Find fixed point x n + 1 := 0.5 ⋅ ( x n 2 + 2 ⋅ y n − 2 ⋅ x n ⋅ y n − 2 ⋅ y n 2 ) y n + 1 := − 0.8 ⋅ ( x n 2 + 2 ⋅ y n − 2 ⋅ x n ⋅ y n − 2 ⋅ y n 2 ) How to find x and y?

Question

Find fixed point                              x                      n            +            1                                              :=        0.5        ⋅        (                  x                  n          2                +        2        ⋅                  y                  n                −        2        ⋅                  x                  n                ⋅                  y                  n                −        2        ⋅                  y                  n          2                )                                      y                      n            +            1                                              :=        −        0.8        ⋅        (                  x                  n          2                +        2        ⋅                  y                  n                −        2        ⋅                  x                  n                ⋅                  y                  n                −        2        ⋅                  y                  n          2                )            How to find   x and   y?

Oswaldo Rosales · Accepted Answer

Here is a start. Multiply the first eq. by .8 and the second by .5 and add them to get the nice relation      y          n      +      1        =  −      8    5        x          n      +      1        .Substitute in the first eq. for       y    n   and solve for       x    n  . I think you can take it from here.After substituting for       y    n   in the first equation, assume       lim          n      →      ∞            x    n    =  x which implies       lim          n      →      ∞            x          n      +      1        =  x and solve the quadratic equation in   x. To find   y use the above relation.

dresu9dnjn · Accepted Answer

since you want a fixed point, drop the subscripts and solve. But, first notice that, ratioing the two expressions gives   y  =  −            8      x        5  . So, replace   y in the first equation. After simplification, the equation becomes a very simple quadratic equation in   x that you will not have any problem to solve. Do not forget that you were already told that   (  0  ,  0  ) is alreasy a fixed point; so   x should probably be factored out and let you with a simple linear equation in   x. When you have   x, compute   y according to what we wrote at the beginning.

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