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.See .
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 , the sequence , , , ..., which can be interpreted as a sequence of time steps in the evolution of the system, is obtained.
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..
Another way to understand this sequence is to iterate the logistic map (here represented by ) to the initial state :.
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 ). 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 .
Characterization of the logistic map
The animation shows the behaviour of the sequence 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 (see Exponential function or Characterizations of the exponential function point 4) initial oscillation and then convergence (see Damping and Damped harmonic oscillator) stable oscillations between two values (see Resonance and Simple harmonic oscillator) growing oscillations between a set of values which are multiples of two such as 2,4,8,16 etc. (see Period-doubling bifurcation) Intermittency (i.e. sprouts of oscillations at the onset of chaos) fully developed chaotic oscillations topological mixing (i.e. the tendency of oscillations to cover the full available space).
The first four are also available in standard linear systems, oscillations between two values are available too under resonance, chaotic systems though have typically a large range of resonance conditions. The other phenomena are peculiar to chaos. This progression of stages is strikingly similar to the onset of turbulence. Chaos is not peculiar to non-linear systems alone and it can also be exhibited by infinite dimensional linear systems.
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 is the explanation of how the behavior of the logistic map changes as increases.
Domain, graphs and fixed points
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 the variables are restricted 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 is in the range [0, 1], then the maximum value of is r/4. Thus, when r > 4, the value of can exceed 1. On the other hand, when r is negative, x can take negative values.
A graph of the map can also be used to learn much about its behavior. The graph of the logistic map is the plane curve that plots the relationship between and , with (or x) on the horizontal axis and (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
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 ) at two points. The two intersection points are and , 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. 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. The fixed point will be denoted as . In the case of the logistic map, the fixed point that satisfies equation (2-2) is obtained by solving .
(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 on the horizontal axis, draw a vertical line from the initial value 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 . 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.
By varying the parameter , the following behavior is observed:
Case when 0 ≤ r < 1 First, when the parameter r = 0, , regardless of the initial value . 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, decreases monotonically for any value of between 0 and 1. That is, converges to 0 in the limit n → ∞. The point to which converges is the fixed point 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 move away from as time n increases, the fixed point is called unstable or repulsive.
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 , is asymptotically stable if the following condition is satisfied.
This can be seen by graphing the map: if the slope of the tangent to the curve at is between −1 and 1, then is stable and the orbit around it is attracted to . The derivative of the logistic map is
Therefore, for x = 0 and 0 < r < 1, 0 < f '(0) < 1, so the fixed point = 0 satisfies equation (3-1).
However, the discrimination method using equation (3-1) does not know the range of orbits from that are attracted to . It only guarantees that x within a certain neighborhood of 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 is unstable if
If the parameter lies in the range 0 < r < 1, then the other fixed point 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.
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 , which exists in the negative range for , is . For , that is, as r increases, the value of approaches 0, and just at r = 1 , collides with . This collision gives rise to a phenomenon known as a transcritical bifurcation.