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This article should help: https://www.gardeningknowhow.com/ornamental/flowers/jasmine/growing-jasmine-plants.htm

This should help: https://www.gardenguides.com/85865-plant-night-blooming-jasmine-seeds.html
and: http://www.seedman.com/fragrant.htm

Depending on the size of your arbor, you might want to purche another set and plant on both sides of the arbor. This way, they will meet at the top and your arbor will be evenly covered. Don't worry too much about trying to intertwine the stems, they will do this naturally. This article should help: https://www.gardeningknowhow.com/ornamental/flowers/jasmine/growing-jasmine-plants.htm

The plant could be Carolina Jasmine. This fact sheet shows pictures: https://aggie-horticulture.tamu.edu/ornamentals/nativeshrubs/gelsemiumsemper.htm

Summer jasmine is best pruned just after flowering, in late summer or early autumn. Winter jasmine should be pruned in spring, immediately after flowering. On both types you can cut back flowered stems to a strong sideshoot and thin out crowded, crossing or misplaced branches. You should also remove any weak or thin stems. Overgrown vines can be cut back to about 2 feet from the ground.

That is rather odd. I suspect that there is a combination of pruning and fertilizer that may be causing this.

When you prune the plant, you are cutting away much of the type of growth the plant blooms on, and it takes about a year for it to regrow. Since you are pruning in the cooler months, it has to wait nearly a year to be able to rebloom - and thus the cycle is repeated. If you can, wait until the spring to prune or don't prune at all this year to get the plant back on its normal cycle.

Second, if you are like most people, you probably cut back on your fertilizer in the cooler months. You may be using a fertilizer that is too heavy on the nitrogen. Too much nitrogen will result in a lush, fast growing plant that has few blooms. Then, when you cut back on your fertilizer, the plant is getting less nitrogen and is able to bloom again. If you are fertilizing, look for a fertilizer that is a little heavier in phosphorous than in nitrogen and that will help with blooming as well.