Its name is Dendrobium nobile type Hybrid, Registered Name Unknown.

Culture depends a bit on where you are in the world, but you should wait till the flowers are done before dividing or repotting the plant.

Here is the relevant bit cut from the American Orchid Society Culture notes.

"Group 1 Species such as Den. chrysanthum, Den. friedricksianum, Den. nobile and Den.wardianum
Culture Growth period in summer; give warmth, water and fertilize heavily from when roots appear until top leaf appears on canes. Then give high light, little or no water, no fertilizer, cool nights (40 to 50 F). In other words, forget about them."

But why divide it? The bigger they get the more spectacular they get. Also over time these dendrobiums will grow babies on the canes that will become a whole plant in time. Let it get bigger but if it is time to repot then do the whole plant.

Here is my Dendrobium Love Memory 'Fizz' about 3 weeks ago, now the first flowers are fading. It is very fragrant, kind of like aftershave I think, love it!

Your flowers look just about the same so you might have the same hybrid. And look how nice of a plant it can make if you don`t divide. Mine is outside year round here in Houston unless they predict a freeze. Hope that helps.

Wow! Yours is so beautiful. I also thinks they're the same and I'm happy thinking about having the same result later