|(quoted from A Sensitive Dependence on Initial Conditions)|
So the covering law model is amended yet again. Explanations still require laws, but there are not laws for every event. The task of historical explanation becomes the act of making distinctions, between those parts of an event that can be explained by laws, and those that cannot. The component events that combine to create an explanandum are analyzed each in turn, and the historian then concentrates on the explicable components.
Paul Tibbets flies toward Hiroshima. The nomad steps out of his yurt.
Lyapunov exponents are numbers that measure the conflicting effects of stretching, contracting, and folding in the phase space of an attractor. They set the topological parameters of unpredictability. An exponent greater than zero means stretching, so that each alternative history moves farther and farther apart as time passes. An exponent smaller than zero means contraction, so that alternatives tend to come back together. When the exponent equals zero, a periodic orbit results.
What is history's Lyapunov exponent? This is the law that no one can know.
Frank January flies toward Hiroshima. The nomad stops in his yurt.