IDC402 Non Linear Dynamics

Tuesday, January 18, 2022

Stability of orbits

For the tent map or shift map

$|\lambda| = 2$

Dense set of unstable orbits.

Given $x$ and $\epsilon$, there is an unstable orbit in a ball of radius $\epsilon$ centered at $x$.

Periodic points are countably infinite. They are a measure zero set. Hence “all” points are unstable and chaos is typical.

Consider a histogram of the fraction of time in a finite length orbit originating from a typical initial point falls in N bins of equal size along the x axis.

As $N$ and orbit length $\to\infty$ this probability $\frac{1}{N}$.

Can define a Natural Invariant Density which is the fraction of time typical orbits spend in an interval [x,x+dx].

Logistic Map

\[x_{n+1} = r x_n (1-x_n)\]

Models generational variation of population.

Explanation of terms

If infinite resources,

\[x_{n+1} = r x_n\]

This is an exponential growth.

However, there must be a negative growth rate.

\[x_{n+1} = -rx_n^2\]

Hence we get the logistic map.

Properties

1. Max at $x=1/2$ with value $r/4$
2. Zeros are found at 0 and 1
3. For $0\le r\le4$, orbit is bound in $[0,1]$ if $x_0 \in [0,1]$, else runaway to $-\infty$
4. There are [2022-01-18#Bifurcations|bifurcations]

Bifurcations

Qualitative changes in dynamics which occur as a system parameter varies. Bifurcations are generic i

Three types

• Transcritical Bifurcation: Exchange of Stability.
  • Unstable becomes stable and vice versa.
• Tangent Bifurcation: Creating of new fixed points
  • New fixed points appear with half being stable and half unstable
• Period doubling: Creation of new fixed points and Exchange of Stability.
  • Both happen.

Magnitude of Stability Coefficient is 1 at bifurcation point.

For the logistic map

Solutions-

$x^* = 0 \text{ and } x^* = 1- \frac{1}{r}$. This implies $r \ge 1$ only we have 2 solutions.

For stability, examine $F’(x^*) = r-2r(x^*)^2$.

At $x=0$, $F’(0) = r$

Hence it is stable only if $r<1$.

At $x = 1-1/r$, $F’(1-1/r) = 2-r$ which is stable only if $r \in [1, 3]$