
Growing perennial Paperwhites
Paperwhites are a less cold hardy cousin to our local daffodils, which grow perennially outside. They are only cold hardy to zone 8, so they won't survive our winter's outside. Their relationship to daffodils tells us (local gardeners) two things: the first being their natural growing season, the second being that they are toxic to consume and their freshly cut stems can release irritating sap that can irritate your skin and kill other freshly cut stems if they share the same water. These bulbs are often sold forced in pots with stones and water, but this growing technique does nothing but stimulate already developed energy in the bulb, if you want to grow your bulbs again next year, you'll want to make sure they have food this season, so they can store energy for next spring.
I pot my paperwhites in shallow bowls, so the bulbs sit right at the top of the bowl water line and the bowl serves as a resevoir for the roots. The roots will hold the bulbs into the bowl, but the level will mean that you can make sure the bulb doesn't get emersed in water. Bulbs like to stay high and dry, and many are prone to rot when they get soggy.
As the plants begins to grow and flower, I make sure to water enough to keep the soil saturated. The flowers will last longer if the plants are kept out of full sun at this point, so I put them somewhere nice to enjoy them in bright indirect light. As the paperwhite flowers, it continues growing longer stems and foliage, so I like to add branches and string to support the tidiness of this growth. You can either support further top growth by adding more string or tightening the string towards the top of the plant to gather the loose foliage into a cone shape. The flowers are beautiful even as they dry back and die. I often save mine for little eternity vase decorations around the house. If you want to save your flowers you can let them, dry fully on the plant, then pinch off the stem as far back as the base of the bulb (or anyl length you want for the vase you'll display them in dry.)
When your paperwhite foliage is at full length, this is the peak growing season for your bulbs to take in and store energy. More them to a sunny place, and continue to water them. I like to add a nice liquid plant food to the water, every other week. About a month in, you will notice your paperwhite foliage becoming a little thinner and starting to yellow. Keep watering lightly, until the foliage dies back naturally, and then your bulbs are ready to be stored in a basement or garage (somewhere frost proof, where you can retrieve them to bloom again next winter).

Growing perennial Amaryllis
Amaryllis originate from South Africa. So their growing season needs are bit less conventional than paperwhites. When forcing them in a northern climate, and they will need a period of forced dormany.
This year our amayllis bloomed completely before sending their leaves up to restore the energy they spent on their spectacular floral displays. I can't believe how floriferous they were this year, most of my stems had 3 to 4 blooms and most of my plants threw out at least two stems.
The amaryllis bloom best, when their roots are constricted, so we pot them in shallow bowls, which makes it important to water regularly, during their winter growing season.
As with our paperwhites, amaryllis will need to be cared for, given a sunny position, water and food, throughout the remainder of the winter, spring and summer. I like to pot up my bulbs into deeper pots after they flower, so they hold water for longer and have more food available in their growing medium. Make sure the pot has drainage holes and dish beneath is to catch excess water. At this point, water regularly as you would any house plant. Don't oversaturate the soil, but don't let it dry out for too long either. It helps to feed the bulb every few times you give it water, or dillute feed it with every watering.
I force a 2 month period of dormancy beginning in August, when I stop watering the bulb and move it to a dark place that stays between 45 and 50 degrees. You can take your bulbs completely out of the soil and cut off yellow leaves at this point. A plastic bag with insulation (woodshavings) in a fridge crisper or root cellar would work well for the dormancy period. Your amaryllis bulb should never be exposed to temperatures under 40 degrees.
At the end of October, I pull my bulbs out of dormancy and pot them up again.
Pro tip: Amaryllis leaves can be very delicate, so, during the growing period, be careful not to move them around too much.
Let me know how you do?