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**Flows in phase space**

**1.1 Determinism, phase flows, and Liouville’s theorem**

We begin with the general idea that the state of a dynamical system is defined by *n* variables at time *t, X*(*t*) = (*x*_{1}(*t*), . . . , *x*_{n}(*t*)). Geometrically, this is a point in an *n*-dimensional space called phase space. By determinism, we mean that there is a definite rule whereby, if the state *X*(*t*) occurs, then it arose by a sequence of steps from a definite state *X*(*t*_{0}) at an earlier time *t*_{0} < *t.* In other words, if *X*(*t*) occurs then *X*(*t*_{0}) had to have occurred earlier. As can happen in a nonlinear system, nonuniqueness can arise (one initial condition *X*(*t*_{0}) may give rise to two or more distinct solutions),^{1} but this is only a complication. The main point is that there is no ...

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