Ok. (You're probably not gonna like this)
1. First of all, you need to make sure that the system you want to turn anarchy (I'll call this the Target System from now on) has 6 factions or less. If it doesn't you'll need to make a non-native faction of the Target System retreat by forcing them below 2.5% and keeping them there. Let's assume you're lucky and there are 6 or less.
2. Then you need to find another system (I'll call this the Expansion System) really close to your Target System that has an anarchy faction (ideally the closest one there is) You need to check that the nearest system to the Expansion System with 6 or fewer factions is the Target System. If there is another system closer that has 6 factions or less, the expansion will go there instead. Again, let's assume that you are lucky and the Expansion system anarchy faction will expand into the Target System.
3. You need to raise the anarchy faction up until it becomes the controlling faction through a conflict with the current controlling faction. You can do this by equalising with the controlling faction, or raising the influence above 60%.
4. Once your anarchy faction is in control, you need to raise it's influence to 70%+ (we usually have to get them above 75%) to trigger an expansion.
5. Wait 5 days pending. Wait 5-7 days of active expansion. Your anarchy faction will appear in the Target System at about 8-10%.
6. Raise the anarchy faction in the target system until it's the controlling faction as in step 3.
Simples. [knocked out]