English: This figure shows an orbit on phase plane of Gumowski-Mira map that is given by the following 2-dimensional discrete dynamical system:

${\displaystyle x_{n+1}=y_{n}+\alpha y_{n}(1-\sigma y_{n}^{2})+g(x_{n}),}$

${\displaystyle y_{n+1}=-x_{n}+g(x_{n+1}).}$

Where,

${\displaystyle g(x)=\mu x+{\frac {2(1-\mu )x^{2}}{1+x^{2}}}.}$

The used parameters are α = 0.008, σ = 0.05, μ = 1.00004, the used initial values are x₀ = 0.01, y₀ = 0.01, and the iteration number is 100000 for plotting. The orbit on the figure finally converges to the stable 3-periodic point,

${\displaystyle (691683.0674,72.87648721),\ (691629.3758,-26.0264592),\ (691637.8581,36.14774404).}$

However, interestingly, the orbit of Gumowski-Mira behaves like the bifurcation diagram of Logistic map until it converes to the above 3-periodic point. The figure shows the transient behaviour until the periodic point.

The figure was made with the following reference:

• 高橋 純・増田 尚美・山田 泰司・合原 一幸、1999、「GumowskiとMiraの写像の過渡的振舞い」、『電子情報通信学会論文誌. A, 基礎・境界』82巻10号、電子情報通信学会、NAID 110003313432 pp. 1664–1667