The significance of this question is profound and challenges the classical mechanics, deterministic world view and understanding of nature posed by Isaac Newton (the "clockwork universe").
The Earth's atmosphere is considered unstable, meaning that the weather has a high degree of variability through distance and time.
To read more about atmospheric instability: https://en.wikipedia.org/wiki/Atmospheric_instability
Here are pictures from the 139th meeting the American Association for the Advancement of Science (AAAS):
AAAS is an international non-profit with the stated goals of promoting cooperation among scientists, defending scientific freedom, encouraging scientific responsibility, and supporting scientific education and science outreach for the betterment of all humanity. It is the world's largest general scientific society, with over 120,000 members. https://en.wikipedia.org/wiki/American_Association_for_the_Advancement_of_Science
The following discussions seems to focus on "scale dependent" stability. The stability and its measures such as Lyapunov exponents and eigenvalues of the Jacobian etc. depend on the size of the perturbation. How can these size dependent stability measures be mathematically defined?
Some of these approaches I see are notions of the Scale Dependent Lyapunov Exponents and multi-scale analysis.
As long as the system is stable for perturbations of an intermediate size, intuition would point to the unlikeliness of a butterfly's flutter inducing a tornado.
In 1972, Lorenz stated that “if the flap of a butterfly’s wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado.”
In 2008, the last year of Lorenz's life, he said this when asked whether a butterfly can really cause a tornado. “Even today I am unsure of the proper answer”. The question is valuable because it begs the bigger question that nature is highly sensitive to infinitessimal changes. “The idea has now entered the everyday vision of many scientists across all disciplines,” says Daniel Rothman, a professor of geophysics at MIT. “They understand that some things are chaotic, and that there’s exponential divergence from initial conditions. They may not voice it, but they know it because it’s in the air. That’s the sign of a great achievement.” For a great writeup about the Butterfly Effect and Edward Lorenz: https://www.technologyreview.com/s/422809/when-the-butterfly-effect-took-flight/
Edward N. Lorenz was an American mathematician, meteorologist, and a major pioneer of chaos theory. Edward had a background in math, but after serving as a meteorologist for the United States Army Air Corps during World War 2, he decided to study meteorology at MIT, and later was a professor in their department for the rest of his career.
In the 1950s, Lorenz became skeptical of the appropriateness of the linear statistical models in meteorology and began to realize that most atmospheric phenomena involving weather forecasting was non-linear and depended on initial starting conditions. This thinking led Lorenz to discover the field of chaos theory, when in his 1963 paper "Deterministic Nonperiodic Flow" in the Journal of the Atmospheric Sciences, he states:
Two states differing by imperceptible amounts may eventually evolve into two considerably different states ... If, then, there is any error whatever in observing the present state—and in any real system such errors seem inevitable—an acceptable prediction of an instantaneous state in the distant future may well be impossible....In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-long-range forecasting would seem to be nonexistent.
For more on Edward N. Lorenz: https://en.wikipedia.org/wiki/Edward_Norton_Lorenz