Logistic map - WikiHQ

class=skin-invert-image|thumb|The behavior of the logistic map is shown in [[Cobweb plot form. The animation shows the change in behavior as the parameter (r in the figure) is increased from 1 to 4, starting from an initial value of 0.2.)]]

The logistic map is a discrete dynamical system defined by the quadratic difference equation

It is a recurrence relation and a polynomial mapping of degree 2. It is often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple nonlinear dynamical equations.

The map was initially utilized by Edward Lorenz in the 1960s to showcase properties of irregular solutions in climate systems. It was popularized in a 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation written down by Pierre François Verhulst.

Other researchers who have contributed to the study of the logistic map include Stanisław Ulam, John von Neumann, Pekka Myrberg, Oleksandr Sharkovsky, Nicholas Metropolis, and Mitchell Feigenbaum.

Two introductory examples

Dynamical systems example

In the logistic map, x is a variable, and r is a parameter. It is a map in the sense that it maps a configuration or phase space to itself (in this simple case the space is one dimensional in the variable x):

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 the logistic differential equation, which can be compared to a time evolution equation of the system.

Given an appropriate value for the parameter r and performing calculations starting from an initial condition <math>x_0</math>, we obtain the sequence <math>x_0</math>, <math>x_1</math>, <math>x_2</math>, ..., which can be interpreted as a sequence of time steps in the evolution of the system.

In the field of dynamical systems, this sequence is called an orbit, 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 showing non-periodic fluctuations known as chaos.

\begin{aligned}

x_1 &= f(x_0), \\

x_2 &= f(x_1) = f(f(x_0)), \\

x_3 &= f(x_2) = f(f(f(x_0))), \\

x_4 &= \dots \\

\end{aligned}

</math>

This was the initial approach of Henri Poincaré to study dynamical systems and ultimately chaos starting from the study of fixed points or, in other words, states that do not change over time (i.e. when <math>x_n = ... = x_1 = x_0 = f(x_0)</math>). Many chaotic systems such as the Mandelbrot set emerge from iteration of very simple quadratic nonlinear functions such as the logistic map.

Demographic model example

Taking the biological population model as an example is a number between zero and one, which represents the ratio of existing population to the maximum possible population.

This nonlinear difference equation is intended to capture two effects:

reproduction, where the population will increase at a rate proportional to the current population when the population size is small,
starvation (density-dependent mortality), where the growth 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 parameter are those in the interval , so that remains bounded on . The case of the logistic map is a nonlinear transformation of both the bit-shift map and the case of the tent map. If , this leads to negative population sizes. (This problem does not appear in the older Ricker model, which also exhibits chaotic dynamics.) One can also consider values of in the interval , so that remains bounded on .

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 parameter changes. Depending on the value of , the behavior of the trajectory of the logistic map can be simple or complex. Below, we will explain how the behavior of the logistic map changes as increases.

Domain, graphs and fixed points

class=skin-invert-image|thumb|Graph of the logistic map (the relationship between <math>x_{n +1}</math> and <math>x_n</math>). The graph has the shape of a parabola, and the vertex of the parabola changes as the parameter r changes.

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].

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 <math>x_n</math> is in the range [0, 1], then the maximum value of <math>x_{n+1}</math> is

r/4. Thus, when r > 4, the value of <math>x_{n +1}</math> can exceed 1. On the other hand, when r is negative, x can take negative values.

\right)}</math>|

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 <math>x_{n+1} = 0</math>) at two points. The two intersection points are <math>(x_n, x_{n + 1}) = (0,0)</math> and <math>(x_n, x_{n + 1}) = (1,0)</math>, and the positions of these intersection points are constant and do not depend on the value of r.

class=skin-invert-image|thumb|An example of a spider web projection of a trajectory on the graph of the logistic map, and the locations of the fixed points <math>x_{f1}</math> and <math>x_{f2}</math> on the graph.

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

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

</math> |

(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 the spider web projection. After determining an initial value <math>x_0</math> on the horizontal axis, draw a vertical line from the initial value <math>x_0</math> 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 the spider web diagram created represents the trajectory starting from <math>x_0</math>. This projection allows the overall behavior of the trajectory to be seen at a glance.

Behavior dependent on

The image below shows the amplitude and frequency 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 the time domain (left) but especially in the frequency domain or spectrum (right), i.e. chaos is present at all scales as it is in the case of Energy cascade of Kolmogorov and it even propagates from one scale to another.

class=skin-invert-image

By varying the parameter , the following behavior is observed:

Case when 0 ≤ r < 1

First, when the parameter r = 0, <math>x_1 = 0</math>, regardless of the initial value <math>x_0</math>. 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, <math>x_n</math> decreases monotonically for any value of <math>x_0</math> between 0 and 1. That is, <math>x_n</math> converges to 0 in the limit n → ∞.

The point to which <math>x_n</math> converges is the fixed point <math>x_{f1}</math> 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 <math>x_f</math> move away from <math>x_f</math> as time n increases, the fixed point <math>x_f</math> is called unstable or repulsive.

class=skin-invert-image|thumb|284x284px| Spider plot (left) and time series (n vs. x n) (right) for parameter r = 0.9. The trajectory converges monotonically to 0.

A common and simple way to know whether a fixed point is asymptotically stable is to take the derivative of the map f. This derivative is expressed as <math>f'(x)</math>,

<math>x_f</math> is asymptotically stable if the following condition is satisfied.

class=skin-invert-image|thumb|284x284px|Tangent slopes of an asymptotically stable fixed point (left) and an unstable fixed point (right) and the state of the surrounding orbits

We can see this by graphing the map: if the slope of the tangent to the curve at <math>x_f</math> is between −1 and 1, then <math>x_f</math> is stable and the orbit around it is attracted to <math>x_f</math>. The derivative of the logistic map is

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

However, the discrimination method using equation (3-1) does not know the range of orbits from <math>x_f</math> that are attracted to <math>x_f</math>. It only guarantees that x within a certain neighborhood of <math>x_f</math> 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 <math>x_f</math> is unstable if

If the parameter lies in the range 0 < r < 1, then the other fixed point <math>x_{f2} = 1 - 1/a</math>

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 between 1 and 2, the population will quickly approach the value , independent of the initial population.

class=skin-invert-image|thumb|center|650px|[[Transcritical bifurcation of the logistic map occurring at r = 1. For r < 1, <math>x_{f2}</math> exists outside [0, 1] as an unstable fixed point, but for r = 1, the two fixed points collide, and for r > 1, <math>x_{f2}</math> appears between [0, 1] as a stable fixed point.]]

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 <math>x_{f2} = 1 - 1/r</math>, which exists in the negative range for <math>0 < r < 1</math>, is <math>x_{f2}=0</math>.

For <math>x_{f2} = 0</math>, that is, as r increases, the value of <math>x_{f2}</math> approaches 0, and just at r = 1 , <math>x_{f2}</math> collides with <math>x_{f1} = 0</math>. 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, <math>x_{f1}</math> is stable and <math>x_{f2}</math> is unstable, but when r is greater than 1, <math>x_{f1}</math> is unstable and <math>x_{f2}</math> is stable. The parameter values at which bifurcation occurs are called bifurcation points. In this case, r = 1 is the bifurcation point.

As a result of the bifurcation, the orbit of the logistic map converges to the limit point <math>x_{f2} = 1 - 1/r</math> instead of <math>x_{f1} = 0</math>. In particular, if the parameter <math>1 < r \le 2</math>, then the trajectory starting from a value <math>x_0</math>in the interval (0, 1), exclusive of 0 and 1, converges to <math>x_{f2}</math> by increasing or decreasing monotonically. The difference in the convergence pattern depends on the range of the initial value. <math>0 < x_0 < 1 - 1/r</math>

In the case of <math>1 - 1/r< x_0 < 1/r</math>

Then, it converges monotonically,

<math>1/r< x_0 < 1</math>, the function converges monotonically except for the first step.

Furthermore, the fixed point <math>x_{f1} = 0</math> 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 <math>x_{f1}</math> itself that can reach this unstable fixed point <math>x_{f1}</math>. This is <math>x_0 = 1</math>, and since the logistic map satisfies f (1) = 0 regardless of the value of r, applying the map once to <math>x_0 = 1</math> maps it to <math>x_{f1} = 0</math>. 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 between 2 and 3, the population will also eventually approach the same value , but first will fluctuate around that value for some time. The rate of convergence is linear, except for , 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 <math>x_{f2} = 1 - 1/r</math> is the same as when 1 < r ≤ 2.

However, in this case the convergence is not monotonic. As the variable approaches <math>x_{f2}</math>, it becomes larger and smaller than <math>x_{f2}</math> repeatedly, and follows a convergent trajectory that oscillates around <math>x_{f2}</math>.

<!-- TODO: obscure translation to be fixed, maybe using a ref

Tentative:

In this parameter range, <math>1/2<x_{f2}<1</math>,

The oscillations around the fixed point of the orbit are bounded and given it is an attractor, they are convergent. (approximately 3.82843) there is a range of parameters that show oscillation among three values, and for slightly higher values of oscillation among 6 values, then 12 etc.

At <math>r = 1 + \sqrt 8 = 3.8284...</math>, the stable period-3 cycle emerges.
The development of the chaotic behavior of the logistic sequence as the parameter 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. There are other ranges that yield oscillation among 5 values etc.; all oscillation periods occur for some values of . A period-doubling window with parameter is a range of -values consisting of a succession of subranges. The th subrange contains the values of for which there is a stable cycle (a cycle that attracts a set of initial points of unit measure) of period . This sequence of sub-ranges is called a cascade of harmonics.
At <math>r = 3.678..., x = 0.728...</math>, two chaotic bands of the bifurcation diagram intersect in the first Misiurewicz point for the logistic map. It satisfies the equations <math>r^3 - 2r^2 - 4r -8 = 0, x = 1-1/r</math>.
Beyond , almost all initial values eventually leave the interval and diverge. The set of initial conditions which remain within form a Cantor set and the dynamics restricted to this Cantor set is chaotic.

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

class=skin-invert-image|thumb|right|[[Bifurcation diagram for the logistic map.

The attractor for any value of the parameter is shown on the vertical line at that .]]

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

The bifurcation diagram is a self-similar: if we zoom in on the above-mentioned value 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 between chaos and fractals.

We can also consider negative values of :

For between -2 and -1 the logistic sequence also features chaotic behavior.
With between -1 and 1 -  and for <sub>0</sub> between 1/ and 1-1/, the population will approach permanent oscillations between two values, as with the case of between 3 and 1 + , and given by the same formula. (see Chaotic 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.
The system exhibits dense periodic orbits

These are properties of the logistic map for most values of between about 3.57 and 4 (as noted above).

The following figure illustrates the stretching and folding over a sequence of iterates of the map. Figure (a), left, shows a two-dimensional Poincaré plot of the logistic map's state space for , and clearly shows the quadratic curve of the difference equation (). However, we can embed 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 of corresponding to the steeper sections of the plot.

class=skin-invert-image|center|upright=1.8|Two- and three-dimensional [[Poincaré plots show the stretching-and-folding structure of the logistic map]]

This stretching-and-folding does not just produce a gradual divergence of the sequences of iterates, but an exponential divergence (see Lyapunov exponents), evidenced also by the complexity and unpredictability 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 a pseudo-random number generator in early computers.

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 a correlation dimension of (Grassberger, 1983), a Hausdorff dimension of about 0.538 (Grassberger 1981), and an information dimension of approximately 0.5170976 (Grassberger 1983) for (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 the likelihood of a future state in a chaotic system. If a (possibly chaotic) dynamical system has an attractor, then there exists a probability 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 parameter and an initial state in , the attractor is also the interval and the probability measure corresponds to the beta distribution with parameters and . Specifically, the invariant measure is

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 inform decisions based on the state of the system.

class=skin-invert-image|thumb|Logistic map with [[Lyapunov exponent function]]

Graphical representation

The bifurcation diagram for the logistic map can be visualized with the following Python code:

import numpy as np

import matplotlib.pyplot as plt

interval = (2.8, 4) # start, end

accuracy = 0.0001

reps = 600 # number of repetitions

numtoplot = 200

lims = np.zeros(reps)

fig, biax = plt.subplots()

fig.set_size_inches(16, 9)

lims[0] = np.random.rand()

for r in np.arange(interval[0], interval[1], accuracy):

for i in range(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()

</syntaxhighlight>

Special cases of the map

Upper bound when

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 . 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 , and the fast initial decay when is close to 1, driven by the term in the recurrence relation. The following bound captures both of these effects:

<math display="block"> \forall n \in \{0, 1, \ldots \} \quad \text{and} \quad x_0, r \in [0, 1], \quad x_n \le \frac{x_0}{r^{-n} + x_0n}. </math>

Solution when

The special case of can in fact be solved exactly, as can the case with ;

The solution when is:

<math display="block">x_{n}=\sin^{2}\left(2^{n} \theta \pi\right),</math>

where the initial condition parameter is given by

<math display="block">\theta = \tfrac{1}{\pi}\sin^{-1}\left(\sqrt{x_0}\right).</math>

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

For an equivalent solution in terms of complex numbers instead of trigonometric functions is And <math>f^\infty_r</math>converges to the fixed point to <math display="block">f(x) \mapsto - \alpha f(f( f(-x/\alpha ) ))

</math>As another example, period-4-pling has a pair of Feigenbaum constants distinct from that of period-doubling, even though period-4-pling is reached by two period-doublings. In detail, define <math>r_1, r_2, \dots</math> such that <math>r_n</math> is the lowest value in the period-<math>4^n</math> window of the bifurcation diagram. Then we have <math>r_1 =3.960102, r_2 = 3.9615554, \dots</math>, with the limit <math>r_\infty = 3.96155658717\dots</math>. This has a different pair of Feigenbaum constants <math>\delta= 981.6\dots, \alpha = 38.82\dots</math>.

In general, each period-multiplying route to chaos has its own pair of Feigenbaum constants. In fact, there are typically more than one. For example, for period-7-pling, there are at least 9 different pairs of Feigenbaum constants.

The gradual increase of <math>G</math> at interval <math>[0, \infty)</math> changes dynamics from regular to chaotic one with qualitatively the same bifurcation diagram as those for logistic map.

Renormalization estimate

The Feigenbaum constants can be estimated by a renormalization argument. (Section 10.7,

The logistic map as a model of biological populations

Discrete population model

While Lorenz used the logistic map in 1964, is laser gain as bifurcation parameter.

Hofstadter sequences are an example of one dimensional quasi-random, aperiodic, chaotic sequences again defined by recursion, a very special case is the logistic map

Notes

Citations

References

External links

The Chaos Hypertextbook. An introductory primer on chaos and fractals.
An interactive visualization of the logistic map as a Jupyter notebook
The Logistic Map and Chaos by Elmer G. Wiens
Complexity & Chaos (audiobook) by Roger White. Chapter 5 covers the Logistic Equation.
"History of iterated maps," in A New Kind of Science by Stephen Wolfram. Champaign, IL: Wolfram Media, p. 918, 2002.
"A very brief history of universality in period doubling" by P. Cvitanović
"A not so short history of Universal Function" by P. Cvitanović
Discrete Logistic Equation by Marek Bodnar after work by Phil Ramsden, Wolfram Demonstrations Project.
Multiplicative coupling of 2 logistic maps by C. Pellicer-Lostao and R. Lopez-Ruiz after work by Ed Pegg Jr, Wolfram Demonstrations Project.
Using SAGE to investigate the discrete logistic equation