Since <math>d^4 x\!</math> and all higher order <math>d^j x\!</math> are fixed, the entire dynamics can be plotted in the extended space <math>\operatorname{E}^3 X = \langle x, dx, d^2 x, d^3 x \rangle.</math> Thus, there is just enough room in a planar venn diagram to plot both orbits and to show how they partition the points of <math>\operatorname{E}^3 X.</math> As it turns out, there are exactly two possible orbits, of eight points each, as illustrated in Figures 16-a and 16-b. See here: | Since <math>d^4 x\!</math> and all higher order <math>d^j x\!</math> are fixed, the entire dynamics can be plotted in the extended space <math>\operatorname{E}^3 X = \langle x, dx, d^2 x, d^3 x \rangle.</math> Thus, there is just enough room in a planar venn diagram to plot both orbits and to show how they partition the points of <math>\operatorname{E}^3 X.</math> As it turns out, there are exactly two possible orbits, of eight points each, as illustrated in Figures 16-a and 16-b. See here: |