You can find a solution using this systematic approach:
Define each area with crossroad between the circles to be a single waypoint A B C D E.
Then list all possible ways through the labyrinth, avoid loops, so do not go to a waypoint/color combination you already have in your path.
- Start - Dy (D after yellow) - Eg - Cy - loop (only loop continuations; Eg is already in this path)
- Start - Dy - Ag - By - Bg - Ay - Dg - Ey - exit
- Start - Dy - Ag - By - Bg - Ay - Eg - Cy - loop
- Start - Dy - Ag - By - Bg - Cy - Eg - loop
- Start - Dy - Ag - By - Bg - Eg - Cy - loop
You can see, only the second try leads to the exit. So the only solution path is
Start - D - A - B - green Circle - A - D - E - exit
From this path you can derive any other solution by adding additional loops.
For instance Ey - Cg - Ey ...