📂Dynamics

Chaotic Transition

Definition

The phenomenon where a system becomes chaotic or remains non-chaotic depending on the change of parameters is called chaotic transition.

Examples

For example, if we consider the logistic family, a system formed by $g_{a} = ax (1-x)$ shows different behaviors depending on the parameter $a$, and it can be observed that it has a chaotic orbit when $a=4$. The next question then is ‘what happens when it’s $a>4$?’

Firstly, if it’s $a>4$ then $g_{a}(x) = ax(1-x)$ which means its maximum is greater than $1$, and $g_{a}$ maps $x_{n} \notin [-1,1]$ to negative. Moreover, since it’s a quadratic function, once it goes beyond a certain $\tau \in \mathbb{N}$ for $x_{ \tau } > 1$ to occur, it diverges at an alarming rate afterwards. Before a chaotic orbit comes, it has to be a bounded orbit, but divergence means the chaotic nature was lost as the parameter $a$ changed, which according to the definition above, can be called a chaotic transition.

Such phenomenon will happen more quickly as the value of $a>4$ increases. Hence, we can guess there is some relation between $\tau$, which makes $x_{ \tau } > 1$ occur, and $a$. It is not very difficult to verify this through simulation, just by running the code, the following plot is drawn.

In the code, it slightly increases the value of $a$ while finding $n = \tau$ that satisfies $x_{n} >1$ for a random initial value $x_{0}$, and repeats this $N=1000$ times to calculate the average of $\tau$, $\overline{\tau}$. Looking at the actual plot, one can observe the linear relationship between $a = 4 + \varepsilon$ and $\log \varepsilon \sim \log \left( \overline{ \tau } \right)$.

Such chaotic transitions are not only found in logistic map systems but are phenomena discovered in many systems.

Code

ga<-function(a,x,k=1) {
  tmp <- a*x *(1 - x)
  if(k==1) {return( tmp ) }
  else {return(ga(a,tmp,k-1))}
}
 
N<-1000
y<-numeric(0)
A<-4+10^(-(80:120)/20)
for(a in A){
  taubar<-0
  for(i in 1:N){
    tau<-0
    x<-runif(1)
    while(x>0){
      x<-ga(a,x)
      tau=tau+1
    }
    taubar<-taubar+tau
  }
  y<-c(y,taubar/N)
}
 
taubar<-y
a<-A
 
win.graph(); plot(log(a-4),log(taubar),type='l',main='a>4 일 때 로지스틱 맵의 Chaotic Transient')