# Chaos Theory: Sensitivity to Initial Conditions

As stated in the previous posts, sensitivity to initial conditions is popularly known as the butterfly effect. It is one of the properties of chaos theory mostly affiliated by scientists when determining if a given system is chaotic. It means that each point in a system is arbitrarily closely approximated by other points with distinguishably different orbits in the future. In other words, a small change to the initial conditions may lead to significantly different behaviour. However, just because systems are sensitive to initial conditions it doesn’t imply that they are chaotic. To illustrate we will look at the mapping given below:

$g_{t+1}=g_{t}^2, \quad t \, \epsilon \, \mathbb{R}$

Starting at two initial conditions $\it{t_{0}}$ and $\it{t_{0}^*}$ with minute differences the orbits diverge quickly from one another. Starting with a small difference of approximately 0.01 after 10 iterations. However, after 20 iterations, the difference is in its billions. The graph above shows the map between 14 and 19 iterations with the orbits represented by different colours, red and blue. Clearly, the system is dependent on initial conditions, however, the system isn’t chaotic. The distinction of chaotic systems is that, given two states that are minute in difference, the orbits distance from one another continuously varies; i.e. sometimes becoming arbitrarily close or far from one another with respect to the attractor. For the quadratic system, this doesn’t occur, and so, it isn’t chaotic. To illustrate this we will look at the logistic map for different initial conditions.

Like before we represent the two orbits of the logistic map with different colours (graph above) . As shown, the orbits remain the same for the first few iterations. However, eventually the orbits diverge from each other and become non-identical, and so exhibit behaviour of sensitivity to initial conditions. To support this we plot the corresponding (absolute) distance between the orbits. In contrast to the quadratic mapping before, the difference between the two orbits vary. Clearly, the difference is arbitrarily small sometimes and then arbitrarily large with respect to the attractor, and so demonstrates behaviour of a chaotic system. Note the rate at which nearby orbits diverge from each other with time is characterised by a statistic called the Lyapunov exponent. This helps determine chaotic behaviour, and is a statistic we will be looking at later in the blog.