Editing Logistic map (section)

=====The Emergence of Chaos=====
<div class=skin-invert-image>{{multiple image
| align             = center
| direction         = horizontal
| total_width       = 620
| image1            = パラメータ3.82のロジスティック写像における微差の初期値から出発する2つの軌道.png
| caption1          = Chaotic orbits of the logistic map when r = 3.82. The orange squares are orbits starting from <math>x_0 = 0.1234</math>, and the blue-green circles are orbits starting from <math>\hat{x}_0 = 0.1234 + 10^{-9}</math>.
| image2            = パラメータ3.82のロジスティック写像におけるズレの進展.png
| caption2          = The trajectory starting from x 0 = 0.1234 and ˆx The difference in orbits starting from <math>x_0 = 0.1234 + 10^{-9}</math> grows exponentially. The vertical axis is <math>\Delta x_n = | x_n - \hat{x}_n |</math>,shown on a [[logarithmic scale]].
}}</div>

When the parameter r exceeds <math>r_{\infty} = 3.56994...</math>, the logistic map exhibits chaotic behavior.<!--[ 145 ]--> 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.<!--[ 158 ]--> The range of <math>r > r_{\infty}</math> of the logistic map is called the chaotic region.<!--[ 159 ]-->

One of the properties of chaos is its unpredictability, symbolized by the term [[butterfly effect]].<!--[ 160 ]--> 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.<!--[ 160 ]--> In terms of a discrete dynamical system, if we have two initial values <math>x_0</math> and <math>\hat{x}_0</math>
No matter how close they are, once time n has progressed to a certain extent, each destination <math>x_n</math> and <math>\hat{x}_n</math> can vary significantly.<!--[ 161 ]--> For example, use <math>r = 3.95, x_0 = 0.1, \hat{x}_0=x_0+10^{-9}</math>
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.<!--[ 162 ]-->

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<!--[ 163 ]-->:

{{NumBlk|:|<math>{\displaystyle \lambda =\lim _{N\to \infty }{\frac {1}{N}}\sum _{i=0}^{N-1}\log \left|f^{\prime }(x_{i})\right\vert }</math>|{{EquationRef|3-14}}}}

Here, log means natural logarithm. This λ is the distance between the two orbits (<math>x_n</math> and
<math>\hat{x}_n</math>).
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.<!--[ 164 ]--> When calculating λ of numerically, it can be confirmed λ remains in the range of zero or negative values in the range <math>r < r_{\infty}</math>, and that λ can take positive values in the range <math>r > r_{\infty}</math>.<!--[ 165 ]-->

'''Window, intermittent'''

Even beyond <math>r_{\infty}</math>, the behavior does not depend simply on the parameter r.<!--[ 166 ]--> Many sophisticated mathematical structures lurk in the chaotic region for <math>r > r_{\infty}</math>.<!--[ 167 ]--> In this region, chaos does not persist forever; stable periodic orbits reappear.<!--[ 166 ]--> The behavior for <math>r_{\infty}< a \le 4</math> can be broadly divided into two types:<!--[ 168 ]-->

* 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 <math>r_{\infty} < r \le 4</math> is called a periodic window, or simply a window.<!--[ 169 ]--> 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.<!--[ 170 ]-->

{{Image frame|width=620|content=[[File:Logistic orbit map 3.55 to 4.png|class=skin-invert-image|620px]]
|caption=Orbit diagram of the logistic map from r = 3.55 to r = 4 (parameter is denoted as r in the diagram)|align=center}}

In each window, the cascade of [[period-doubling]] bifurcations that occurred before <math>r_{\infty} = 3.56994...</math> occurs again.<!--[ 171 ]--> 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.<!--[ 172 ]--> 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..<!--[ 173 ]--> 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, ....<!--[ 174 ]-->

[[File:ロジスティック写像の過渡カオス.png|class=skin-invert-image|thumb|Transient chaos at a = 3.8285. The system behaves chaotically until it is attracted into a periodic 3 orbit.]]

In the window region, chaos does not disappear but exists in the background.<!--[ 175 ]--> However, this chaos is unstable, so only stable periodic orbits are observed.<!--[ 175 ]--> In the window region, this potential chaos appears before the orbit is attracted from its initial state to a stable periodic orbit.<!--[ 176]--> Such chaos is called transient chaos.<!--[177 ]--> In this potential presence of chaos, windows differ from the periodic orbits that appeared before a∞.<!--[ 175 ]-->

There are an infinite number of windows in the range a∞ < a < 4.<!--[ 178 ]--> The windows have various periods, and there is a window with a period for every natural number greater than or equal to three.<!--[ 179 ] --> However, each window does not occur exactly once.<!-- [ 180 ]--> The larger the value of p, the more often a window with that period occurs.<!--[ 181 ]--> A window with period 3 occurs only once, while a window with period 13 occurs 315 times.<!--[ 182 ]--> 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.<!--[ 183 ]-->

If we restrict ourselves to the case where p is a prime number, the number of windows with period p is

{{NumBlk|:|<math>{\displaystyle N_{p}={\frac {2^{p-1}-1}{p}}}</math>|{{EquationRef|3-15}}}}

<!--[ 184 ]--> 

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.<!--[ 181 ]-->

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.<!--[ 185 ]--> For example, the window width for a window with period 13 is about 3.13 × 10−6.<!--[ 186 ]--> Rough estimates suggest that about 10% of <math>[ r_{\infty}, 4]</math> is in the window region, with the rest dominated by chaotic orbits.<!--[ 187 ]-->

The change from chaos to a window as r is increased is caused by a tangent bifurcation,<!--[ 188 ]--> 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.<!--[ 189 ]--> For a window of period p, the iterated map <math>f^p(x)</math> exhibits tangent bifurcation, resulting in stable p-periodic orbits.<!--[ 168 ]--> 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 <math>r_3</math>, then <math>r_3 = 1 + \sqrt{8} = 3.828427...</math>.<!--[ 190 ]--> The outline of this bifurcation can be understood by considering the graph of <math>f^3(x)</math> (vertical axis <math>x_{n+3}</math>, horizontal axis <math>x_n</math>).<!--[ 191 ]-->  

<div class=skin-invert-image>{{multiple image
| align             = center
| direction         = horizontal
| total_width       = 620
| image1            = ロジスティック写像3回反復グラフの接線分岐の様子 (0.99xa).png
| caption1          = Graph of <math>f^3(x)</math> when r is slightly less than 3. The graph is not tangent except at the fixed points, and there are no 3-periodic points.
| image2            =ロジスティック写像3回反復グラフの接線分岐の様子 (1.00xa).png
| caption2          = When a is exactly 3, the graph touches the diagonal at exactly three points, resulting in three periodic points.
| image3            = ロジスティック写像3回反復グラフの接線分岐の様子 (1.01xa).png
| caption3          = When a is slightly greater than 3, the graph passes the diagonal and splits into stable and unstable 3-periodic points.}}</div>

When we look at the behavior of <math>x_n</math> when r = 3.8282, which is slightly smaller than the branch point <math>r_3</math>, we can see that in addition to the irregular changes, there is also a behavior that changes periodically with approximately three periods, and these occur alternately.<!--[ 192 ]--> This type of periodic behavior is called a "laminar", and the irregular behavior is called a burst, in analogy with  fluids.<!--[ 193 ]--> There is no regularity in the length of the time periods of the bursts and laminars, and they change irregularly.<!--[ 194 ]--> However, when we observe the behavior at r = 3.828327, which is closer to <math>r_3</math>, the average length of the laminars is longer and the average length of the bursts is shorter than when r = 3.8282.<!--[194]--> If we further increase r, the length of the laminars becomes larger and larger, and at <math>r_3</math> it changes to a perfect three- period.<!--[ 195 ]-->

<div class=skin-invert-image>{{multiple image
| align             = center
| direction         = horizontal
| total_width       = 620
| image1            = Logistic map time evolution a=3.8282.png
| caption1          = Time series when r = 3.8282
| image2            = Logistic map time evolution a=3.828327.png
| caption2          = Time series when r = 3.828327
| footnote          = The intermittency that occurs just before <math>r_3 = 3.828427...</math> The part where the three almost identical values continue periodically is a laminar, and the part where chaotic irregular changes occur is a burst.
}}</div>

The phenomenon in which orderly motions called laminars and disorderly motions called bursts occur intermittently is called intermittency or intermittent chaos.<!--[ 196 ]--> If we consider the parameter a decreasing from a3, this is a type of emergence of chaos.<!--[ 197 ]--> As the parameter moves away from the window, bursts become more dominant, eventually resulting in a completely chaotic state.<!--[ 198 ]--> 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.<!--[ 199 ]-->

[[File:Laminar channel in three times iterated logistic map.png|class=skin-invert-image|thumb|Channel patterns appearing in the graph of f3 (x)]]

The mechanism of intermittency can also be understood from the graph of the map.<!--[ 194 ]--> When <math>r</math> is slightly smaller than <math>r_3</math>, there is a very small gap between the graph of <math>f^3(x)</math> and the diagonal<!--[ 190  ]-->. This gap is called a channel, and many iterations of the map occur as the orbit passes through the narrow channel.<!--[ 200 ]--> During the passage through this channel, <math>x_n</math> and <math>x_{n + 3}</math> become very close, and the variables change almost like a periodic three orbit.<!--[ 198 ]--> This corresponds to a laminar.<!--[ 201 ]--> The orbit eventually leaves the narrow channel, but returns to the channel again as a result of the global structure of the map.<!--[ 202 ]--> While leaving the channel, it behaves chaotically.<!--[ 202 ]--> This corresponds to a burst.<!--[ 201 ]-->

'''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.<!--[ 203 ]--> 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.<!--[ 204 ]--> The lower limit of the attractor is given by the point f(r/4) where r/4 is mapped.<!--[ 204 ]--> Ultimately, the maximum and minimum values at which xn moves on the orbital diagram depend on the parameter r

{{NumBlk|:|<math>{\displaystyle {\frac {r^{2}(4-r)}{16}}\leq x_{n}\leq {\frac {r}{4}}}</math>|{{EquationRef|3-16}}}}

<!--[ 203 ]-->
Finally, for r = 4, the orbit spans the entire range [0, 1].<!--[ 205 ]-->

When observing an orbital map, the distribution of points has a characteristic shading.<!--[ 206 ]--> 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.{{clarify inline| date=March 2025}}<!--[ 206 ]--> These differences in the frequency of the points are due to the shape of the graph of the logistic map.<!--[ 206 ]--> 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 <math>f^2(r/4)</math> that is mapped from there also becomes  highly frequent, and so on.<!--[ 206 ]--> 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.<!--[ 207 ]-->

Looking at the beginning of the chaotic region of the orbit diagram, just beyond the accumulation point <math>r_{\infty} = 3.56994</math> of the first period - doubling cascade, one can see that the orbit is divided into several subregions.<!--[ 208 ]--> These subregions are called bands.<!--[ 209 ]--> When there are multiple bands, the orbit moves through each band in a regular order, but the values within each band are irregular.<!--[ 210 ]--> Such chaotic orbits are called band chaos or periodic chaos, and chaos with k bands is called k -band chaos.<!--[ 211 ]--> Two-band chaos lies in the range 3.590 < r < 3.675, approximately.<!--[ 212 ]-->

{{Image frame|width=620|content=[[File:Bands of logistic map from 1 to 8.png|class=skin-invert-image|620px]]
|caption=Band structure. Because the <math>e_p</math> 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).|align=center}}

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.<!--[ 212 ]--> Let <math>e_p</math> (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 → ∞.<!--[ 213 ]--> At this accumulation point <math>e_{\infty}</math>, the number of bands becomes infinite, and the value of <math>e_{\infty}</math> is equal to the value of <math>r_{\infty}</math>.<!--[ 214 ]-->

[[File:ロジスティック写像の全体の自己相似.png|class=skin-invert-image|thumb|Self-similar hierarchical structure of the entire trajectory map of the logistic map]]

Similarly, for the bifurcation points of the period-doubling bifurcation cascade that appeared before a∞, let a p (where p = 1, 2, 4, ..., 2k, ...) denote the bifurcation points where p stable periodic orbits branch into p + 1 stable periodic orbits. In this case, if we look at the orbital diagram from <math>r_2</math> to <math>e_2</math>, there are two reduced versions of the global orbital diagram from <math>r_1</math> to <math>e_1</math> in the orbital diagram from <math>r_2</math> to <math>e_2</math><!--[215]-->. Similarly, if we look at the orbital diagram from <math>r_4</math> to <math>e_4</math>, there are four reduced versions of the global orbital diagram from a1 to e1 in the orbital diagram from <math>r_4</math> to <math>e_4</math>.<!--[ 215 ]--> 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.<!--[ 215 ]-->

[[File:ロジスティック写像の窓の自己相似.png|class=skin-invert-image|thumb|Self-similar hierarchical structure of windows of the logistic map]]

A self-similar hierarchy of bifurcation structures also exists within windows.<!--[ 216 ]--> The period-doubling bifurcation cascades within a window follow the same path as the cascades of period-2k bifurcations.<!--[ 217 ]--> That is, there are an infinite number of period-doubling bifurcations within a window, after which the behavior becomes chaotic again.<!--[ 217 ]--> For example, in a window of period 3, the cascade of stable periodic orbits ends at <math>a_{3\infty}</math> ≈ 3.8495.<!--[ 218 ]--> After <math>a_{3\infty}</math> ≈ 3.8495, the behavior becomes band chaos of multiples of three.<!--[ 218 ]--> As a increases from <math>a_{3\infty}</math>, these band chaos also merge by twos, until at the end of the window there are three bands.<!--[ 219 ]--> Within such bands within a window, there are an infinite number of windows.<!--[ 220 ]--> 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.<!--[ 221 ]-->

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.<!--[ 222 ]--> However, this change is discontinuous, and the 3-band chaotic attractor suddenly changes size and turns into a 1-band.<!--[ 223 ]--> Such discontinuous changes in attractor size are called crises.<!--[ 224 ]--> Crises of this kind, which occur at the end of a window, are also called internal crises.<!--[ 225 ]--> 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.<!--[ 226 ]--> This creates an exit point through which the periodic orbits can escape, resulting in an internal crisis.<!--[ 227 ]--> 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.<!--[ 197 ]-->