If it were me, I would get that stuff off of the walls and out of the yard ASAP. I've lived with it when I rented and I've seen what it can do to wood siding, window screens, gutters, eaves, and how fast it can happen. I also don't like spiders and spiders LOVE to live in ivy. That stuff was tearing apart a low rock wall, too by exploiting every little crack in the mortar. Obviously this mortar needed some maintenance, but it was necessary to remove the ivy to do so.

What kind of wall is it? Even brick, I still wouldn't let ivy grow because you can't have just a little bit. It will never stop going up and out, and any which way it can. If you want to keep it, there's not really any necessary maintenance except to try to keep it from eating the window screens and door jambs. It's cold enough here, and probably there, to turn the leaves ugly in the winter. It will green up in the summer.

Ivy can be very invasive, but some varieties are less aggressive than others.

I have a varigated ivy planted to grow on lattice on the foundation of my home and love it. It hasn't created any problems, however it does have to be kept in check. I just wack it off where it doens't belong and it needs to be done at least once a year in this area. I don't allow it to grow under the siding or on top of it.

I also have a larger ivy growing as a ground cover and it get weed wacked about two times per year (enforcing edges and cutting down to reduce height and make look nice). It looks a little ragged for a couple weeks and then snaps back out of it. I have found this gets rid of the big leaves and exposes nicer smaller ones.

I don't understand if the two plants are in pots or in ground. If your home is regular siding I'd get it off the house. It adheres by feelers and will make a mess of things. I have seen it grow on many old homes with cement/brick and it looks lovely when kept in bounds. You might do a seach and see what you can find about ivy and brick, etc. Find out if it damages.

If you don't remove it I would use shears or hedge trimer and cut it back and make it short but still looking good. New growth will fill in and look nicer. I have never had my ivy turn brown though. sounds like it might be dying out.