A lot will depend on the herd mix. If you have a 1 buck to 3 doe or better (1-2 or 1-1) you will see a lot of chasing (and catching)starting as early as the last week of October and stretching into around Thanksgiving. If your population is more of a mix like some of the Hill Country area (1 - 4,5,6 or more) you'll most likely see the chasing starting around the 1st of November and lasting a week or two until the majority of the does come in to estrus. As a buck locks down with a doe, he'll disappear for a day or so, then find another and repeat.
After the rut is over, some bucks will continue to look and some will just crawl into the bush to recover. Some of those 'bush-bucks' will be found the next spring. About midway through December they start coming back out piling on the food trying to fatten up for January/February.
There may also be a 'second' rut in some of the areas with high doe density where does that didn't get caught the first time will come back into estrus around the end of December.
This all varies according to population densities, make-up and many other factors.
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Rifles are similar to boats and young women...there's no end to how much money you can pour into them without making them any more useful.