for the logistic map. It shows the
attractor values, like x_* and x'_*, as a function of the parameter r.
Logistic map The
logistic map is :x_{n+1} = r x_n (1 - x_n) where x_n is a function of the (discrete) time n = 0, 1, 2, \ldots. The parameter r is assumed to lie in the interval [0,4], in which case x_n is bounded on [0,1]. For r between 1 and 3, x_n converges to the stable fixed point x_* = (r-1)/r. Then, for r between 3 and 3.44949, x_n converges to a permanent oscillation between two values x_* and x'_* that depend on r. As r grows larger, oscillations between 4 values, then 8, 16, 32, etc. appear. These period doublings culminate at r \approx 3.56995, beyond which more complex regimes appear. As r increases, there are some intervals where most starting values will converge to one or a small number of stable oscillations, such as near r=3.83, where there is a stable period-three solution. In the interval where the period is 2^n for some positive integer n, not all the points actually have period 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.
Kuramoto–Sivashinsky equation 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. The one-dimensional Kuramoto–Sivashinsky equation is : u_t + u u_x + u_{xx} + \nu \, u_{xxxx} = 0 A common choice for boundary conditions is spatial periodicity: u(x + 2 \pi, t) = u(x,t). For large values of \nu, u(x,t) evolves toward steady (time-independent) solutions or simple periodic orbits. As \nu is decreased, the dynamics eventually develops chaos. The transition from order to chaos occurs via a cascade of period-doubling bifurcations, 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: \pi_{t} = f(u_{t}) + b \pi_{t}^e \pi_{t+1} = \pi_{t}^e + c (\pi_{t} - \pi_{t}^e) f(u) = \beta_{1} + \beta_{2} e^{-u} \, b > 0, 0 \leq c \leq 1, \frac {df} {du} where : • \pi is the actual
inflation • \pi^e is the expected inflation, • u is the level of unemployment, • m - \pi is the
money supply growth rate. Keeping \beta_{1} = -2.5, \ \beta_{2} = 20, \ c = 0.75 and varying b, the system undergoes period-doubling bifurcations and ultimately becomes chaotic. ==Experimental observation==