📂Dynamics

Logistic Family

Definition ¹

Equation $a \ge 0$ is referred to as the Logistic Map, and $\left\{ g_{a} \mid a > 0 \right\}$ is called the Logistic Family.

Properties

• [1]: $x \in [0,1] \iff g_{a} (x) \ge 0$
• [2]: $g'_{a} (x) = a ( 1 - 2x)$
• [3]: If $1 < a \le 4$, then $\displaystyle x_{1} = {{ a - 1} \over { a }}$ is a fixed point of $g_{a} (x)$.
• [4]: If $1 < a < 3$, then $$ \lim_{ k \to \infty} f^{k} (x) = x_{1} = {{a-1} \over {a}} $$

Explanation

The logistic model can be usefully applied to model various phenomena such as population growth and decline. For populations, each generation can be seen as applying the logistic map $g_{a}$ once. Although it is a quadratic function and the function value becomes negative unless the initial value is $x_{1} \in [0,1]$, we are not interested in those cases to begin with, so we only need to pay attention to $a \ge 0$ and $x_{1} \in [0,1]$.

Mathematical Understanding

• If $0 \le a \le 1$, $g_{a}$ has a unique sink $x_{1} =0$. Applying the map results in all $a , x, (1-x)$ being less than or equal to $1$, so their product continuously decreases. Thus, applying the map gradually approaches $0$. If considered from a population perspective, it’s a case where more die than are born, eventually leading to extinction. For example, on a small island teeming with only male bears and a single female bear, although there’s still a chance of prospering again, that probability is extremely low. The condition $a<1$ almost certainly means extinction, and there’s not much else of interest.

If $4 < a $, the maximum value of $a x (1-x)$ is $\displaystyle {{a} \over {4}} > 1$, so each time the map is applied, it eventually exceeds $x_{n } = g^{k}_{a} (x_{n-1})$ which is greater than $1$.

• Therefore, by [1] $x_{n+1} = g^{k+1}_{a} ( x_{n} )$ is negative, and thereafter it must diverge in the negative direction. This is too obvious to arouse any curiosity. In summary, only when $1 \le a \le 4$ does it bear any significance.
• According to [2] and [3], as well as the Sink and Source Judgement Method, if $1 < a < 3$ then $\displaystyle | g'_{a} ( x_{1} )| = | 2 - a | < 1$, hence the fixed point $x_{0}$ becomes a sink, and (4) is obtained. If we continue to apply the map until we hit a point sufficiently close to the sink, the values will then stabilize. For populations, this means there’s a stable equilibrium state where it neither grows nor declines.

On the other hand, if $a$ becomes greater than $3$, by solving equation $g_{a}^{2^{n}} (x) = x$, a periodic-$2^{n}$ sink can be found. In this case, several numbers repeat as the map is applied, and if it stops at a random timing, one of them will be obtained. However, this doesn’t hold around $a=3.84$, so it cannot be definitively concluded as a theorem.

Bifurcation

Let’s look at the Bifurcation Diagram below. The bifurcation diagram is drawn according to the following sequence starting from $a=1$.

• Step 1. Increase $a$ a little bit.
• Step 2. Pick a random $x \in [ 0 , 1 ]$.
• Step 3. Place point $(a, g_{a}^{101} (x) )$ and return to Step 1.

The curve part of $1 < a < 3$ indicates that no matter what the initial value $x$ is, applying the map about a hundred times stabilizes and becomes a consistent value. Thereafter, a periodic-$2^{n}$ sink occurs, causing the numbers to oscillate between two, and crossing the red line will result in the existence of a periodic $2^2$-sink where oscillation occurs between four values. Then, suddenly, a huge number of possibilities arises, creating movements that are difficult to predict, referred to as Chaotic Attractors.

Natural Measure

Meanwhile, even though periodic orbits may move chaotically, they can stay longer in specific regions. After discarding the initial values, this characteristic of the system itself allows the calculation of an ‘average’ distribution, irrespective of initial conditions. This probability density function $\rho (x)$, known as Natural Invariant Density, can be expressed mathematically for sufficiently large $N$ as $\displaystyle p \left( x_{N} \in [a,b] \right) = \int_{a}^{b} \rho (x) dx$.

In fact, plotting $\rho (x)$ of the logistic map $g_{4} = 4 x (1 - x )$ looks like the above. Although $\left\{ x_{1} , x_{2} , \cdots \right\}$ is chaotic, it’s still possible to notice that the probabilities are highest near $0$ and $1$.

Code

Below is the R code for plotting the bifurcation diagram.

ga<-function(a,x,k=1) {
  tmp <- a*x *(1 - x)
  if(k==1) {return( tmp ) }
  else {return(ga(a,tmp,k-1))}
}
 
result<-numeric(0)
for(a in seq(1,4.1,by=0.0001))
{
  result<-rbind(result,c(a,ga(a,runif(1),101)))
}
 
win.graph(8,4)
plot(result,main='bifurcation diagram',xlim=c(1,4),ylim=c(0,1),pch=20, cex=0.1,
     xlab='a',ylab='101번째 값')
abline(v=c(3,4),lty=2); abline(h=0)
abline(v=3.45,col='red')

Below is the R code for plotting the shape of natural invariant density.

set.seed(150421)
logistic<-function(x) {4*x*(1-x)}
 
record<-runif(1)
for(i in 1:10^5) {record<-c(record,logistic(record[length(record)]))}
out<-hist(record[-(1:10000)])
win.graph(4,4); plot(x=seq(0,1,len=20),out$density,type='l',
                  xlab='x',ylab='ρ',main='Naturla Invariant Density ρ(x)')