The Worldwide Air-Transportation Network

The Logistic Map

The logistic map is a one-dimensional discrete time dynamical system that is defined by the equation (For more information about this dynamical system check out the Wikipedia article):

$$x_{n+1}=f(x_n)=\lambda x_n(1-x_n)$$

For an initial value $0\le x_0\le 1$ this map generates a sequence of values $x_0,x_1,...x_n,x_{n+1},...$. The growth parameter is chosen to be

0 < \lambda\leq 4 $$ which implies that for all $n$ the state variable will remain bounded in the unit interval. Despite its simplicity this famous dynamical system can exhibit an unbelievable dynamic richness. The logistic map has two stationary points $x^{\star}=0$ and $x^{\star}=1-\lambda^{-1}$, for those values we have $x^{\star}=f(x^{\star})$. The second, non-trivial stationary point only exists for $\lambda>1$. Also when $\lambda>1$ the origin is unstable and the nontrivial stationary state is stable. When $\lambda>3$ the second stationary state loses stability, the attractor of the dynamics is a period-2 cycle. The two points that make up this period-2 cycle, are stable fixpoints of the map $g=f\circ f$. This cycle also becomes unstable eventually at $\lambda=1+\sqrt{6}\approx 3.4495$. Beyond this point the attractor is a period-4 cycle (all points in it are stable fixpoints of $f\circ f\circ f\circ f$) until, again, stability is lost. This period-doubling occurs more and more as we increase $\lambda$ until a critical point is reached (try identifying it) and the dynamics becomes chaotic. ## Orbits You can explore the dynamics of the logistic map with the tools below. The first interactive figure illustrates the orbits of the logistic map for two different initial conditions. You can switch on and off each orbit as well as the transients to look at the structure of the attractor. 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transform="translate(580,145)"><circle r="7" style="fill: rgb(253, 141, 60);"></circle><text style="fill: rgb(253, 141, 60);" transform="translate(20,6)">transient</text></g> <g class="olabel" transform="translate(580,225)"><circle r="7" style="fill: red;"></circle><text style="fill: red;" transform="translate(20,6)">orbit 1</text></g> <g class="olabel" transform="translate(580,245)"><circle r="7" style="fill: green;"></circle><text style="fill: green;" transform="translate(20,6)">orbit 2</text></g> <g transform="translate(60,575)" class="x saxis"><g class="tick" style="opacity: 1;" transform="translate(0,0)"><line y2="0" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="8" x="0">1</text></g> <g class="tick" style="opacity: 1;" transform="translate(83.33333333333333,0)"><line y2="0" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="8" x="0">1.5</text></g> <g class="tick" style="opacity: 1;" transform="translate(166.66666666666666,0)"><line y2="0" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="8" x="0">2</text></g> <g class="tick" style="opacity: 1;" transform="translate(250,0)"><line y2="0" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="8" x="0">2.5</text></g> <g class="tick" style="opacity: 1;" transform="translate(333.3333333333333,0)"><line y2="0" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="8" x="0">3</text></g> <g class="tick" style="opacity: 1;" transform="translate(416.6666666666667,0)"><line y2="0" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="8" x="0">3.5</text></g> <g class="tick" style="opacity: 1;" transform="translate(500,0)"><line y2="0" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="8" x="0">4</text></g> <path class="domain" d="M0,0V0H500V0"></path><path class="halo" d="M0,0V0H500V0"></path></g><g transform="translate(60,575)" class="slider" style="pointer-events: all;"><circle class="handle" r="5" style="fill: rgb(31, 119, 180);" cx="283.33333333333337"></circle><text transform="translate(0,-15)" style="text-anchor: middle;" class="tag" x="283.33333333333337">λ = 2.7</text> <rect class="extent" style="cursor: move;" width="0" x="283.33333333333337"></rect></g><g transform="translate(60,505)" class="slider" style="pointer-events: all;"><circle class="handle" r="5" style="fill: red;" cx="49.59919839679359"></circle><text transform="translate(5,-10)" class="tag" x="49.59919839679359">x0 = 0.1</text> <rect class="extent" style="cursor: move;" width="0" x="49.59919839679359"></rect></g><g transform="translate(60,490)" class="slider" style="pointer-events: all;"><circle class="handle" r="5" style="fill: green;" cx="99.6993987975952"></circle><text transform="translate(5,-10)" class="tag" x="99.6993987975952">x0 = 0.2</text><rect class="extent" style="cursor: move;" width="0" x="99.6993987975952"></rect></g></svg> ## Bifurcation Analysis The figure below illustrates the bifurcation diagram of the logistic map. For each $\lambda$ the map is iterarated for some time $N\gg1$. After this initial transient period, points $x_{n}$ of the trajectory with $n>N$ are recorded and depicted. This way a stable stationary state is just a single point, a period-2 cycle corresponds to two points etc. This way one can see the structure of period-doubling events and the onset of determinstic chaos. <svg xmlns="http://www.w3.org/2000/svg" width="660" height="455"><g transform="translate(60,5)"><g class="x axis" transform="translate(0,400)"><g class="tick" style="opacity: 1;" transform="translate(2.239105259748214e-13,0)"><line y2="6" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="9" x="0">2.8</text></g> <g class="tick" style="opacity: 1;" transform="translate(50.420168067227145,0)"><line y2="6" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="9" x="0">2.9</text></g> <g class="tick" style="opacity: 1;" transform="translate(100.84033613445406,0)"><line y2="6" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="9" x="0">3.0</text></g> <g class="tick" style="opacity: 1;" transform="translate(151.26050420168096,0)"><line y2="6" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="9" x="0">3.1</text></g> <g class="tick" style="opacity: 1;" transform="translate(201.68067226890767,0)"><line y2="6" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="9" x="0">3.2</text></g> <g class="tick" style="opacity: 1;" transform="translate(252.10084033613435,0)"><line y2="6" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="9" x="0">3.3</text></g> <g class="tick" style="opacity: 1;" transform="translate(302.5210084033613,0)"><line y2="6" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="9" x="0">3.4</text></g> <g class="tick" style="opacity: 1;" transform="translate(352.9411764705882,0)"><line y2="6" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="9" x="0">3.5</text></g> <g class="tick" style="opacity: 1;" transform="translate(403.36134453781517,0)"><line y2="6" x2="0"></line><text dy=".71em" style="text-anchor: middle;" y="9" x="0">3.6</text></g> <g class="tick" 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