= will aid in categorization. Do not translate text that appears unreliable or low-quality. If possible, verify the text with references provided in the foreign-language article. Youmust providecopyright attribution in theedit summary accompanying your translation by providing aninterlanguage link to the source of your translation. A model attribution edit summary isContent in this edit is translated from the existing Japanese Wikipedia article at [[:ja:ロジスティック写像]]; see its history for attribution. You may also add the template{{Translated|ja|ロジスティック写像}} to thetalk page. For more guidance, seeWikipedia:Translation.

The animation shows the behaviour of the sequence${\displaystyle x_n}$ over different values of the parameter r. A first observation is that the sequence does not diverge and remains finite for r between 0 and 4. It is possible to see the following qualitative phenomena in order of time:

• exponential convergence to zero
• convergence to a non-zero fixed value (seeExponential function orCharacterizations of the exponential function point 4)
• initial oscillation and then convergence (seeDamping andDamped harmonic oscillator)
• stable oscillations between two values (seeResonance andSimple harmonic oscillator)
• growing oscillations between a set of values which are multiples of two such as 2,4,8,16 etc. (seePeriod-doubling bifurcation)
• Intermittency (i.e. sprouts of oscillations at the onset of chaos)
• fully developedchaotic oscillations
• topological mixing (i.e. the tendency of oscillations to cover the full available space).

The first four are also available in standardlinear systems, oscillations between two values are available too underresonance, chaotic systems though have typically a large range of resonance conditions.The other phenomena are peculiar tochaos. This progression of stages is strikingly similar to the onset ofturbulence.Chaos is not peculiar to non-linear systems alone and it can also be exhibited by infinite dimensional linear systems.^[7]

As mentioned above, the logistic map itself is an ordinary quadratic function. An important question in terms of dynamical systems is how the behavior of the trajectory changes when the parameterr changes. Depending on the value ofr, the behavior of the trajectory of the logistic map can be simple or complex.^{[Thompson & Stewart 1]} Below, we will explain how the behavior of the logistic map changes asr increases.

As mentioned above, the logistic map can be used as a model to consider the fluctuation of population size. In this case, the variable x of the logistic map is the number of individuals of an organism divided by the maximum population size, so the possible values of x are limited to 0 ≤ x ≤ 1. For this reason, the behavior of the logistic map is often discussed by limiting the range of the variable to the interval [0, 1].^{[Hirsch,Smale & Devaney 1]}

If we restrict the variables to 0 ≤ x ≤ 1, then the range of the parameter r is necessarily restricted to 0 to 4 (0 ≤ r ≤ 4). This is because if${\displaystyle x_n}$ is in the range [0, 1], then the maximum value of${\displaystyle x_{n+1}}$ isr/4. Thus, when r > 4, the value of${\displaystyle x_{n+1}}$ can exceed 1. On the other hand, when r is negative, x can take negative values.^{[Hirsch,Smale & Devaney 1]}

A graph of the map can also be used to learn much about its behavior. The graph of the logistic map${\displaystyle x_{n+1}=rx(1-x_n)}$ is the plane curve that plots the relationship between${\displaystyle x_n}$ and${\displaystyle x_{n+1}}$, with${\displaystyle x_n}$ (or x) on the horizontal axis and${\displaystyle x_{n+1}}$ (or f (x)) on the vertical axis. The graph  of the logistic map looks like this, except for the case r = 0:

It has the shape of a parabola with a vertex at^{[Gulick 1]}

${\displaystyle {\displaystyle (x_n,x_{n+1})=\left(0.5,\frac{r}{4}\right)}}$

2-1

When r is changed, the vertex moves up or down, and the shape of the parabola changes. In addition, the parabola of the logistic map intersects with the horizontal axis (the line where${\displaystyle x_{n+1}=0}$) at two points. The two intersection points are${\displaystyle (x_n,x_{n+1})=(0,0)}$ and${\displaystyle (x_n,x_{n+1})=(1,0)}$, and the positions of these intersection points are constant and do not depend on the value of r.

Graphs of maps, especially those of one variable such as the logistic map, are key to understanding the behavior of the map. One of the uses of graphs is to illustrate fixed points, called points. Draw a line y = x (a 45° line) on the graph of the map. If there is a point where this 45° line intersects with the graph, that point is a fixed point. In mathematical terms, a fixed point is

${\displaystyle f(x)=x}$

2-2

It means a point that does not change when the map is applied. We will denote the fixed point as${\displaystyle x_f}$. In the case of the logistic map, the fixed point that satisfies equation (2-2) is obtained by solving${\displaystyle rx(1-x)=x}$.

${\displaystyle {\displaystyle x_{f1}=0}}$

2-3

${\displaystyle x_{f2}=1-\frac{1}{r}}$

2-4

(except for r = 0). The concept of fixed points is of primary importance in discrete dynamical systems.

Another graphical technique that can be used for one-variable mappings is thespider web projection. After determining an initial value${\displaystyle x_0}$ on the horizontal axis, draw a vertical line from the initial value${\displaystyle x_0}$ to the curve of f(x). Draw a horizontal line from the  point where the curve of f(x) meets the 45° line of y = x, and then draw a vertical line from the point where the curve meets the 45° line to the curve of f(x). By repeating this process, a spider web or staircase-like diagram is created on the plane. This construction is in fact equivalent to calculating the trajectory graphically, and thespider web diagram created represents the trajectory starting from${\displaystyle x_0}$. This projection allows the overall behavior of the trajectory to be seen at a glance.

The image below shows theamplitude andfrequency content of a logistic map that iterates itself for parameter values ranging from 2 to 4. Again one can see initial linear behaviours then chaotic behaviour not only in thetime domain (left) but especially in the frequency domain orspectrum (right), i.e. chaos is present at all scales as it is in the case ofEnergy cascade ofKolmogorov and it even propagates from one scale to another.^{[Thompson & Stewart 2]}

By varying the parameterr, the following behavior is observed:

First, when the parameter r = 0,${\displaystyle x_1=0}$, regardless of the initial value${\displaystyle x_0}$. In other words, the trajectory of the logistic map when a = 0 is a trajectory in which all values after the initial value are 0, so there is not much to investigate in this case.

Next, when the parameter r is in the range 0 < r < 1,${\displaystyle x_n}$ decreases monotonically for any value of${\displaystyle x_0}$ between 0 and 1. That is,${\displaystyle x_n}$ converges to 0 in the limit n → ∞.^{[Gulick 2]} The point to which${\displaystyle x_n}$ converges is the fixed point${\displaystyle x_{f1}}$ shown in equation (2-3). Fixed points of  this type, where orbits around them converge, are called asymptotically stable, stable, or attractive. Conversely, if orbits around${\displaystyle x_f}$ move away from${\displaystyle x_f}$ as time n increases, the fixed point${\displaystyle x_f}$ is called unstable or repulsive.^{[Gulick 3]}

A common and simple way to know whether a fixed point is asymptotically stable is to take the derivative of the map f.^{[Gulick 4]} This derivative is expressed as${\displaystyle f^{\prime }(x)}$,${\displaystyle x_f}$ is asymptotically stable if the following condition is satisfied.

3-1

We can see this by graphing the map: if the slope of the tangent to the curve at${\displaystyle x_f}$ is between −1 and 1, then${\displaystyle x_f}$ is stable and the orbit around it is attracted to${\displaystyle x_f}$. The derivative of the logistic map is

${\displaystyle f^{\prime }(x)=r(1-2x)}$

3-2

Therefore, for x = 0 and 0 < r < 1, 0 < f  '(0) < 1, so the fixed point${\displaystyle x_{f1}}$ = 0  satisfies equation (3-1).

However, the discrimination method using equation (3-1) does not know the range of orbits from${\displaystyle x_f}$ that are attracted to${\displaystyle x_f}$. It only guarantees that x within a certain neighborhood of${\displaystyle x_f}$ will converge. In this case, the domain of initial values that converge to 0 is the entire domain [0, 1], but to know this for certain, a separate study is required.

The method for determining whether a fixed point is unstable can be found by similarly differentiating the map. For r<1 if a fixed point${\displaystyle x_f}$ is unstable if

${\displaystyle \left|f^{\prime }(x_f)\right|>1}$

3-3

If the parameter lies in the range 0 < r < 1, then the other fixed point${\displaystyle x_{f2}=1-1/a}$is negative and therefore does not lie in the range [0, 1], but it does exist as an unstable fixed point.

In the general case withr between 1 and 2, the population will quickly approach the value⁠r − 1/r⁠, independent of the initial population.

When the parameter r = 1, the trajectory of the logistic map converges to 0 as before, but the convergence speed is slower at r = 1. The fixed point 0 at r = 1 is asymptotically stable, but does not satisfy equation (3-1). In fact, the discrimination method based on equation (3-1) works by approximating the map to the first order near the fixed point. When r = 1, this approximation does not hold, and stability or instability is determined by the quadratic (square) terms of the map, or in order words the second order perturbation.

When r = 1 is graphed, the curve is tangent to the 45° diagonal at x = 0. In this case, the fixed point${\displaystyle x_{f2}=1-1/r}$, which exists in the negative range for, is${\displaystyle x_{f2}=0}$.For${\displaystyle x_{f2}=0}$, that is, as r increases, the value of${\displaystyle x_{f2}}$ approaches 0, and just at r = 1  ,${\displaystyle x_{f2}}$ collides with  ${\displaystyle x_{f1}=0}$. This collision gives rise to a phenomenon known as atranscritical bifurcation.Bifurcation is a term used to describe a qualitative change in the behavior of a dynamical system. In this case, transcritical bifurcation is when the stability of fixed points alternates between each other. That is, when r is less than 1,${\displaystyle x_{f1}}$ is stable and${\displaystyle x_{f2}}$ is unstable, but when r is greater than 1,${\displaystyle x_{f1}}$ is unstable and${\displaystyle x_{f2}}$ is stable. The parameter  values at which bifurcation occurs are called bifurcation points. In this case, r = 1 is the bifurcation point.

Fixed point${\displaystyle x_{f2}=1-1/r}$ Example of monotonically decreasing convergence to (r = 1.2, x 0 = 0.6)

Fixed point${\displaystyle x_{f2}=1-1/a}$ Example of monotonically increasing convergence to (r = 1.8, x 0 = 0.2)

As a result of the bifurcation, the orbit of the logistic map converges to the limit point${\displaystyle x_{f2}=1-1/r}$ instead of${\displaystyle x_{f1}=0}$. In particular, if the parameter, then the trajectory starting from a value${\displaystyle x_0}$in the interval (0, 1), exclusive of 0 and 1, converges to${\displaystyle x_{f2}}$ by increasing or decreasing monotonically. The difference in the convergence pattern depends on the range of the initial value.

In the case ofThen, it converges monotonically,, the function converges monotonically except for the first step.

Furthermore, the fixed point${\displaystyle x_{f1}=0}$ becomes unstable due to bifurcation, but continues to exist as a fixed point even after r > 1. This does not mean that there is no initial value other than${\displaystyle x_{f1}}$ itself that  can reach this unstable fixed point  ${\displaystyle x_{f1}}$. This is${\displaystyle x_0=1}$, and since the logistic map satisfies f (1) = 0 regardless of the value of r,  applying the map once to${\displaystyle x_0=1}$ maps it to${\displaystyle x_{f1}=0}$. A point such as x = 1 that can be reached directly as a fixed point by a  finite number of iterations of the map is called a final fixed point.

Withr between 2 and 3, the population will also eventually approach the same value⁠r − 1/r⁠, but first will fluctuate around that value for some time. Therate of convergence is linear, except forr = 3, when it is dramatically slow, less than linear (seeBifurcation memory).

When the parameter 2 < r < 3, except for the initial values 0 and 1, the fixed point${\displaystyle x_{f2}=1-1/r}$ is the same as when 1 < r ≤ 2.However, in this case the convergence is not monotonic. As the variable approaches${\displaystyle x_{f2}}$, it becomes larger and smaller than${\displaystyle x_{f2}}$ repeatedly, and follows a convergent trajectory that oscillates around${\displaystyle x_{f2}}$.

. The value that is mapped to${\displaystyle x_{f2}}$ by applying the mapping once is${\displaystyle f({\tilde{x}}_{f2})=x_{f2}}$-->

In general,bifurcation diagrams are useful for understanding bifurcations. These diagrams are graphs of fixed points (or periodic points, as described below) x as a function of a parameter a, with a on the horizontal axis and x on the vertical axis. To distinguish between stable and unstable fixed points, the former curves are sometimes drawn as solid lines and the latter as dotted lines. When drawing a bifurcation diagram for the logistic map, we have a straight line representing the fixed point${\displaystyle x_{f1}=0}$  and a straight line representing the fixed point${\displaystyle x_{f2}=1-1/a}$It can be seen that the curves representing a and b intersect at r = 1, and that stability is switched between the two.

In the general case Withr between 3 and 1 + √6 ≈ 3.44949 the population will approach permanent oscillations between two values. These two values are dependent onr and given by^[6]${\displaystyle x_{\pm }=\frac{1}{2r}\left(r+1\pm \sqrt{(r-3)(r+1)}\right)}$.

When the parameter is exactly r = 3, the orbit also has a fixed point${\displaystyle x_{f2}=1-1/r}$. However, the variables converge more slowly than when. When${\displaystyle r=3}$, the derivative${\displaystyle f^{\prime }(x_{f2})}$ reaches −1 and no longer satisfies equation (3-1). When r exceeds 3,, and${\displaystyle x_{f2}}$ becomes  an unstable fixed point. That is, another bifurcation occurs at${\displaystyle r=3}$.

For${\displaystyle r=3}$ a type of bifurcation known as aperiod doubling bifurcation occurs. For${\displaystyle r>3}$, the orbit no longer converges to a single point, but instead alternates between large and small values even after a sufficient amount of time has passed. For example, for${\displaystyle r=3.3}$, the variable alternates between the values 0.4794... and 0.8236....

Spider diagram and time series for a = 3.3. The orbit is attracted to a stable 2-periodic point.

An orbit that cycles through the same values periodically is called a periodic orbit. In this case, the final behavior of the variable as n → ∞ is a periodic orbit with two periods. Each value (point) that makes up a periodic orbit is called a periodic point. In the example where a = 3.3, 0.4794... and 0.8236... are periodic points. If a certain x is a periodic point, then in the case of two periodic points, applying the map twice to x will return it to its original state, so

${\displaystyle f(f(x))=f^2(x)=x}$

3-4

If we apply the logistic map equation (1-2) to this equation, we get

${\displaystyle r^{2}x(1-x)(1-rx(1-x))=x}$

3-5

This gives us the following fourth-order equation. The solutions of this equation are the periodic points. In fact, there are two fixed points${\displaystyle x_{f1}=0}$ and${\displaystyle x_{f2}=1-1/r}$also satisfies equation (3-4). Therefore, of the solutions to equation (3-5), two correspond to${\displaystyle x_{f1}}$ and${\displaystyle x_{f2}}$, and the remaining two solutions are 2-periodic points. Let the 2-periodic points be denoted as${\displaystyle x_{f1}^{(2)}}$ and${\displaystyle x_{f2}^{(2)}}$, respectively. By solving equation (3-5), we can obtain them as follows

${\displaystyle x_{f1}^{(2)},x_{f2}^{(2)}=\frac{r+1\pm \sqrt{(r+1)(r-3)}}{2r}}$

3-6

A similar theory about the stability of fixed points can also be applied to periodic points. That is, a periodic point that attracts surrounding orbits is called an asymptotically stable periodic point, and a periodic point where the surrounding orbits move away is called an unstable periodic point. It is possible to determine the stability of periodic points in the same way as for fixed points. In the general case, consider${\displaystyle f^k(x)}$ after k iterations of the map. Let${\displaystyle (f^k{)}^{\prime }(x)}$ be the derivative ${\displaystyle d{f}^k(x)/dx}$ of the k-periodic point${\displaystyle x_f^{(k)}}$. If${\displaystyle x_f^{(k)}}$ satisfies:

3-7

then${\displaystyle x_f^{(k)}}$ is asymptotically stable.

${\displaystyle \left|(f^k{)}^{\prime }(x_f^{(k)})\right|>1}$

3-8

then${\displaystyle x_f^{(k)}}$ is unstable.

The above discussion of the stability of periodic points can be easily understood by drawing a graph, just like the fixed points. In this diagram, the horizontal axis is xn and the vertical axis is${\displaystyle x_{n+2}}$, and a curve is drawn that shows the relationship between${\displaystyle x_{n+2}}$ and${\displaystyle x_n}$. The intersections of this curve and the 45° line are points that satisfy equation (3-4), so the intersections represent fixed points and 2-periodic points. If we draw a graph of the logistic map${\displaystyle f^2(x)}$, we can observe that the slope of the tangent at the fixed point${\displaystyle x_{f2}}$ exceeds  1 at the boundary${\displaystyle r=3}$ and becomes unstable. At the same time, two new intersections appear, which are the periodic points${\displaystyle x_{f1}^{(2)}}$ and${\displaystyle x_{f2}^{(2)}}$.

The relationship between${\displaystyle x_{n+2}}$ and${\displaystyle x_n}$ when r = 2.7, before the period doubling bifurcation occurs. The orbit converges to a fixed point${\displaystyle x_{f2}}$.

The relationship between${\displaystyle x_{n+2}}$ and${\displaystyle x_n}$ when r = 3. The tangent slope at the fixed point${\displaystyle x_{f2}}$ is exactly 1, and a period doubling bifurcation occurs.

The relationship between${\displaystyle x_{n+2}}$ and${\displaystyle x_n}$ when r = 3.3.${\displaystyle x_{f2}}$ becomes unstable and the orbit converges to the periodic points${\displaystyle x_{f1}^{(2)}}$ and${\displaystyle x_{f2}^{(2)}}$.

When we actually calculate the differential coefficients of two periodic points for the logistic map, we get

${\displaystyle (f^2{)}^{\prime }(x_f^{(2)})=4+2r-r^2}$

3-9

When this is applied to equation (3-7), the parameter a becomes:

3-10

It can be seen that the 2-periodic points are asymptotically stable when this range is, i.e., when r exceeds${\displaystyle 1+\sqrt{6}=\mathrm{3.44949...}}$, the 2-periodic points are no longer asymptotically stable and their behavior changes.

Almost all initial values in [0, 1] are attracted to the 2-periodic points, but${\displaystyle x_{f1}=0}$ and${\displaystyle x_{f2}=1-1/a}$remains as an unstable fixed point in [0,1]. These unstable fixed points continue to remain in [0,1] even if r is increased. Therefore, when the initial value is exactly${\displaystyle x_{f1}}$ or${\displaystyle x_{f2}}$, the orbit  does not attract to a 2-periodic point. Moreover, when the initial value is the final fixed point for${\displaystyle x_{f1}}$ or the final fixed point for${\displaystyle x_{f2}}$, the orbit does not attract to a 2-periodic point. There are an infinite number of such final fixed points in [0, 1]. However, the number of such points is negligibly small compared to the set of real numbers [ 0, 1].

Withr between 3.44949 and 3.54409 (approximately), from almost all initial conditions the population will approach permanent oscillations among four values. The latter number is a root of a 12th degree polynomial (sequenceA086181 in theOEIS).

Withr increasing beyond 3.54409, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc. The lengths of the parameter intervals that yield oscillations of a given length decrease rapidly; the ratio between the lengths of two successive bifurcation intervals approaches theFeigenbaum constantδ ≈ 4.66920. This behavior is an example of aperiod-doubling cascade.

When the parameter r exceeds${\displaystyle 1+\sqrt{6}=\mathrm{3.44949...}}$, the previously stable 2-periodic points become unstable, stable 4-periodic points are generated, and the orbit gravitates toward a 4-periodic oscillation. That is, a period-doubling bifurcation occurs again at${\displaystyle r=\mathrm{3.44949...}}$. The value of x at the 4-periodic point is also

${\displaystyle f(f(f(f(x))))=f^4(x)=x}$

3-11

satisfies, so that solving this equation allows the values of x at the 4-periodic points to be found. However, equation (3-11) is a 16th-order equation, and even if we factor out the four solutions for the fixed points and the 2-periodic points, it is still a 12th-order equation. Therefore, it is no longer possible to solve this equation to obtain an explicit function of a that represents the values of the 4-periodic points in the same way as for the 2-periodic points.

As a becomes larger, the stable 4-periodic point undergoes another period doubling, resulting in a stable 8-periodic point. As an increases, period doubling bifurcations occur infinitely: 16, 32, 64, ..., and so on, until an infinite period, i.e., an orbit that never returns to its original value. This infinite series of period doubling bifurcations is called a cascade. While these period doubling bifurcations occur infinitely, the intervals between a at which they occur decrease in a geometric progression. Thus, an infinite number of period doubling bifurcations occur before the parameter a reaches a finite value. Let the bifurcation from period 1 to period 2 that occurs at r = 3 be counted as the first period doubling bifurcation. Then, in this cascade of period doubling bifurcations, a stable 2k-periodic point occurs at the k-th bifurcation point. Let the k-th bifurcation point a be denoted as a k. In this case, it is known that${\displaystyle r_k}$ converges to the following value as k → ∞. (sequenceA098587 in theOEIS)

${\displaystyle r_{\mathrm{\infty }}=\underset{k\to \mathrm{\infty }}{lim}{r}_k=\mathrm{3.56994...}}$

3-12

Furthermore, it is known that the rate of decrease of a k reaches a constant value in the limit, as shown in the following equation.

${\displaystyle \delta =\underset{k\to \mathrm{\infty }}{lim}\frac{r_k-r_{k-1}}{r_{k+1}-r_k}=\mathrm{4.66920...}}$

3-13

This value of δ is called the Feigenbaum constant because it was discovered by mathematical physicist Mitchell Feigenbaum.${\displaystyle r_{\mathrm{\infty }}}$ is called the Feigenbaum point.^{[citation needed]} In the period doubling cascade,${\displaystyle f^m}$  and${\displaystyle f^{2m}}$ have the property that they are locally identical after an appropriate scaling transformation. The Feigenbaum constant can be found by a technique called renormalization that exploits this self-similarity. The properties that the logistic map exhibits in the period doubling cascade are also universal in a broader class of maps, as will be discussed later. 

To get an overview of the final behavior of an orbit for a certain parameter, an approximate bifurcation diagram, orbital diagram, is useful. In this diagram, the horizontal axis is the parameter r and the vertical axis is the variable x, as in the bifurcation diagram. Using a computer, the parameters are determined and, for example, 500 iterations are performed. Then, the first 100 results are ignored and only the results of the remaining 400 are plotted. This allows the initial transient behavior to be ignored and the asymptotic behavior of the orbit remains. For example, when one point is plotted for r, it is a fixed point, and when m points are plotted for r, it corresponds to an m-periodic orbit. When an orbital diagram is drawn for the logistic map, it is possible to see how the branch representing the stable periodic orbit splits, which represents a cascade of period-doubling bifurcations.

When the parameter${\displaystyle r=r_{\mathrm{\infty }}}$ is exactly the accumulation point of the period-doubling cascade, the variable${\displaystyle x_n}$ is attracted to aperiodic orbits that never close. In other words, there exists a periodic point with infinite period at${\displaystyle r_{\mathrm{\infty }}}$. This aperiodic orbit is called the Feigenbaum attractor. The critical${\displaystyle 2^{\mathrm{\infty }}}$ attractor. An attractor is a term used to refer to a region that has the property of attracting surrounding orbits, and is the orbit that is eventually drawn into and continues. The attractive fixed points and periodic points mentioned above are also members of the attractor family.

The structure of the Feigenbaum attractor is the same as that of a fractal figure called theCantor set. The number of points that compose the Feigenbaum attractor is infinite and their cardinality is equal to the real numbers. However, no matter which two of the points are chosen, there is always an unstable periodic point between them, and the distribution of the points is not continuous. The fractal dimension of the Feigenbaum attractor, the Hausdorff dimension or capacity dimension, is known to be approximately 0.54.

• Atr ≈ 3.56995 (sequenceA098587 in theOEIS) is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions, we no longer see oscillations of finite period. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
• This number shall be compared and understood as the equivalent of theReynolds number for the onset of other chaotic phenomena such asturbulence and similar to thecritical temperature of aphase transition. In essence thephase space contains a full subspace of cases with extra dynamical variables to characterize the microscopic state of the system, these can be understood asEddies in the case of turbulence andorder parameters in the case ofphase transitions.
• Most values ofr beyond 3.56995 exhibit chaotic behaviour, but there are still certain isolated ranges ofr that show non-chaotic behavior; these are sometimes calledislands of stability. For instance, beginning at 1 + √8^[8] (approximately 3.82843) there is a range of parametersr that show oscillation among three values, and for slightly higher values ofr oscillation among 6 values, then 12 etc.
• At${\displaystyle r=1+\sqrt{8}=\mathrm{3.8284...}}$, the stable period-3 cycle emerges.^[9]
• The development of the chaotic behavior of the logistic sequence as the parameterr varies from approximately 3.56995 to approximately 3.82843 is sometimes called thePomeau–Manneville scenario, characterized by a periodic (laminar) phase interrupted by bursts of aperiodic behavior. Such a scenario has an application in semiconductor devices.^[10] There are other ranges that yield oscillation among 5 values etc.; all oscillation periods occur for some values ofr. Aperiod-doubling window with parameterc is a range ofr-values consisting of a succession of subranges. Thekth subrange contains the values ofr for which there is a stable cycle (a cycle that attracts a set of initial points of unit measure) of period2^kc. This sequence of sub-ranges is called acascade of harmonics.^{[May, Robert M. (1976) 1]} In a sub-range with a stable cycle of period2^k*c, there are unstable cycles of period2^kc for allk <k*. Ther value at the end of the infinite sequence of sub-ranges is called thepoint of accumulation of the cascade of harmonics. Asr rises there is a succession of new windows with differentc values. The first one is forc = 1; all subsequent windows involving oddc occur in decreasing order ofc starting with arbitrarily largec.^{[May, Robert M. (1976) 1][11]}
• At${\displaystyle r=\mathrm{3.678...},x=\mathrm{0.728...}}$, two chaotic bands of the bifurcation diagram intersect in the firstMisiurewicz point for the logistic map. It satisfies the equations${\displaystyle r^3-2r^2-4r-8=0,x=1-1/r}$.^[12]
• Beyondr = 4, almost all initial values eventually leave the interval[0,1] and diverge. The set of initial conditions which remain within[0,1] form aCantor set and the dynamics restricted to this Cantor set is chaotic.^[13]

For any value ofr there is at most one stable cycle. If a stable cycle exists, it is globally stable, attracting almost all points.^[14]: 13 Some values ofr with a stable cycle of some period have infinitely many unstable cycles of various periods.

Thebifurcation diagramat rightabove summarizes this. The horizontal axis shows the possible values of the parameterr while the vertical axis shows the set of values ofx visited asymptotically from almost all initial conditions by the iterates of the logistic equation with thatr value.

The bifurcation diagram is aself-similar: if we zoom in on the above-mentioned valuer ≈ 3.82843 and focus on one arm of the three, the situation nearby looks like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points. This is an example of the deep and ubiquitous connection betweenchaos andfractals.

Magnification of the chaotic region of the map

Stable regions within the chaotic region, where a tangent bifurcation occurs at the boundary between the chaotic and periodic attractor, giving intermittent trajectories as described in thePomeau–Manneville scenario

We can also consider negative values ofr:

• Forr between -2 and -1 the logistic sequence also features chaotic behavior.^[6]
• Withr between -1 and 1 - √6 and forx₀ between 1/r and 1-1/r, the population will approach permanent oscillations between two values, as with the case ofr between 3 and 1 + √6, and given by the same formula.^[6]

Chaotic orbits of the logistic map when r = 3.82. The orange squares are orbits starting from${\displaystyle x_0=0.1234}$, and the blue-green circles are orbits starting from${\displaystyle {\hat{x}}_0=0.1234+{10}^{-9}}$.

The trajectory starting from x 0 = 0.1234 and ˆx The difference in orbits starting from${\displaystyle x_0=0.1234+{10}^{-9}}$ grows exponentially. The vertical axis is${\displaystyle \mathrm{\Delta }{x}_n=|x_n-{\hat{x}}_n|}$,shown on alogarithmic scale.

When the parameter r exceeds${\displaystyle r_{\mathrm{\infty }}=\mathrm{3.56994...}}$, the logistic map exhibits chaotic behavior. Roughly speaking, chaos is a complex and irregular behavior that occurs despite the fact that the difference equation describing the logistic map has no probabilistic ambiguity and the next state is completely and uniquely determined. The range of${\displaystyle r>r_{\mathrm{\infty }}}$ of the logistic map is called the chaotic region.

One of the properties of chaos is its unpredictability, symbolized by the termbutterfly effect. This is due to the property of chaos that a slight difference in the initial state can lead to a huge difference in the later state. In terms of a discrete dynamical system, if we have two initial values${\displaystyle x_0}$ and${\displaystyle {\hat{x}}_0}$No matter how close they are, once time n has progressed to a certain extent, each destination${\displaystyle x_n}$ and${\displaystyle {\hat{x}}_n}$ can vary significantly. For example, use${\displaystyle r=3.95,x_0=0.1,{\hat{x}}_0=x_0+{10}^{-9}}$If the orbits are calculated using two very similar initial values, 0 = 0.1000000001, the difference grows to macroscopic values that are clearly visible on the graph after about 29 iterations.

This property of chaos, called initial condition sensitivity, can be quantitatively expressed by theLyapunov exponent. For a one-dimensional map, the Lyapunov exponent λ can be calculated as follows:

${\displaystyle {\displaystyle \lambda =\underset{N\to \mathrm{\infty }}{lim}\frac{1}{N}\sum _{i=0}^{N-1}\log \left|f^{\mathrm{\prime }}(x_i)\right|}}$

3-14

Here, log means natural logarithm. This λ is the distance between the two orbits (${\displaystyle x_n}$ and${\displaystyle {\hat{x}}_n}$).A positive value of λ indicates that the system is sensitive to initial conditions, while a zero or negative value indicates that the system is not sensitive to initial conditions. When calculating λ of numerically, it can be confirmed λ remains in the range of zero or negative values in the range, and that λ can take positive values in the range${\displaystyle r>r_{\mathrm{\infty }}}$.

Window, intermittent

Even beyond${\displaystyle r_{\mathrm{\infty }}}$, the behavior does not depend simply on the parameter r. Many sophisticated mathematical structures lurk in the chaotic region for${\displaystyle r>r_{\mathrm{\infty }}}$. In this region, chaos does not persist forever; stable periodic orbits reappear. The behavior for can be broadly divided into two types:

• Stable periodic point: In this case, the Lyapunov exponent is negative.
• Aperiodic orbits: In this case, the Lyapunov exponent is positive.

The region of stable periodic points that exists for r is called a periodic window, or simply a window. If one looks at a chaotic region in an orbital diagram, the region of nonperiodic orbits looks like a cloud of countless points, with the windows being the scattered blanks surrounded by the cloud.

Orbit diagram of the logistic map from r = 3.55 to r = 4 (parameter is denoted as r in the diagram)

In each window, the cascade ofperiod-doubling bifurcations that occurred before${\displaystyle r_{\mathrm{\infty }}=\mathrm{3.56994...}}$ occurs again. However, instead of the previous stable periodic orbits of 2^k, new stable periodic orbits such as 3×2^k and 5×2^k are generated. The first window has a period of p, and the windows from which the period-doubling cascade occurs are called windows of period p, etc.. For example, a window of period 3 exists in the region around 3.8284 < a < 3.8415, and within this region the period doublings are: 3, 6, 12, 24, ..., 3×2^k, ....

In the window region, chaos does not disappear but exists in the background. However, this chaos is unstable, so only stable periodic orbits are observed. In the window region, this potential chaos appears before the orbit is attracted from its initial state to a stable periodic orbit. Such chaos is called transient chaos. In this potential presence of chaos, windows differ from the periodic orbits that appeared before a∞.

There are an infinite number of windows in the range a∞ < a < 4. The windows have various periods, and there is a window with a period for every natural number greater than or equal to three. However, each window does not occur exactly once. The larger the value of p, the more often a window with that period occurs. A window with period 3 occurs only once, while a window with period 13 occurs 315 times. When a periodic orbit of 3 occurs in the window with period 3, the Szarkovsky order is completed, and all orbits with all periods have been seen.

If we restrict ourselves to the case where p is a prime number, the number of windows with period p is

${\displaystyle {\displaystyle N_p=\frac{2^{p-1}-1}{p}}}$

3-15

This formula was derived for p to be a prime number, but in fact it is possible to calculate with good accuracy the number of stable p- periodic points for non-prime p as well.

The window width (the difference between a where the window begins and a where the window ends) is widest for windows with period 3 and narrows for larger periods. For example, the window width for a window with period 13 is about 3.13 × 10−6. Rough estimates suggest that about 10% of${\displaystyle [r_{\mathrm{\infty }},4]}$ is in the window region, with the rest dominated by chaotic orbits.

The change from chaos to a window as r is increased is caused by a tangent bifurcation, where the map curve is tangent to the diagonal of y = x at the moment of bifurcation, and further parameter changes result in two fixed points where the curve and the line intersect. For a window of period p, the iterated map${\displaystyle f^p(x)}$ exhibits tangent bifurcation, resulting in stable p-periodic orbits. The exact value of the bifurcation point for a window of period 3 is known, and if the value of this bifurcation point r is${\displaystyle r_3}$, then${\displaystyle r_3=1+\sqrt{8}=\mathrm{3.828427...}}$. The outline of this bifurcation can be understood by considering the graph of${\displaystyle f^3(x)}$ (vertical axis${\displaystyle x_{n+3}}$, horizontal axis${\displaystyle x_n}$).  

Graph of${\displaystyle f^3(x)}$ when r is slightly less than 3. The graph is not tangent except at the fixed points, and there are no 3-periodic points.

When a is exactly 3, the graph touches the diagonal at exactly three points, resulting in three periodic points.

When a is slightly greater than 3, the graph passes the diagonal and splits into stable and unstable 3-periodic points.

When we look at the behavior of${\displaystyle x_n}$ when r = 3.8282, which is slightly smaller than the branch point${\displaystyle r_3}$, we can see that in addition to the irregular changes, there is also a behavior that changes periodically with approximately three periods, and these occur alternately. This type of periodic behavior is called a "laminar", and the irregular behavior is called a burst, in analogy with fluids. There is no regularity in the length of the time periods of the bursts and laminars, and they change irregularly. However, when we observe the behavior at r = 3.828327, which is closer to${\displaystyle r_3}$, the average length of the laminars is longer and the average length of the bursts is shorter than when r = 3.8282. If we further increase r, the length of the laminars becomes larger and larger, and at${\displaystyle r_3}$ it changes to a perfect three- period.

Time series when r = 3.8282

Time series when r = 3.828327

The phenomenon in which orderly motions called laminars and disorderly motions called bursts occur intermittently is called intermittency or intermittent chaos. If we consider the parameter a decreasing from a3, this is a type of emergence of chaos. As the parameter moves away from the window, bursts become more dominant, eventually resulting in a completely chaotic state. This is also a general route to chaos, like the period doubling bifurcation route mentioned above, and routes characterized by the emergence of intermittent chaos due to tangent bifurcations are called intermittency routes.

The mechanism of intermittency can also be understood from the graph of the map. When${\displaystyle r}$ is slightly smaller than${\displaystyle r_3}$, there is a very small gap between the graph of${\displaystyle f^3(x)}$ and the diagonal. This gap is called a channel, and many iterations of the map occur as the orbit passes through the narrow channel. During the passage through this channel,${\displaystyle x_n}$ and${\displaystyle x_{n+3}}$ become very close, and the variables change almost like a periodic three orbit. This corresponds to a laminar. The orbit eventually leaves the narrow channel, but returns to the channel again as a result of the global structure of the map. While leaving the channel, it behaves chaotically. This corresponds to a burst.

Band, window finish

Looking at the entire chaotic domain, whether it is chaotic or windowed, the maximum and minimum values on the vertical axis of the orbital diagram (the upper and lower limits of the attractor) are limited to a certain range. As shown in equation (2-1), the maximum value of the logistic map is given by r/4, which is the upper limit of the attractor. The lower limit of the attractor is given by the point f(r/4) where r/4 is mapped. Ultimately, the maximum and minimum values at which xn moves on the orbital diagram depend on the parameter r

${\displaystyle {\displaystyle \frac{r^2(4-r)}{16}\le x_n\le \frac{r}{4}}}$

3-16

Finally, for r = 4, the orbit spans the entire range [0, 1].

When observing an orbital map, the distribution of points has a characteristic shading. Darker areas indicate that the variable takes on values in the vicinity of the darker areas, whereas lighter areas indicate that the variable takes on values in the vicinity of the darker areas.^{[clarification needed]} These differences in the frequency of the points are due to the shape of the graph of the logistic map. The top of the graph, near r/4, attracts orbits with high frequency, and the area near f(r/4) that is mapped from there also becomes highly frequent, and the area near${\displaystyle f^2(r/4)}$ that is mapped from there also becomes  highly frequent, and so on. The density distribution of points generated by the map is characterized by a quantity called an invariant measure or distribution function, and the invariant measure of the attractor is reproducible regardless of the initial value.

Looking at the beginning of the chaotic region of the orbit diagram, just beyond the accumulation point${\displaystyle r_{\mathrm{\infty }}=3.56994}$ of the first period - doubling cascade, one can see that the orbit is divided into several subregions. These subregions are called bands. When there are multiple bands, the orbit moves through each band in a regular order, but the values within each band are irregular. Such chaotic orbits are called band chaos or periodic chaos, and chaos with k bands is called k -band chaos. Two-band chaos lies in the range 3.590 < r < 3.675, approximately.

Band structure. Because the${\displaystyle e_p}$ spacing rapidly decreases, it is not possible to show more than eight bands. The top and bottom lines, which contain the orbitals, are within the range of equation (3-16).

As the value of r is further decreased from the left-hand end of two-band chaos, r = 3.590, the number of bands doubles, just as in the period doubling bifurcation. Let${\displaystyle e_p}$ (for p = 1, 2, 4, ..., 2k, ...) denote the bifurcation points where p − 1 band chaos splits into p band chaos, or where p band chaos merges into p − 1 band chaos. Then, just as in the period doubling bifurcation, e p accumulates to a value as p → ∞. At this accumulation point${\displaystyle e_{\mathrm{\infty }}}$, the number of bands becomes infinite, and the value of${\displaystyle e_{\mathrm{\infty }}}$ is equal to the value of${\displaystyle r_{\mathrm{\infty }}}$.

Similarly, for the bifurcation points of the period-doubling bifurcation cascade that appeared before a∞, let a p (where p = 1, 2, 4, ..., 2k, ...) denote the bifurcation points where p stable periodic orbits branch into p + 1 stable periodic orbits. In this case, if we look at the orbital diagram from${\displaystyle r_2}$ to${\displaystyle e_2}$, there are two reduced versions of the global orbital diagram from${\displaystyle r_1}$ to${\displaystyle e_1}$ in the orbital diagram from${\displaystyle r_2}$ to${\displaystyle e_2}$. Similarly, if we look at the orbital diagram from${\displaystyle r_4}$ to${\displaystyle e_4}$, there are four reduced versions of the global orbital diagram from a1 to e1 in the orbital diagram from${\displaystyle r_4}$ to${\displaystyle e_4}$. Similarly, there are p reduced versions of the global orbital diagram in the orbital diagram from ap to ep, and the branching structure of the logistic map has an infinite self-similar hierarchy.

A self-similar hierarchy of bifurcation structures also exists within windows. The period-doubling bifurcation cascades within a window follow the same path as the cascades of period-2k bifurcations. That is, there are an infinite number of period-doubling bifurcations within a window, after which the behavior becomes chaotic again. For example, in a window of period 3, the cascade of stable periodic orbits ends at${\displaystyle a_{3\mathrm{\infty }}}$ ≈ 3.8495. After${\displaystyle a_{3\mathrm{\infty }}}$ ≈ 3.8495, the behavior becomes band chaos of multiples of three. As a increases from${\displaystyle a_{3\mathrm{\infty }}}$, these band chaos also merge by twos, until at the end of the window there are three bands. Within such bands within a window, there are an infinite number of windows. Ultimately, the window contains a miniature version of the entire orbital diagram for 1 ≤ a ≤ 4, and within the window there exists a self-similar hierarchy of branchings.

At the end of the window, the system reverts to widespread chaos. For a period 3 window, the final 3-band chaos turns into large-area 1-band chaos at a ≈ 3.857, ending the window. However, this change is discontinuous, and the 3-band chaotic attractor suddenly changes size and turns into a 1-band. Such discontinuous changes in attractor size are called crises. Crises of this kind, which occur at the end of a window, are also called internal crises. When a crisis occurs at the end of a window, a stable periodic orbit just touches an unstable periodic point that is not visible on the orbit diagram. This creates an exit point through which the periodic orbits can escape, resulting in an internal crisis. Immediately after the internal crisis, there are periods of widespread chaos, and periods of time when the original band chaotic behavior reoccurs, resulting in a kind of intermittency similar to that observed at the beginning of a window.

Spider diagram of the logistic map with parameter r = 4 (left) and time series up to n = 500 (right) for the initial value${\displaystyle x_0}$ = 0.3.

When the parameter r = 4, the behavior becomes chaotic over the entire range [0, 1]. At this time, the Lyapunov exponent λ is maximized, and the state is the most chaotic. The value of λ for the logistic map at r = 4 can be calculated precisely, and its value is λ = log 2. Although a strict mathematical definition of chaos has not yet been unified, it can be shown that the logistic map with r = 4 is chaotic on [0, 1] according to one well-known definition of chaos.

The invariant measure of the density of points, ρ(x), can also be given by the exact function ρ(x) for r = 4:

${\displaystyle {\displaystyle \rho (x)=\frac{1}{\pi \sqrt{x(1-x)}}}}$

3-17

Here, ρ(x) means that the fraction of points xn that fall in the infinitesimal interval [x,x+dx] when the map is iterated is given by ρ(x) dx. The frequency distribution of the logistic map with r = 4 has high density near both sides of [0, 1] and is least dense at x = 0.5.

When r = 4, apart from chaotic orbits, there are also periodic orbits with any period. For a natural number n, the graph of${\displaystyle f_{r=4}^n(x)}$ is a curve with${\displaystyle 2^{n-1}}$ peaks and${\displaystyle 2^{n-1}-1}$ valleys, all of which are tangent to 0 and 1. Thus, the number of intersections between the diagonal and the graph is${\displaystyle 2^n}$, and there are${\displaystyle 2^n}$ fixed points of${\displaystyle f^n(x)}$. The n-periodic points are always included in these${\displaystyle 2^n}$ fixed points, so any n-periodic orbit exists for${\displaystyle f_{r=4}^n(x)}$. Thus,when r = 4, there are an infinite number of periodic points on [0, 1], but all of these periodic points are unstable. Furthermore,the uncountably infinite set in the interval [0, 1], the number of periodic points is countably infinite, and so almost all orbits starting from initial values are not periodic but non-periodic. 

One of the important aspects of chaos is its dual nature: deterministic and stochastic. Dynamical systems are deterministic processes, but when the range of variables is appropriately coarse-grained, they become indistinguishable from stochastic processes. In the case of the logistic map with r = 4, the outcome of every coin toss can be described by the trajectory of the logistic map. This can be elaborated as follows.

Assume that a coin is tossed with a probability of 1/2 landing on heads or tails, and the coin is tossed repeatedly. If heads is 0 and tails is 1, then the result of heads, tails, heads, tails, etc. will be a symbol string such as 01001.... On the other hand, for the trajectory${\displaystyle x_0,x_1,x_2,...}$ of the logistic map, values less than x = 0.5 are converted to 0 and values greater than x = 0.5 are converted to 1, and the trajectory is replaced with a symbol string consisting of 0s and 1s. For example, if the initial value is${\displaystyle x_0=0.2}$, then${\displaystyle x_1=0.64}$,${\displaystyle x_2=0.9216}$,${\displaystyle x_3=0.28901}$, ..., so the trajectory will be the symbol string 0110.... Let${\displaystyle S_C}$ be the symbol string resulting from the former coin toss, and${\displaystyle S_L}$ be the symbol string resulting from the latter logistic map. The symbols in the symbol string${\displaystyle S_C}$ were determined by random coin tossing, so any number sequence patterns are possible. So, whatever the string${\displaystyle S_L}$ of the logistic map, there is an identical one in${\displaystyle S_C}$. And, what is "remarkable" is that the converse is also true: whatever the string of${\displaystyle S_C}$, it can be realized by a logistic map trajectory${\displaystyle S_L}$ by choosing the appropriate initial values. That is, for any${\displaystyle S_C}$, there exists a unique point${\displaystyle x_0}$ in [0, 1] such that${\displaystyle S_C=S_L}$.

When the parameter r exceeds 4, the vertex r /4 of the logistic map graph exceeds 1. To the extent that the graph penetrates 1, trajectories can escape [0, 1].

The bifurcation at r = 4 is also a type of crisis, specifically a boundary crisis. In this case, the attractor at [0, 1] becomes unstable and collapses, and since there is no attractor outside it, the trajectory diverges to infinity.

On the other hand, there are orbits that remain in [0, 1] even if r > 4. Easy-to-understand examples are fixed points and periodic points in [0, 1], which remain in [0, 1]. However, there are also orbits that remain in [0, 1] other than fixed points and periodic points.

Let${\displaystyle A_0}$ be the interval of x such that f  (x) > 1. As mentioned above,once a variable${\displaystyle x_n}$ enters${\displaystyle A_0}$, it diverges to minus infinity. There is also${\displaystyle r_n}$ x in [0, 1] that maps to${\displaystyle A_0}$ after one application of the map. This interval of x is divided into two, which are collectively called${\displaystyle A_1}$. Similarly, there are four intervals that map to${\displaystyle A_1}$ after one application of the map, which are collectively called${\displaystyle A_2}$. Similarly,there are 2n intervals${\displaystyle A_n}$ that reach${\displaystyle A_0}$ after n iterations. Therefore, the interval${\displaystyle \mathrm{\Lambda }}$ obtained by removing${\displaystyle A_n}$ from [0, 1] an infinite number of times as follows is a collection of orbits that remain in I.

${\displaystyle {\displaystyle \mathrm{\Lambda }=\left[0,1\right]-\bigcup _{n=0}^{\mathrm{\infty }}{A}_n}}$

3-18

The process of removing${\displaystyle A_n}$ from [0, 1] is similar to the construction of the Cantor set mentioned above, and in fact Λ exists in [0, 1] as a Cantor set (a closed, completely disconnected, and complete subset of [0, 1]). Furthermore, on${\displaystyle \mathrm{\Lambda }}$, the logistic map${\displaystyle f_{r>4}}$ is chaotic.

Since the logistic map has been often studied as an ecological model, the case where the parameter r is negative has rarely been discussed. As a decreases from 0, when −1 < r < 0, the map asymptotically approaches a stable fixed point of xf = 0, but when a exceeds −1, it bifurcates into two periodic points, and as in the case of positive values, it passes through a period doubling bifurcation and reaches chaos. Finally, when a falls below −2, the map diverges to plus infinity.

Orbit diagram for parameter r from −2 to 4. The orbit diverges when the parameter a goes beyond this range, both on the negative and positive sides.

For a logistic map with a specific parameter${\displaystyle a}$, an exact solution that explicitly includes the time${\displaystyle n}$ and the initial value${\displaystyle x_0}$ has been obtained as follows.

When r = 4

${\displaystyle {\displaystyle x_n=\frac{1-\cos \left[2^n\arccos (1-2x_0)\right]}{2}}}$

3-19

When r = 2

${\displaystyle {\displaystyle x_n=\frac{1-\exp \left[2^n\log (1-2x_0)\right]}{2}}}$

3-20

When r = −2

${\displaystyle {\displaystyle x_n=\frac{1}{2}-\cos \left\{\frac{1}{3}\left[\pi -(-2{)}^n\left(\pi -3\arccos (\frac{1}{2}-x_0)\right)\right]\right\}}}$

3-21

Considering the three exact solutions above, all of them are

${\displaystyle {\displaystyle x_n=\frac{1}{2}\left\{1-f\left[a^n{f}^{-1}(1-2x_0)\right]\right\}}}$

3-22

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 exhibits:^{[Devaney 1989 3]} (seeChaotic dynamics)

• Great sensitivity on initial conditions: i.e. for a small or infinitesimal variation in the initial conditions you may have a large finite effect.
• Topologically transitive: i.e. the system tends to occupy all available states in a similar sense to fluid mixing.^[15]
• The system exhibitsdenseperiodic orbits

These are properties of the logistic map for most values ofr between about 3.57 and 4 (as noted above).^{[May, Robert M. (1976) 1]} 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, thequadraticdifference equation describing it may be thought of as a stretching-and-folding operation on the interval(0,1).^[4]

The following figure illustrates the stretching and folding over a sequence of iterates of the map. Figure (a), left, shows a two-dimensionalPoincaré plot of the logistic map'sstate space forr = 4, and clearly shows the quadratic curve of the difference equation (1). However, we canembed the same sequence in a three-dimensional state space, in order to investigate the deeper structure of the map. Figure (b) demonstrates this, showing how initially nearby points begin to diverge, particularly in those regions ofx_t 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 (seeLyapunov exponents), evidenced also by thecomplexity andunpredictability 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 apseudo-random number generator in early computers.^[4]

Atr = 2, the function${\displaystyle rx(1-x)}$ intersects${\displaystyle y=x}$ precisely at the maximum point, so convergence to the equilibrium point is on the order of${\displaystyle {\delta }^{2^n}}$. Consequently, the equilibrium point is called "superstable". Its Lyapunov exponent is${\displaystyle -\mathrm{\infty }}$. A similar argument shows that there is a superstable${\displaystyle r}$ value within each interval where the dynamical system has a stable cycle. This can be seen in the Lyapunov exponent plot as sharp dips.^[16]

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 acorrelation dimension of0.500±0.005 (Grassberger, 1983), aHausdorff dimension of about 0.538 (Grassberger 1981), and aninformation dimension of approximately 0.5170976 (Grassberger 1983) forr ≈ 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 thelikelihood of a future state in a chaotic system. If a (possibly chaotic)dynamical system has anattractor, then there exists aprobability 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 parameterr = 4 and an initial state in(0,1), the attractor is also the interval(0,1) and the probability measure corresponds to thebeta distribution with parametersa = 0.5 andb = 0.5. Specifically,^[17] the invariant measure is

$${\displaystyle \frac{1}{\pi \sqrt{x(1-x)}}.}$$

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 informdecisions based on the state of the system.

Thebifurcation diagram for the logistic map can be visualized with the followingPython code:

importnumpyasnpimportmatplotlib.pyplotaspltinterval=(2.8,4)# start, endaccuracy=0.0001reps=600# number of repetitionsnumtoplot=200lims=np.zeros(reps)fig,biax=plt.subplots()fig.set_size_inches(16,9)lims[0]=np.random.rand()forrinnp.arange(interval[0],interval[1],accuracy):foriinrange(reps-1):lims[i+1]=r*lims[i]*(1-lims[i])biax.plot([r]*numtoplot,lims[reps-numtoplot:],"b.",markersize=0.02)biax.set(xlabel="r",ylabel="x",title="logistic map")plt.show()

Although exact solutions to the recurrence relation are only available in a small number of cases, a closed-form upper bound on the logistic map is known when0 ≤r ≤ 1.^[18] There are two aspects of the behavior of the logistic map that should be captured by an upper bound in this regime: the asymptotic geometric decay with constantr, and the fast initial decay whenx₀ is close to 1, driven by the(1 −x_n) term in the recurrence relation. The following bound captures both of these effects:

$${\displaystyle \mathrm{\forall }n\in \{0,1,\dots \}\text{and}{x}_0,r\in [0,1],x_n\le \frac{x_0}{r^{-n}+x_{0}n}.}$$

The special case ofr = 4 can in fact be solved exactly, as can the case withr = 2;^[19] however, the general case can only be predicted statistically.^[20] The solution whenr = 4 is:^[19][21]

$${\displaystyle x_n={\sin }^2\left(2^n\theta \pi \right),}$$

where the initial condition parameterθ is given by

$${\displaystyle \theta =\frac{1}{\pi }{\sin }^{-1}\left(\sqrt{x_0}\right).}$$

For rationalθ, after a finite number of iterationsx_n maps into a periodic sequence. But almost allθ are irrational, and, for irrationalθ,x_n never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor2ⁿ shows the exponential growth of stretching, which results insensitive dependence on initial conditions, while the squared sine function keepsx_n folded within the range[0,1].

Forr = 4 an equivalent solution in terms ofcomplex numbers instead of trigonometric functions is^[19]

$${\displaystyle x_n=\frac{-{\alpha }^{2^n}-{\alpha }^{-2^n}+2}{4}}$$

whereα is either of the complex numbers

$${\displaystyle \alpha =1-2x_0\pm \sqrt{{\left(1-2x_0\right)}^2-1}}$$

withmodulus equal to 1. Just as the squared sine function in the trigonometric solution leads to neither shrinkage nor expansion of the set of points visited, in the latter solution this effect is accomplished by the unit modulus ofα.

By contrast, the solution whenr = 2 is^[19]

$${\displaystyle x_n=\frac{1}{2}-\frac{1}{2}{\left(1-2x_0\right)}^{2^n}}$$

forx₀ ∈ [0,1). Since(1 − 2x₀) ∈ (−1,1) for any value ofx₀ other than the unstable fixed point 0, the term(1 − 2x₀)²ⁿ goes to 0 asn goes to infinity, sox_n goes to the stable fixed point⁠1/2⁠.

For ther = 4 case, from almost all initial conditions the iterate sequence is chaotic. Nevertheless, there exist an infinite number of initial conditions that lead to cycles, and indeed there exist cycles of lengthk forall integersk > 0. We can exploit the relationship of the logistic map to thedyadic transformation (also known as thebit-shift map) to find cycles of any length. Ifx follows the logistic mapx_{n + 1} = 4x_n(1 −x_n) andy follows thedyadic transformation

then the two are related by ahomeomorphism

$${\displaystyle x_n={\sin }^2\left(2\pi y_n\right).}$$

The reason that the dyadic transformation is also called the bit-shift map is that wheny is written in binary notation, the map moves the binary point one place to the right (and if the bit to the left of the binary point has become a "1", this "1" is changed to a "0"). A cycle of length 3, for example, occurs if an iterate has a 3-bit repeating sequence in its binary expansion (which is not also a one-bit repeating sequence): 001, 010, 100, 110, 101, or 011. The iterate 001001001... maps into 010010010..., which maps into 100100100..., which in turn maps into the original 001001001...; so this is a 3-cycle of the bit shift map. And the other three binary-expansion repeating sequences give the 3-cycle 110110110... → 101101101... → 011011011... → 110110110.... Either of these 3-cycles can be converted to fraction form: for example, the first-given 3-cycle can be written as⁠1/7⁠ →⁠2/7⁠ →⁠4/7⁠ →⁠1/7⁠. Using the above translation from the bit-shift map to the${\displaystyle r=4}$ logistic map gives the corresponding logistic cycle 0.611260467... → 0.950484434... → 0.188255099... → 0.611260467.... We could similarly translate the other bit-shift 3-cycle into its corresponding logistic cycle. Likewise, cycles of any lengthk can be found in the bit-shift map and then translated into the corresponding logistic cycles.

However, since almost all numbers in[0,1) are irrational, almost all initial conditions of the bit-shift map lead to the non-periodicity of chaos. This is one way to see that the logisticr = 4 map is chaotic for almost all initial conditions.

The number of cycles of (minimal) lengthk = 1, 2, 3,… for the logistic map withr = 4 (tent map withμ = 2) is a known integer sequence (sequenceA001037 in theOEIS): 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161.... This tells us that the logistic map withr = 4 has 2 fixed points, 1 cycle of length 2, 2 cycles of length 3 and so on. This sequence takes a particularly simple form for primek:2 ⋅⁠2^{k − 1} − 1/k⁠. For example: 2 ⋅ ⁠2^{13 − 1} − 1/13⁠ = 630 is the number of cycles of length 13. Since this case of the logistic map is chaotic for almost all initial conditions, all of these finite-length cycles are unstable.

The bifurcation pattern shown above for the logistic map is not limited to the logistic map . It appears in a number of maps that satisfy certain conditions . The following dynamical system using sine functions is one example :

${\displaystyle {\displaystyle x_{n+1}=b\sin \pi x_n}}$

4-1

Here, the domain is 0 ≤ b ≤ 1 and 0 ≤ x ≤ 1 . The sine map ( 4-1 ) exhibits qualitatively identical behavior to the logistic map ( 1-2 ) : like the logistic map, it also becomes chaotic via a period doubling route as the parameter b increases, and moreover, like the logistic map, it also exhibits a window in the chaotic region .

Both the logistic map and the sine map are one-dimensional maps that map the interval [0, 1] to [0, 1] and satisfy the following property, called unimodal .

${\displaystyle f(0)=f(1)=0}$.The map is differentiable and there exists a unique critical point c in [0, 1] such that${\displaystyle f^{\prime }(c)=0}$.In general, if a one-dimensional map with one parameter and one variable is unimodal and the vertex can be approximated by a second-order polynomial, then, regardless of the specific form of the map, an infinite period-doubling cascade of bifurcations will occur for the parameter range 3 ≤ r ≤ 3.56994... , and the ratio δ defined by equation ( 3-13 ) is equal to the Feigenbaum constant, 4.669... .

The pattern of stable periodic orbits that emerge from the logistic map is also universal . For a unimodal map,${\displaystyle x_{n+1}=cf(x_n)}$ , with parameter c, stable periodic orbits with various periods continue to emerge in a parameter interval where the two fixed points are unstable, and the pattern of their emergence (the number of stable periodic orbits with a certain period and the order of their appearance) is known to be common . In other words, for this type of map, the sequence of stable periodic orbits is the same regardless of the specific form of the map . For the logistic map, the parameter interval is 3 < a < 4, but for the sine map ( 4-1 ), the parameter interval for the common sequence of stable periodic orbits is 0.71... < b < 1 . This universal sequence of stable periodic orbits is called the U sequence .

In addition, the logistic map has the property that its Schwarzian derivative is always negative on the interval [0, 1] . The Schwarzian derivative of a map f (of class C3 ) is 

${\displaystyle {\displaystyle Sf(x)=\frac{f^{‴}(x)}{f^{\prime }(x)}-\frac{3}{2}{\left(\frac{f^{″}(x)}{f^{\prime }(x)}\right)}^2}}$

4-2

In fact, when we calculate the Schwarzian derivative of the logistic map, we get

4-3

where the Schwarzian derivative is negative regardless of the values of a and x . It is known that if a one-dimensional mapping from [0, 1] to [0, 1] is unimodal and has a negative Schwarzian derivative, then there is at most one stable periodic orbit .

Let the symbol ∘ denote the composition of maps . In general, for  a topological space X, Y, two maps f  : X → X and g  : Y → Y are composed by a homeomorphism h : X → Y.

${\displaystyle {\displaystyle h\circ f=g\circ h}}$

4-4

f and g are said to be phase conjugates if they satisfy the relation . The concept of phase conjugation plays an important role in the study of dynamical systems . Phase conjugate f and g exhibit essentially identical behavior, and if the behavior of f is periodic, then g is also periodic, and if the behavior of f is chaotic, then g is also chaotic .

In particular, if a homeomorphism h is linear, then f and g are said to be linearly conjugate . Every quadratic function is linearly conjugate with every other quadratic function . Hence,

${\displaystyle {\displaystyle x_{n+1}=x_n^2+b}}$

4-5

${\displaystyle {\displaystyle x_{n+1}=1-c{x}_n^2}}$

4-6

${\displaystyle {\displaystyle x_{n+1}=d-x_n^2}}$

4-7

are linearly conjugates of the logistic map for any parameter a . Equations ( 4-6 ) and ( 4-7 ) are also called logistic maps . In particular, the form ( 4-7 ) is suitable for time-consuming numerical calculations, since it requires less computational effort .

Moreover, the logistic map${\displaystyle f_{a=4}}$ for${\displaystyle r=4}$ is topologically conjugate to the following tent map T  ( x ) and Bernoulli shift map B  ( x ) .

4-8

4-9

These phase conjugate relations can be used to prove that the logistic map${\displaystyle f_{a=4}}$ is strictly chaotic and to derive the exact solution ( 3-19 ) of${\displaystyle f_{r=4}}$ .

Alternatively, introducing the concept of symbolic dynamical systems, consider the following shift map σ defined on the symbolic string space consisting of strings of 0s and 1s as introduced above :

${\displaystyle {\displaystyle \sigma (s_0{s}_1{s}_2\cdots )=(s_1{s}_2\cdots )}}$

4-10

Here,${\displaystyle s_i}$ is 0 or 1. On the set${\displaystyle \mathrm{\Lambda }}$ introduced in equation ( 3-18 ), the logistic map${\displaystyle f_{r>4}}$ istopologically conjugate to the shift map, so we can use this to derive that${\displaystyle f_{r>4}}$ on${\displaystyle \mathrm{\Lambda }}$ is chaotic .

In the logistic map, we have a function${\displaystyle f_r(x)=rx(1-x)}$, and we want to study what happens when we iterate the map many times. The map might fall into a fixed point, a fixed cycle, or chaos. When the map falls into a stable fixed cycle of length${\displaystyle n}$, we would find that the graph of${\displaystyle f_r^n}$ and the graph of${\displaystyle x\mapsto x}$ intersects at${\displaystyle n}$ points, and the slope of the graph of${\displaystyle f_r^n}$ is bounded in${\displaystyle (-1,+1)}$ at those intersections.

For example, when${\displaystyle r=3.0}$, we have a single intersection, with slope bounded in${\displaystyle (-1,+1)}$, indicating that it is a stable single fixed point.

As${\displaystyle r}$ increases to beyond${\displaystyle r=3.0}$, the intersection point splits to two, which is a period doubling. For example, when${\displaystyle r=3.4}$, there are three intersection points, with the middle one unstable, and the two others stable.

As${\displaystyle r}$ approaches${\displaystyle r=3.45}$, another period-doubling occurs in the same way. The period-doublings occur more and more frequently, until at a certain${\displaystyle r\approx 3.56994567}$, the period doublings become infinite, and the map becomes chaotic. This is theperiod-doubling route to chaos.

Relationship between${\displaystyle x_{n+2}}$ and${\displaystyle x_n}$ when${\displaystyle a=2.7}$. Before the period doubling bifurcation occurs. The orbit converges to the fixed point${\displaystyle x_{f2}}$.

Relationship between${\displaystyle x_{n+2}}$ and${\displaystyle x_n}$ when${\displaystyle a=3}$. The tangent slope at the fixed point${\displaystyle x_{f2}}$. is exactly 1, and a period doubling bifurcation occurs.

Relationship between${\displaystyle x_{n+2}}$ and${\displaystyle x_n}$ when${\displaystyle a=3.3}$. The fixed point${\displaystyle x_{f2}}$ becomes unstable, splitting into a periodic-2 stable cycle.

When${\displaystyle r=3.0}$, we have a single intersection, with slope exactly${\displaystyle +1}$, indicating that it is about to undergo a period-doubling.

When${\displaystyle r=3.4}$, there are three intersection points, with the middle one unstable, and the two others stable.

When${\displaystyle r=3.45}$, there are three intersection points, with the middle one unstable, and the two others having slope exactly${\displaystyle +1}$, indicating that it is about to undergo another period-doubling.

When${\displaystyle r\approx 3.56994567}$, there are infinitely many intersections, and we have arrived atchaos via the period-doubling route.

Looking at the images, one can notice that at the point of chaos${\displaystyle r^{\ast }=3.5699\cdots }$, the curve of${\displaystyle f_{r^{\ast }}^{\mathrm{\infty }}}$ looks like a fractal. Furthermore, as we repeat the period-doublings${\displaystyle f_{r^{\ast }}^1,f_{r^{\ast }}^2,f_{r^{\ast }}^4,f_{r^{\ast }}^8,f_{r^{\ast }}^{16},\dots }$, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees.