On Logistic, 9 of 8; Period Two Logistic

9. Period Two Logistic

          [Note to blog readers: this section is 9 of 8 because I wrote it up right after blogging the 8 sections, inspired by them. For now, with this, I catch up to myself; but open questions remain, such as logistic chaos.]

Consider this logistic system:

                    A = 1/A {+} B     =  B / (AB+1)
                    B = 1/B {+} C     =  C / (BC+1)
                    C = 1/C {+} 1/A  =  1 / (C+A)

          Moe: “If I’m right, then Larry is right.”
          Larry: “If I’m right, then Curly is right.”

          Curly: “If I’m right, then Moe is wrong.”

          If you iterate from (1,1,1), then you converge to a wobble between                           (0.5346378109, 0.4619568588, 0.9633840464)   and
                    (0.3704606374, 0.6666824455, 0.6675470022)

          but if you start from (2,2,2), then you converge to a wobble between
                    (0.786389671, 0.3140677107, 1.417025223)       and
                    (0.2518627488, 0.9806118803, 0.4538409914)

          and I conjecture that any initial conditions converge to a period 2 wobble. Of course conjecture is not proof.  So do we get a pair of strange attractors in 3-space? Perhaps someone could do a computer graphic of this.

          The second iterate of a period-2 function has period 1; a fixedpoint. So second-iterate the function:

           A = 1/(1/A {+} B) {+} (1/B {+} C)   =   (A + 1/B) {+} 1/B {+} C
           B = 1/(1/B {+} C) {+} (1/C {+} 1/A)  =  (B + 1/C) {+} 1/C {+} 1/A
           C = 1/(1/C {+} 1/A) {+} 1/(1/A {+} B) =   (C + A) {+} (A + 1/B)

          To simplify, let b = 1/B, so:

            b   =  1 / ((B + 1/C) {+} 1/C {+} 1/A)    =   (1/B {+} C) + C + A

          Then we get this system:

A    =   (A + b) {+} (b {+} C)
          b    =   (b {+} C) + (C + A)
          C    =   (C + A) {+} (A + b)

          Experiment on a hand calculator shows that this system does converge to fixedpoints, from a wide range of initial values for A,b,C; but not the same fixedpoint for different initial values. The range of all fixedpoints might be a strange attractor. Would graphing it be feasible?

          Note the sub-sums and sub-reduction (A+b), (C+A), (b{+}C). Call these D, E, F; then:

                    A = D {+} E
                    b =  E + F
                    C = F {+} D
                    D = A + b
                    E = b {+} C
                    F = C + A

          If you start with D = 1/A, E = 1/b, F = 1/C, then these relations continue with this system’s iteration. Strange to say, this system iterates to period two! Its  second iterate is

                    A = (A + b) {+} (b {+} C)
                    b =  (b {+} C) + (C+A)
                    C = (C+A) {+} (A + b)
                    D = (D {+} E) + (E + F)
                    E = (E + F) {+} (F {+} D)
                    F = (F {+} D) + (D {+} E)

          In the second iterate, AbC and DEF are separate and reciprocal systems, matching additions with reductions and vice versa; both iterating to fixedpoints in a strange attractor.

          You could redo the entire argument above with + and {+} swapped, starting with the period-2 system:

                    A = 1/A + B
                    B = 1/B + C
                    C = 1/C + 1/A

          Moe: “I’m wrong, and Larry is right.”
          Larry: “I’m wrong, and Curly is right.”

Curly: “I’m wrong, and Moe’s wrong.”

          which reiterates to this fixedpoint system:

                    A    =   (A {+} b) + (b + C)
                    b     =   (b + C) {+} (C {+} A)
                    C    =   (C {+} A) + (A {+} b)

          which expands to this period-2 system:

                    A = D + E
                    b =  E {+} F
                    C = F + D
                    D = A {+} b
                    E = b + C
                    F = C {+} A

          which is the previous such system, re-labeled. Much symmetry here.

          I have been trying to make a chaotic logistic system, but have yet to get period higher than two. Maybe two is the highest period?