Period-doubling bifurcation

Bifurcation diagram for the logistic map. It shows the attractor values, like ${\displaystyle x_{*}}$ and ${\displaystyle x'_{*}}$, as a function of the parameter ${\displaystyle r}$.

Consider the following simple dynamics: ${\displaystyle x_{n+1}=rx_{n}(1-x_{n})}$ where ${\displaystyle x_{n}}$, the value of ${\displaystyle x}$ at time ${\displaystyle n}$, lies in the ${\displaystyle [0,1]}$ interval and changes over time according to the parameter ${\displaystyle r\in (0,4]}$. This classic example is a simplified version of the logistic map.

For ${\displaystyle r}$ between 1 and 3, ${\displaystyle x_{n}}$ converges to the stable fixed point ${\displaystyle x_{*}=(r-1)/r}$. Then, for ${\displaystyle r}$ between 3 and 3.44949, ${\displaystyle x_{n}}$ converges to a permanent oscillation between two values ${\displaystyle x_{*}}$ and ${\displaystyle x'_{*}}$ that depend on ${\displaystyle r}$. As ${\displaystyle r}$ grows larger, oscillations between 4 values, then 8, 16, 32, etc. appear. These period-doublings culminate at ${\displaystyle r\approx 3.56995}$ from where more complex regimes appear. As ${\displaystyle r}$ increases, there are some intervals where most starting values will converge to one or a small number of stable oscillations, such as near ${\displaystyle r=3.83}$. See figure.

In the interval where the period is ${\displaystyle 2^{n}}$ for some positive integer ${\displaystyle n}$, not all the points actually have period ${\displaystyle 2^{n}}$. These are single points, rather than intervals. These points are said to be in unstable orbits, since nearby points do not approach the same orbit as them. See Sharkovskii's theorem.

Bifurcation diagram for the modified Phillips curve.

Consider the following logistical map for a modified Phillips curve:

${\displaystyle \pi _{t}=f(u_{t})+b\pi _{t}^{e}}$

${\displaystyle \pi _{t+1}=\pi _{t}^{e}+c(\pi _{t}-\pi _{t}^{e})}$

${\displaystyle f(u)=\beta _{1}+\beta _{2}e^{-u}\,}$

${\displaystyle b>0,0\leq c\leq 1,{\frac {df}{du}}<0}$

where :

• ${\displaystyle \pi }$ is the actual inflation
• ${\displaystyle \pi ^{e}}$ is the expected inflation,
• u is the level of unemployment,
• ${\displaystyle m-\pi }$ is the money supply growth rate.

Keeping ${\displaystyle \beta _{1}=-2.5,\ \beta _{2}=20,\ c=0.75}$ and varying ${\displaystyle b}$, the system undergoes period doubling bifurcations, and after a point becomes chaotic, as illustrated in the bifurcation diagram on the right.