So AP Udyr is incredibly strong. I know this sounds like a troll build, but it isn't. When built properly, AP Udyr has massive single target / AOE burst, good defense, and great mobility. With the new adjustments to Udyr's R, attack speed is no longer necessary to do good damage, as his R procs immediately. Rather, cooldown reduction is better, as it lets you cycle between your abilities much faster. You can easily apply the q dot, r aoe, e stun, and w shield at the right pace with just cdr boots.

When playing AP Udyr, you want to make sure to maximize Phoenix stance and Turtle Stance first. This is where most of your damage will come from as well as survive ability. Build order wise, start with Dorans Ring for mana sustain and a boost to your AP. Rush Lich Bane next. When coupled with your Phoenix Stance you will do a big first hit and then moderate damage over time from your pulsing flames. The nice thing about this too is that it gives you damage even while defending. When you switch you your shield, you will activate Lich again.

Your general skill order should be Turtle right before a fight to bring up shields, Bear stance to rush in and stun while your shield persists, Phoenix to apply your AOE, and Q to apply your final dot. Repeat this order when done, or cycle between Phoenix and Turtle. All your damage comes off your Phoenix, as your Phoenix stance scales with AP, so make sure this is your primary stance used for damage. Do not focus on Tiger, as Tiger is only for the added dot effect. While in Phoenix stance, do not feel compelled to stay in the stance until the third hit, this just wastes time. Rather switch through your stances as quickly as possible. By cycling stances, you proc Lich more frequently, and with a 4.9 sec CD, you will most likely be able to re activate Phoenix again before you would through auto attacks. The only time you can would want to stay in Phoenix and proc it on third hit is while farming creeps in order to conserve your mana.

To maximize the effectiveness of AP Udyr, you want to increase AP as much as possible while also increasing resistances. This is because AP not only increases your damage, but it also increases the shield size of Turtle stance. In this manner, AP is used for both offense and defense. Building health is not needed and is actually discouraged. As AP Udyr, you will not be able to heal very well as you have no single target AP damage for spell vamp and no means of proc'ing lifesteal efficiently. Therefore, when you lose health, it is gone for the duration of the fight. This is where your shield comes in. At 500/550 shield by end game, this is what should be soaking up all your damage and on a 4.5 sec CD with blue buff, it is highly spammable. Therefore, to maximize the efficiency of your big shield, you want resistances. Abyssal Scepter is a must, as it increases your AP, your resistances, and makes your AOE more efficient. To this extent, Zhonyas is also a very good pick as it gives you AP and armor, plus an emergency stasis. Rabadons and Lich is also a must to ensure you have proper damage. Picking CDR boots is also the best option, as spamming your abilities is better than magic pen. As for you last item, see the next chapter.

Generally I like to finish my last item with Runic Bulwark. It offers good team support, health, and rounds off your resistances to bring both Magic Resist and Armor above 150. This makes your shield very strong. If you are fighting a very AP heavy team, Spirit Visage is nice for the added MR, health, and CDR. Nashors can be used as well if you want to proc Phoenix more frequently as well. If the enemy team is more AD heavy, both Sunfire and Randuins are good options, as well as Thornmail. If you do feel like you want some health (which I don't recommend), Warmogs and Rod of Ages could both work. If you need CC, Rylais can also serve the function. Basically, your last item truly is a institutional, however, I believe Runic Bulwark is your best bet.