Editing Logistic map (section)

== Applications ==
===Coupled map system===
The degree of freedom or dimension of a one-variable logistic map as a system is one <!--[ 306 ]-->. On the other hand, in the real natural world, it is thought that there are many chaotic systems with many degrees of freedom, not only in time but also in space <!--[ 307 ]-->. Alternatively, the synchronization phenomenon of oscillators performing chaotic motion is also a research subject <!--[ 308 ]--><!--[ 309 ]-->. To investigate such things, there is a method of coupled maps that couples many difference equations (maps) <!--[ 310 ]--><!--[ 309 ]-->. The logistic map is often used as a subject of coupled map model research <!--[ 311 ]-->. The reason for this is that the logistic map itself has already been well investigated as a typical model of chaos, and there is an accumulation of research on it <!--[ 312 ]-->.

There are various methods for the specific coupling in the coupled map model <!--[ 313 ]-->. Suppose a total of N maps are coupled, and the state of the i-th map at time n is represented by <math>x_n( i )</math> . In a method called globally coupled maps, <math>x_{n + 1}( i )</math> is formulated as follows <!--[ 314 ]-->:

{{NumBlk|:|<math>{\displaystyle x_{n+1}(i)=(1-\epsilon )f(x_{n}(i))+{\frac {\epsilon }{N}}\sum _{j=1}^{N}f(x_{n}(j))}</math>|{{EquationRef|6-1}}}}

In the current field of coupled oscillators, the simplest model is the following, in which two oscillators, x and y, are coupled by a difference in variables : <!--[ 315 ]-->

{{NumBlk|:|<math>{\displaystyle {\begin{cases}x_{n+1}=f(x_{n})+D(f(y_{n})-f(x_{n}))\\y_{n+1}=f(y_{n})+D(f(x_{n})-f(y_{n}))\end{cases}}}</math>|{{EquationRef|6-2}}}}

In these equations, f( x ) is the specific map to incorporate into the coupled map model, and applies here if the logistic map is used <!--[ 316 ]-->.

[[File:Synchronization and intermittency of coupled logistic maps.png|class=skin-invert-image|thumb|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.]]
In equations ( 6-1 ) and ( 6-2 ), ε and D are parameters called coupling coefficients, which indicate the strength of the coupling between the maps <!--[ 317 ]-->. On the other hand, when the logistic map is incorporated into a coupled map model, the parameter a of the logistic map indicates the strength of the nonlinearity of the model <!--[ 318 ]-->. By changing the value of a and the value of ε or D, various phenomena appear in the coupled map system of logistic maps. For example, in model ( 6-2 ), when D is increased to a value Dc or more, x and y oscillate chaotically while synchronously <!--[ 319 ]-->. Even below Dc, not only do chaotic oscillations occur in a continuous manner <!--[ 320 ]-->. When D is in a certain range, x and y oscillate with two periods even though r = 4 <!--[ 320 ]-->. When a = 3.8, behavior in which synchronous and asynchronous states alternate continuously is also observed <!--[ 321 ]-->.

In a study of the application of the logistic map to a globally coupled map with a large degree of freedom ( 6-1 ), a phenomenon called chaotic itinerancy was found <!--[ 322 ]-->. This is a phenomenon in which the orbit traverses a region in phase space that is said to be the remains of an attractor, repeating the cycle from an orderly state in which several clusters oscillate together to a disordered state, then to another cluster state, then back to the disordered state again, and so on . <!--[ 323 ]-->

===Pseudorandom number generator===
In the fields of computer simulation and information security, the creation of pseudorandom numbers using a computer is an important technique, and one of the methods for generating pseudorandom numbers is the use of chaos. <!--[ 324 ]--> Although a pseudorandom number generator based on chaos with sufficient performance has not yet been realized, several methods have been proposed. <!--[ 324 ]--> Several researchers have also investigated the possibility of creating a pseudorandom number generator based on chaos for the logistic map. <!--[ 234 ]--><!--[ 325 ]--><!--[ 326 ]-->

Parameter r = 4 is often used for pseudorandom number generation using the logistic map.<!--[ 327 ]--><!--[ 328 ]--><!--[ 329 ]--> 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.<!--[ 330 ]--> However, the distribution of points for the logistic map <math>f_{r = 4}</math> is as shown in equation ( 3-17 ), and the numbers that are generated are biased toward 0 and 1.<!--[ 234 ]--> Therefore, some processing is required to obtain unbiased uniform random numbers.<!--[ 234 ]--> Methods for doing so include:

A method for converting the obtained values to a uniform distribution using the tent map ( 4-8 ). <!--[ 327 ]-->
The resulting number is converted to either 0 or 1 using a threshold, as in the coin tossing analogy above, and this process is repeated to obtain a uniformly random bit stringm <!--[ 329 ]-->
In addition, the sequences <math>x_n</math> and <math>x_n +1</math> obtained by the logistic map are strongly correlated, which makes it problematic for pseudorandom sequences. <!--[ 234 ]--> One way to solve this is to generate the sequence <math>x_0, x_1, x_2, ... </math> for each iteration of the map, rather than generating the sequence <math>x_0, x_{\tau}, x_{2 \tau} ,...</math> for some number of iterations τ > 1  <!--[ 234 ]--> For example, it is said that good pseudorandom numbers can be obtained for method 1 with τ > 10 or τ > 13, <!--[ 234 ]--> and for method 2 with τ > 16. <!--[ 329 ]-->

A common problem with digitally calculating chaos using a computer is that, because a computer has a finite calculation precision, it is in principle impossible to obtain a truly aperiodic sequence, which is the nature of chaos, and instead outputs a finite periodic sequence. <!--[ 325 ]--> Even if aperiodic sequences cannot be obtained in principle, sequences with as long a period as possible are desirable for generating pseudorandom numbers. <!--[ 325 ]--> However, when the periodicity of the sequence actually output by the logistic map <math>f_{r=4}</math> in single-precision floating-point calculations was investigated, it was reported that the period of the sequence actually output is much smaller than the maximum period possible from the number of bits allocated, and from this point of view, it has been pointed out that pseudorandom number generation by the logistic map is inferior to existing pseudorandom number generators such as the Mersenne Twister. <!--[ 325 ]--> In addition, with the logistic map, <math>f_{r=4}</math> there is a risk that the value will fall to the fixed point 0 during the calculation and remain constant. <!--[ 331 ]--> On the other hand, the logistic map always takes values in the open interval (0, 1), so it can be calculated without problems not only with floating point but also with fixed point, and can enjoy the advantages of fixed point arithmetic. <!--[ 331 ]--> It has been pointed out that fixed point has a longer period than floating point for the same number of bits, and that unintended convergence to 0 can be eliminated. <!--[ 331 ]-->

=== Extension to complex numbers ===
[[File:Verhulst-Mandelbrot-Bifurcation.jpg|thumb|Correspondence between the orbit diagram of a variation of the logistic map (top) and the Mandelbrot set (bottom)]]

Dynamical systems defined by complex analytic functions are also of interest.<!--[ 332 ]-->LP An example is the dynamical system defined by the quadratic function: <!--[ 333 ]-->

{{NumBlk|:|<math>{\displaystyle z_{n+1}=z_{n}^{2}+c}</math>|{{EquationRef|6-3}}}}

where the parameter c and the variable z are complex numbers. <!--[ 333 ]--> This map is essentially the same as the logistic map (1–2). <!--[ 334 ]--> As mentioned above, the map (6–3) is topologically conjugate to the logistic map (1–2) through a linear function. <!--[ 335 ]-->

When the iteration of the map (6–3) is calculated with a fixed parameter c and varying the initial value <math>z_0</math>, a set of <math>z_0</math> such that <math>z_n</math> does not diverge to infinity as n → ∞ is called a filled Julia set.<!--[ 336 ]--> Furthermore, the boundary of a filled Julia set is called a Julia set. <!--[ 336 ]--> When the iteration of the map (6–3) is calculated with a fixed initial value <math>z_0 = 0</math> and varying the parameter {{mvar|c}}, a set of {{mvar|c}} such that {{mvar|z}} does not diverge to infinity is called a Mandelbrot set. <!--[ 337 ]--> The Julia sets and Mandelbrot sets of the map (6–3) generate fractal figures that are described as "mystical looking" and "extremely mysterious".{{attribution needed|date=May 2025}} <!--[ 338 ]--> 

In particular, in the Mandelbrot set, each disk in the diagram corresponds to a region of asymptotically stable periodic orbits of a certain period. <!--[ 339 ]--> By juxtaposing the logistic map orbit diagram with the Mandelbrot set diagram, it is possible to see that the asymptotically stable fixed points, period doubling bifurcations, and period-three windows of the logistic map orbit diagram correspond on the real axis to the Mandelbrot set diagram. <!--[ 340 ]-->

=== When there is a time delay ===
[[File:遅延ロジスティック写像.png|class=skin-invert-image|thumb|The trajectory of the delayed logistic map. The initial values <math>(x_0 , y_0)</math> 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).]]
If we interpret the logistic map as a model of the population of each generation of organisms, it is possible that the population of the next generation will affect not only the population of the current generation, but also the population of the generation before that. <!--[ 341 ]--> An example of such a case is

{{NumBlk|:|<math>{\displaystyle x_{n+1}=ax_{n}(1-x_{n-1})}</math>|{{EquationRef|6-4}}}}

where the number of individuals in the previous generation, <math>x_{n-1}</math>, is included in the equation as a negative density effect <!--[ 341 ]-->. If <math>x_{n+1} = y_n</math>, then equation ( 6-4 ) can be replaced by the following two-variable difference equation <!--[ 342 ]-->.

{{NumBlk|:|<math>{\displaystyle {\begin{cases}x_{n+1}=y_{n}\\y_{n+1}=ay_{n}(1-x_{n})\end{cases}}}</math>|{{EquationRef|6-5}}}}

This dynamical system is used to study bifurcation of quasi-periodic attractors and is called the delayed logistic map <!--[ 342 ]--> <!--[ 343 ]-->. The delayed logistic map exhibits a Neimark–Sakher bifurcation at r = 2, where the asymptotically stable fixed point becomes unstable and an asymptotically stable invariant curve forms around the unstable fixed point <!--[ 344 ]-->.