Logistic map

[2] 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.

Many chaotic systems such as the Mandelbrot set are emerging from iteration of very simple quadratic non linear functions such as the logistic map.

[May, Robert M. (1976) 2] This nonlinear difference equation is intended to capture two effects: The usual values of interest for the parameter r are those in the interval [0, 4], so that xn remains bounded on [0, 1].

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.

, 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.

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.

For a logistic map with a specific parameter a, an exact solution that explicitly includes the time n and the initial value x 0 has been obtained as follows.

When r = 4 When r = 2 When r = −2 Considering the three exact solutions above, all of them are 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][19] These are properties of the logistic map for most values of r 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.

Figure (b), right, demonstrates this, showing how initially nearby points begin to diverge, particularly in those regions of xt corresponding to the steeper sections of the plot.

This quality of unpredictability and apparent randomness led the logistic map equation to be used as a pseudo-random number generator in early computers.

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... .

By universality, we can use another family of functions that also undergoes repeated period-doubling on its route to chaos, and even though it is not exactly the logistic map, it would still yield the same Feigenbaum constants.

The logistic map originated from the research of British mathematical biologist Robert May and became widely known as a formula for considering changes in populations of organisms.

Since there is a limit to the number of individuals that an environment can support, it seems natural that the growth rate α decreases as the population N n increases .

Alternatively, we can assume a maximum population size K that the environment can support, and use this to The logistic map can be derived by considering a difference equation that incorporates density effects in the form

However, unlike the laws of physics, the logistic map as a model of biological population size is not derived from direct experimental results or universally valid principles .

May, who made the logistic map famous, did not claim that the model he was discussing accurately represented the increase and decrease in population size .

Generally speaking, mathematical models can provide important qualitative information about population dynamics, but their results should not be taken too seriously without experimental support .

Historically, as described below, in 1947, shortly after the birth of electronic computers, Stanisław Ulam and John von Neumann also pointed out the possibility of a pseudorandom number generator using the logistic map with r = 4 .

In 1947, mathematicians Stanislaw Ulam and John von Neumann wrote a short paper entitled "On combination of stochastic and deterministic processes" in which they They pointed out that pseudorandom numbers can be generated by the repeated composition of quadratic functions such as .

At that time, the word "chaos" had not yet been used, but Ulam and von Neumann were already paying attention to the generation of complex sequences using nonlinear functions .

Between 1958 and 1963, Finnish mathematician Pekka Mylberg developed the This line of research is essential for dynamical systems, and Mühlberg has also investigated the period-doubling branching cascades of this map, showing the existence of an accumulation point λ = 1.401155189...[352 ] .

Others, such as the work of the Soviet Oleksandr Sharkovsky in 1964, the French Igor Gumowski and Christian Mila in 1969, and Nicholas Metropolis in 1973, have revealed anomalous behavior of simple one-variable difference equations such as the logistic map.

Numerical experiments were performed on the logistic map to investigate the change in its behavior depending on the parameter r. In 1976, he published a paper in Nature entitled "Simple mathematical models with very complicated dynamics ".

This paper in particular caused a great stir and was accepted by the scientific community due to May's status as a mathematical biologist, the clarity of his research results, and above all, the shocking content that a simple parabolic equation can produce surprisingly complex behavior.

Li and York completed the paper in 1973, but when they submitted it to The American Mathematical Monthly, they were told that it was too technical and that it should be significantly rewritten to make it easier to understand, and it was rejected .

However, the following year, in 1974, May came to give a special guest lecture at the University of Maryland where Lee and York were working, and talked about the logistic map .

Mathematician Robert Devaney states the following before explaining the logistic map in his book : This means that by simply iterating the quadratic function

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 sequence behaviour from r=0.02 to r=4, one can visualize the horizontal coordinate as time, and the vertical coordinate either as a position in space at time t or as the population size at time t
Graph of the logistic map (the relationship between and ). The graph has the shape of a parabola, and the vertex of the parabola changes as the parameter r changes.
An example of a spider web projection of a trajectory on the graph of the logistic map, and the locations of the fixed points and on the graph.
Spider plot (left) and time series (n vs. x n) (right) for parameter r = 0.9. The trajectory converges monotonically to 0.
Tangent slopes of an asymptotically stable fixed point (left) and an unstable fixed point (right) and the state of the surrounding orbits
Transcritical bifurcation of the logistic map occurring at r = 1. For r < 1, exists outside [0, 1] as an unstable fixed point, but for r = 1, the two fixed points collide, and for r > 1, appears between [0, 1] as a stable fixed point.
Animation of the spider projection at a = 2.8, converging around a fixed point.
Bifurcation diagram of the logistic map for parameters 0 to 3. The blue line represents the fixed point , and the red line represents the fixed point Represents.
Bifurcation diagram of the period-doubling bifurcation cascade occurring between parameters and . After 64 periods ( ), the spacing becomes very narrow and almost collapses.
An example of the construction of a Cantor set: if you keep removing the central third of a line segment infinitely, you will end up with a shape that appears to have zero length but has an uncountably infinite number of points, each of which has an infinitesimal neighborhood of other points.
Evolution of different initial conditions as a function of r (The parameter k from the figure corresponds to the parameter r from the definition in the article.)
Evolution of different initial conditions as a function of r with bias (The parameter k from the figure corresponds to the parameter r from the definition in the article.)
Bifurcation diagram for the logistic map. The attractor for any value of the parameter r is shown on the vertical line at that r .
Transient chaos at a = 3.8285. The system behaves chaotically until it is attracted into a periodic 3 orbit.
Channel patterns appearing in the graph of f3 (x)
Self-similar hierarchical structure of the entire trajectory map of the logistic map
Self-similar hierarchical structure of windows of the logistic map
Graph of the invariant measure ρ(x) for r = 4. The dot plot shows the actual frequency of points obtained over 10,000 iterations (with height scaled to ρ (x)).
If we convert the orbit of the logistic map into a string of 0s and 1s, we can reproduce any string of symbols.
For the logistic map with r = 4.5, trajectories starting from almost any point in [0, 1] go towards minus infinity.
A cobweb diagram of the logistic map, showing chaotic behaviour for most values of r > 3.57
Logistic function f (blue) and its iterated versions f 2 , f 3 , f 4 and f 5 for r = 3.5 . For example, for any initial value on the horizontal axis, f 4 gives the value of the iterate four iterations later.
Two- and three-dimensional Poincaré plots show the stretching-and-folding structure of the logistic map
Two- and three-dimensional Poincaré plots show the stretching-and-folding structure of the logistic map
Logistic map with Lyapunov exponent function
Graph of the sine map ( 4-1 )
Orbit diagram of the sine map ( 4-1 )
Orbital view of the tent map ( 4-8 ). It has a topological conjugate relationship with the a = 4 logistic map.
Approach to the scaling limit as approaches from below.
At the point of chaos , as we repeat the period-doublings , the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees, converging to a fractal.
For the wrong values of scaling factor , the map does not converge to a limit, but when , it converges.
At the point of chaos , as we repeat the functional equation iteration with , we find that the map does converge to a limit.
In the chaotic regime, , the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.
Logistic map approaching the period-doubling chaos scaling limit from below. At the limit, this has the same shape as that of , since all period-doubling routes to chaos are the same (universality).
An example of a solution to the logistic equation. After time t, the population size N converges to the carrying capacity K regardless of the initial value.
Changes in two variables (top) and their difference (bottom) in the coupled map model ( 6-2 ) with a = 3.8 and D = 0.43 . The two variables suddenly become out of sync after synchronization, and then return to the sync state.
Correspondence between the orbit diagram of the logistic map (top) and the Mandelbrot set (bottom)
The trajectory of the delayed logistic map. The initial values are the same in both figures, but at the bifurcation point r = 2, the trajectory is attracted to a closed curve (left) and a point (right).
Stanislaw Ulam
John von Neumann
Robert May (photographed in 2009)
Mitchell Feigenbaum (photographed in 2006)