Certified GKH Gardening Expert
This is standard behavior for zoysia and is the reason it is not used much in areas that get cold. The brown patches are where the grass is going dormant when the weather starts to get colder. The green grass is likely not zoysia, but another type of grass. Sometimes, in colder areas, zoysia is mixed with a grass that does not go dormant so that the lawn still looks nice in the fall and winter. But, over time, the two grasses compete and push each other out, so in some areas you will get patches of the non-dormant grass and in others patches of zoysia.
You can try overseeding with another kind of grass in the spring, but eventually the two grasses will go back to patches. Other than that, there is no way to keep the zoysia from browning in the fall if you live somewhere where it gets cold.