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Chaos, Dynamics, and Fractals by Joseph L. McCauley

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1

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) = (x1(t), . . . , xn(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(t0) at an earlier time t0 < t. In other words, if X(t) occurs then X(t0) had to have occurred earlier. As can happen in a nonlinear system, nonuniqueness can arise (one initial condition X(t0) 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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