Logistic map

The logistic map is

${\displaystyle x_{n+1}=rx_{n}(1-x_{n})}$

where ${\displaystyle x_{n}}$ is a function of the (discrete) time ${\displaystyle n=0,1,2,\ldots }$.^[3] The parameter ${\displaystyle r}$ is assumed to lie in the interval ${\displaystyle [0,4]}$, in which case ${\displaystyle x_{n}}$ is bounded on ${\displaystyle [0,1]}$.

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 ${\displaystyle r\approx 3.56995}$, beyond which 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 ${\displaystyle r=3.83}$.

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.

quadratic map

Real version of complex quadratic map is related with real slice of the Mandelbrot set.

• 
  Period-doubling cascade in an exponential mapping of the Mandelbrot set
• 
  1D version with an exponential mapping
• 
  period doubling bifurcation

Kuramoto–Sivashinsky equation

Period doubling in the Kuramoto–Sivashinsky equation with periodic boundary conditions. The curves depict solutions of the Kuramoto–Sivashinsky equation projected onto the energy phase plane (E, dE/dt), where E is the L²-norm of the solution. For ν = 0.056, there exists a periodic orbit with period T ≈ 1.1759. Near ν ≈ 0.0558, this solution splits into 2 orbits, which further separate as ν is decreased. Exactly at the transitional value of ν, the new orbit (red-dashed) has double the period of the original. (However, as ν increases further, the ratio of periods deviates from exactly 2.)

The Kuramoto–Sivashinsky equation is an example of a spatiotemporally continuous dynamical system that exhibits period doubling. It is one of the most well-studied nonlinear partial differential equations, originally introduced as a model of flame front propagation.^[4]

The one-dimensional Kuramoto–Sivashinsky equation is

${\displaystyle u_{t}+uu_{x}+u_{xx}+\nu \,u_{xxxx}=0}$

A common choice for boundary conditions is spatial periodicity: ${\displaystyle u(x+2\pi ,t)=u(x,t)}$.

For large values of ${\displaystyle \nu }$, ${\displaystyle u(x,t)}$ evolves toward steady (time-independent) solutions or simple periodic orbits. As ${\displaystyle \nu }$ is decreased, the dynamics eventually develops chaos. The transition from order to chaos occurs via a cascade of period-doubling bifurcations,^[5][6] one of which is illustrated in the figure.

Logistic map for a 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 ultimately becomes chaotic.