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We define a Lyapunov exponent for a class of quantum dissipative systems which in the classical limit can undergo a cascade of period-doubling bifurcations into chaos. We do so by computing the average of a functional over a semiclassical trajectory for a dynamical system whose Poincar\'e section corresponds to the H\'enon map. In the strongly dissipative limit we establish a scaling law which determines the way in which chaos can set in for finite values of Planck's constant.
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Cerdeira et al. (1988) studied this question.
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