Is it viable for a highly elliptic orbit to rotate around one of its foci?

I was experimenting with various star system designs with a gravity simulator, and one of the patterns I ended up with is best described as a planet with a very elongated elliptic orbit that itself rotates around an axis located at/near one of the orbit's foci. This gif illustrates it pretty accurately:

I believe I created this by simulating a low-mass body moving near the large-mass body and falling into orbit, but at a slight angle so as to create an offset that shifted the orbit every revolution. Would this be possible with an actual planet and a star? If so, would there be any noteworthy side effects aside from the obvious (e.g. having extremely varying climates at different points during the year)?

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Is it viable for a highly elliptic orbit to rotate around one of its foci?

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