Varying the control parameter in the logistic map

Let us study how the orbits of the logistic map depend on the control parameter that is restricted to $0 \leq r \leq 1$. That range of $r$ is selected because it keeps all future values of $x$ between $0$ and $1$ if initially it was between those values. If $r=0$ then the population does not reproduce at all and becomes immediately extinct at the next time step. For $0 < r \leq 1/4$, it is easy to see that irrespective of the initial population, the population will monotonically decrease as time progresses, and will eventually become extinct. That is, the origin is the only fixed point for $0 \leq r \leq 1/4$ and it is attractive (stable) for all $x_0$. Figure(3.2) shows the state plot for $r=0.2$ and $x_0=0.6$.

Consider now a larger value of $r$, say $r=0.7$. For an intial value $x_0 =0.1$, the corresponding time series is shown in Figure (3.3a). The initial behaviour shown in that figure is called the transient orbit, while the final sequence to which the orbit tends to is called the post-transient orbit. It is the post-transient orbit which is of interest. As the state-space plot in Fig(3.3b) confirms, the fixed point at the origin is now unstable, and the system is attracted to a new stable period-one fixed point. Thus we have our first example of a phenomena whereby as the control parameter in an iterative map is changed, new fixed points can appear, and the stability properties of old fixed points can change. Indeed, for $r > 1/4$ the new fixed point is at $x^{*} = 1-1/4r$ and this point is stable for $1/4 < r < 3/4$.

Next consider a value of $r$ slightly above $3/4$, say at $r=0.8$. We already know from the above discussion that the period-one fixed point at $x^{*} =22/32$ will be unstable. In fact as Fig.(3.4) shows, the final orbit does not converge to any fixed point but rather alternates between two points. This orbit is called a period two orbit (or period two limit cycle), and the two points between which the orbit oscillates are sometimes labelled as period-two fixed points (or period two attractors) because they satisfy the the relation $x_{n+2}=x_n$.

As $r$ is increased further, the system will at some point undergo a period-4 cycle. Fig.(3.5) shows this for $r=0.88$.

These period doublings continue as $r$ is increased until at $r=0.8924865....$ the system shows a completely aperiodic orbit. Such orbits are termed 'chaotic', and though there are no fixed points now, the system is said to have a chaotic or strange attractor. Fig(3.6a) shows the time series for $r=1$ which is in the chaotic regime. Fig(3.6c) shows how in the chaotic regime, small differences in the initial conditions amplify quickly - this was the sensitive dependence on initial conditions mentioned in the introduction as a defining property of chaos.