The Central Island - Diagonal Symmetries

The Central Island - Diagonal Symmetries

The circular orbits are observed to have 6 different types of symmetry. There is a D2 (rectangular) symmetry of the map, generated by (x, y) -> (y, x) and (x, y) -> (-x, -y). This D2 symmetry about the origin induces a D2 symmetry about the corner of the rhombus, which is generated by two reflections that are shown in the figure, namely the upper-right (UR) diagonal, and the down-right (DR) diagonal. The circular orbits are classified and color-coded as to whether they:
    (T) Touch the line of symmetry,
    (S) are Symmetric under the reflection but don't touch, or
    (N) are Not symmetric, in which case there is a pair of orbits.
All of the pairs observed here are obtained from the pair of initial conditions (x, y) and (-x, -y). Note: If we had observed orbits with symmetry type (N,N), then we would have to classify the whether the orbit was S or N under the inversion about the center of the figure. 
UR   DR   Color(s)
T    S    Black
S    T    Blue
S    S    Brown
N    T    Red & Green
T    N    Cyan & Magenta
S    N    Purple & Orange
The `chaotic' orbits whose initial contions are given in `The Central Island' all have symmetry (T,T), meaning they touch both lines of reflection.

Click on the figure (or here) to see `local' reflection symmetries.

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