We have thus far dealt with systems whose time evolution is described by a set of ODEs. Given initial conditions, we can integrate the ODEs and obtain numerical solutions at any point in the future. When the outcome is uniquely determined by the initial condition, we say the system is deterministic. Newtonian systems such as projectile and planetary motion are deterministic. For the most part (except unstable few-body systems alluded to in Section 4.5), we did not have to worry about any unpredictable results, in the sense that if we propagate the system from two nearby initial conditions, we should expect the final results to be close, at least not catastrophically different.

Starting with Poincaré's work on celestial mechanics, that notion of predictability had been challenged. But the field has rapidly accelerated in the last half century with what Edward Lorenz thought was a computation error when the computer generated results showed a totally unexpected outcome. Since then, numerical computation has led to the discovery of an entirely new field of science, *deterministic chaos*, often considered a glory of computational science. The field of chaos covers everything from physical systems to biological ones, from surprisingly simple models to complex ones [35]. When chaos occurs, the system becomes unpredictable and loses long term predictability, in the sense that if we start the evolution of the same system (same ODEs) with slightly different ...

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