Logistic map

Case when 0 ≤ r < 1

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 ${\displaystyle f'(x)}$, ${\displaystyle x_{f}}$ is asymptotically stable if the following condition is satisfied.

${\displaystyle \left|f'(x_{f})\right|<1}$

3-1

This can be seen 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

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

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

${\displaystyle \left|f'(x_{f})\right|>1}$

3-3

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

Case when 1 ≤ r ≤ 2

In the general case with r between 1 and 2, the population will quickly approach the value ⁠r − 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 other 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 ${\displaystyle 0<r<1}$, 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 a transcritical 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 ${\displaystyle 1<r\leq 2}$, 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. ${\displaystyle 0<x_{0}<1-1/r}$

In the case of ${\displaystyle 1-1/r<x_{0}<1/r}$ Then, it converges monotonically, ${\displaystyle 1/r<x_{0}<1}$, 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.

Case when 2 ≤ r ≤ 3

With r between 2 and 3, the population will also eventually approach the same value ⁠r − 1/r⁠, but first will fluctuate around that value for some time. The rate of convergence is linear, except for r = 3, when it is dramatically slow, less than linear (see Bifurcation 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, there is a straight line representing the fixed point ${\displaystyle x_{f1}=0}$  and a straight line representing the fixed point ${\displaystyle x_{f2}=1-1/r}$ It can be seen that the curves representing a and b intersect at r = 1, and that stability is switched between the two.

Case when 3 ≤ r ≤ 3.44949

In the general case With r between 3 and 1 + √6 ≈ 3.44949 the population will approach permanent oscillations between two values. These two values are dependent on r 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 ${\displaystyle 2<r<3}$. When ${\displaystyle r=3}$, the derivative ${\displaystyle f'(x_{f2})}$ reaches −1 and no longer satisfies equation (3-1). When r exceeds 3, ${\displaystyle f'(x_{f2})<-1}$, 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 a period 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 the logistic map equation (1-2) is applied to this equation, the result is

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

3-5

This results in the following fourth-order equation. The solutions of this equation are the periodic points. In fact, the two fixed points ${\displaystyle x_{f1}=0}$ and ${\displaystyle x_{f2}=1-1/r}$ also satisfy 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), they can be obtained 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 ${\displaystyle (f^{k})'(x)}$ be the derivative  ${\displaystyle df^{k}(x)/dx}$ of the k-periodic point ${\displaystyle x_{f}^{(k)}}$. If ${\displaystyle x_{f}^{(k)}}$ satisfies:

${\displaystyle \left|(f^{k})'(x_{f}^{(k)})\right|<1}$

3-7

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

${\displaystyle \left|(f^{k})'(x_{f}^{(k)})\right|>1}$

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. On a graph of the logistic map ${\displaystyle f^{2}(x)}$, 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 the differential coefficients of two periodic points for the logistic map are calculated, the result is

${\displaystyle (f^{2})'(x_{f}^{(2)})=4+2r-r^{2}}$

3-9

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

${\displaystyle \left|4+2r-r^{2}\right|<1}$

3-10

It can be seen that the 2-periodic points are asymptotically stable when this range is ${\displaystyle 3<r<1+{\sqrt {6}}}$, i.e., when r exceeds ${\displaystyle 1+{\sqrt {6}}=3.44949...}$, 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].

Case when 3.44949 ≤ r ≤ 3.56995

With r 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 (sequence A086181 in the OEIS).

With r 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 the Feigenbaum constant δ ≈ 4.66920. This behavior is an example of a period-doubling cascade.

When the parameter r exceeds ${\displaystyle 1+{\sqrt {6}}=3.44949...}$, 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 ${\displaystyle r=3.44949...}$. 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 the four solutions for the fixed points and the 2-periodic points are factored out, 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 → ∞. (sequence A098587 in the OEIS)

${\displaystyle r_{\infty }=\lim _{k\rightarrow \infty }r_{k}=3.56994...}$

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.

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

3-13

This value of δ is called the Feigenbaum constant because it was discovered by mathematical physicist Mitchell Feigenbaum. ${\displaystyle r_{\infty }}$ 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 ${\displaystyle r=r_{\infty }}$ 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 ${\displaystyle r_{\infty }}$. This aperiodic orbit is called the Feigenbaum attractor. The critical ${\displaystyle 2^{\infty }}$ 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 the Cantor 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.

Case when 3.56995 < r < 4

Qualitative Summary

• At r ≈ 3.56995 (sequence A098587 in the OEIS) is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions, there are no longer 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 the Reynolds number for the onset of other chaotic phenomena such as turbulence and similar to the critical temperature of a phase transition. In essence the phase space contains a full subspace of cases with extra dynamical variables to characterize the microscopic state of the system, these can be understood as Eddies in the case of turbulence and order parameters in the case of phase transitions.
• Most values of r beyond 3.56995 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 + √8^[8] (approximately 3.82843) there is a range of parameters r that show oscillation among three values, and for slightly higher values of r oscillation among 6 values, then 12 etc.
• At ${\displaystyle r=1+{\sqrt {8}}=3.8284...}$, the stable period-3 cycle emerges.^[9]
• The development of the chaotic behavior of the logistic sequence as the parameter r varies from approximately 3.56995 to approximately 3.82843 is sometimes called the Pomeau–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 of r. A period-doubling window with parameter c is a range of r-values consisting of a succession of subranges. The kth subrange contains the values of r for which there is a stable cycle (a cycle that attracts a set of initial points of unit measure) of period 2^kc. This sequence of sub-ranges is called a cascade of harmonics.^{[May, Robert M. (1976) 1]} In a sub-range with a stable cycle of period 2^k*c, there are unstable cycles of period 2^kc 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.^{[May, Robert M. (1976) 1][11]}
• At ${\displaystyle r=3.678...,x=0.728...}$, two chaotic bands of the bifurcation diagram intersect in the first Misiurewicz point for the logistic map. It satisfies the equations ${\displaystyle r^{3}-2r^{2}-4r-8=0,x=1-1/r}$.^[12]
• Beyond r = 4, almost all initial values eventually leave the interval [0,1] and diverge. The set of initial conditions which remain within [0,1] form a Cantor set and the dynamics restricted to this Cantor set is chaotic.^[13]

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

The bifurcation diagram at rightabove summarizes this. The horizontal axis shows the possible values of the parameter r while the vertical axis shows the set of values of x visited asymptotically from almost all initial conditions by the iterates of the logistic equation with that r value.

The bifurcation diagram is a self-similar: if one zooms in on the above-mentioned value r ≈ 3.82843 and focuses 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 between chaos and fractals.

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 the Pomeau–Manneville scenario

Negative values of r can also be considered:

• For r between -2 and -1 the logistic sequence also features chaotic behavior.^[6]
• With r between -1 and 1 - √6 and for x₀ between 1/r and 1-1/r, the population will approach permanent oscillations between two values, as with the case of r between 3 and 1 + √6, and given by the same formula.^[6]

The emergence of chaos

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 ${\displaystyle \Delta x_{n}=|x_{n}-{\hat {x}}_{n}|}$, shown on a logarithmic scale.

When the parameter r exceeds ${\displaystyle r_{\infty }=3.56994...}$, 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 ${\displaystyle r>r_{\infty }}$ of the logistic map is called the chaotic region.

One of the properties of chaos is its unpredictability, symbolized by the term butterfly 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 there are 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 the Lyapunov exponent. For a one-dimensional map, the Lyapunov exponent λ can be calculated as follows:

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

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 ${\displaystyle r<r_{\infty }}$, and that λ can take positive values in the range ${\displaystyle r>r_{\infty }}$.

Window, intermittent

Even beyond ${\displaystyle r_{\infty }}$, the behavior does not depend simply on the parameter r. Many sophisticated mathematical structures lurk in the chaotic region for ${\displaystyle r>r_{\infty }}$. In this region, chaos does not persist forever; stable periodic orbits reappear. The behavior for ${\displaystyle r_{\infty }<a\leq 4}$ 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 ${\displaystyle r_{\infty }<r\leq 4}$ 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 of period-doubling bifurcations that occurred before ${\displaystyle r_{\infty }=3.56994...}$ 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 only cases in which p is a prime number are considered, 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 ${\displaystyle [r_{\infty },4]}$ 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 ${\displaystyle r_{3}=1+{\sqrt {8}}=3.828427...}$. 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.

The behavior of ${\displaystyle x_{n}}$ when r = 3.8282, which is slightly smaller than the branch point ${\displaystyle r_{3}}$, shows, in addition to the irregular changes, 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, the behavior at r = 3.828327, which is closer to ${\displaystyle r_{3}}$, shows that the average length of the laminars is longer and the average length of the bursts is shorter than when r = 3.8282. If r is further increased, 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 the parameter is considered to be 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

${\displaystyle {\displaystyle {\frac {r^{2}(4-r)}{16}}\leq x_{n}\leq {\frac {r}{4}}}}$

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 ${\displaystyle r_{\infty }=3.56994}$ 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 ${\displaystyle e_{\infty }}$, the number of bands becomes infinite, and the value of ${\displaystyle e_{\infty }}$ is equal to the value of ${\displaystyle r_{\infty }}$.

Similarly, let ${\displaystyle a_{p}}$, for ${\displaystyle p=1,2,4,\ldots }$, denote the period-doubling bifurcation points before ${\displaystyle a_{\infty }}$. At each of these points, the cycle of period ${\displaystyle p}$ loses stability and a stable cycle of period ${\displaystyle 2p}$ emerges.^[15] In this case, the orbital diagram from ${\displaystyle r_{2}}$ to ${\displaystyle e_{2}}$ shows that 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, the orbital diagram from ${\displaystyle r_{4}}$ to ${\displaystyle e_{4}}$ shows that 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 ${\displaystyle a_{3\infty }}$ ≈ 3.8495. After ${\displaystyle a_{3\infty }}$ ≈ 3.8495, the behavior becomes band chaos of multiples of three. As a increases from ${\displaystyle a_{3\infty }}$, 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.

When r = 4

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 r > 4

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 ${\displaystyle \Lambda }$ 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.

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

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 ${\displaystyle \Lambda }$, the logistic map ${\displaystyle f_{r>4}}$ is chaotic.

When r < 0

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.

Exact solutions for special cases

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

Upper bound when 0 ≤ r ≤ 1

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 when 0 ≤ r ≤ 1.^[19] 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 constant r, and the fast initial decay when x₀ is close to 1, driven by the (1 − x_n) term in the recurrence relation. The following bound captures both of these effects:

$${\displaystyle \forall n\in \{0,1,\ldots \}\quad {\text{and}}\quad x_{0},r\in [0,1],\quad x_{n}\leq {\frac {x_{0}}{r^{-n}+x_{0}n}}.}$$

Solution when r = 4

The special case of r = 4 can in fact be solved exactly, as can the case with r = 2;^[20] however, the general case can only be predicted statistically.^[21] The solution when r = 4 is:^[20][22]

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

where the initial condition parameter θ is given by

$${\displaystyle \theta ={\tfrac {1}{\pi }}\sin ^{-1}\left({\sqrt {x_{0}}}\right).}$$

For rational θ, after a finite number of iterations x_n maps into a periodic sequence. But almost all θ are irrational, and, for irrational θ, x_n never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor 2ⁿ shows the exponential growth of stretching, which results in sensitive dependence on initial conditions, while the squared sine function keeps x_n folded within the range [0,1].

For r = 4 an equivalent solution in terms of complex numbers instead of trigonometric functions is^[20]

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

where α is either of the complex numbers

$${\displaystyle \alpha =1-2x_{0}\pm {\sqrt {\left(1-2x_{0}\right)^{2}-1}}}$$

with modulus 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 of α.

By contrast, the solution when r = 2 is^[20]

$${\displaystyle x_{n}={\tfrac {1}{2}}-{\tfrac {1}{2}}\left(1-2x_{0}\right)^{2^{n}}}$$

for x₀ ∈ [0,1). Since (1 − 2x₀) ∈ (−1,1) for any value of x₀ other than the unstable fixed point 0, the term (1 − 2x₀)²ⁿ goes to 0 as n goes to infinity, so x_n goes to the stable fixed point ⁠1/2⁠.

Finding cycles of any length when r = 4

For the r = 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 length k for all integers k > 0. The relationship of the logistic map can be exploited to the dyadic transformation (also known as the bit-shift map) to find cycles of any length. If x follows the logistic map x_{n + 1} = 4x_n(1 − x_n) and y follows the dyadic transformation

$${\displaystyle y_{n+1}={\begin{cases}2y_{n}&0\leq y_{n}<{\tfrac {1}{2}}\\2y_{n}-1&{\tfrac {1}{2}}\leq y_{n}<1,\end{cases}}}$$

then the two are related by a homeomorphism

$${\displaystyle x_{n}=\sin ^{2}\left(2\pi y_{n}\right).}$$

The reason that the dyadic transformation is also called the bit-shift map is that when y 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 as ⁠1/7⁠ → ⁠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.... The other bit-shift 3-cycle could be similarly translated into its corresponding logistic cycle. Likewise, cycles of any length k 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 logistic r = 4 map is chaotic for almost all initial conditions.

The number of cycles of (minimal) length k = 1, 2, 3,… for the logistic map with r = 4 (tent map with μ = 2) is a known integer sequence (sequence A001037 in the OEIS): 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161.... This means that the logistic map with r = 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 prime k: 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.