Our lives are an ongoing demonstration of this principle. A more rigorous way to express this is that small changes in the initial conditions lead to drastic changes in the results. If the butterfly had not flapped its wings at just the right point in space/time, the hurricane would not have happened. It may take a very long time, but the connection is real.
The Butterfly Effect: This effect grants the power to cause a hurricane in China to a butterfly flapping its wings in New Mexico. By understanding that our ecosystems, our social systems, and our economic systems are interconnected, we can hope to avoid actions which may end up being detrimental to our long-term well-being. For example, by understanding the complex, chaotic dynamics of the atmosphere, a balloon pilot can “steer” a balloon to a desired location. Recognizing the chaotic, fractal nature of our world can give us new insight, power, and wisdom. Many natural objects exhibit fractal properties, including landscapes, clouds, trees, organs, rivers etc, and many of the systems in which we live exhibit complex, chaotic behavior. These phenomena are often described by fractal mathematics, which captures the infinite complexity of nature. 
While most traditional science deals with supposedly predictable phenomena like gravity, electricity, or chemical reactions, Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain states, and so on.
Chaos is the science of surprises, of the nonlinear and the unpredictable.
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