I refer to this article on nytimes.com: Every 202,500 Years, Earth Wanders in a New Direction

TLDR: Scientists believe the earth's orbit oscillates between circular and more elliptical due to the gravitational pull from Jupiter (big) and Venus (close). It takes 405k years to complete a cycle.

The big question: could this affect climate-change? Could it make colder winters, hotter summers or just more extreme weathers? Changing Gulf stream?

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    It's important to note that these changes, while real, take place gradually and continuously, over many thousands of years. – jamesqf May 25 at 19:32

Milankovitch cycles do affect climate on Earth. There are several cycles.

"Variations in eccentricity, axial tilt, and precession of the Earth's orbit resulted in cyclical variation in the solar radiation reaching the Earth, and that this orbital forcing strongly influenced climatic patterns on Earth." source: wikipedia

Scientists of Pleistocene's climate evolution strongly correlate Milankovitch cycles with glacial/interglacial periods, affecting the Gulf Stream Current.
Other factors of Earth's internal system, such as volcanic events do affect climate, but changes in the magnitude of solar radiation are a well known climatic variability generator.

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