We present a route to high-dimensional chaos, that is, chaos with more than
one positive Lyapunov exponent. In this route, as a system parameter chang
es, a subsystem becomes chaotic through, say, a cascade of period-doubling
bifurcations, after which the complementary subsystem becomes chaotic, lead
ing to an additional positive Lyapunov exponent for the whole system. A cha
racteristic feature of this route, as suggested by numerical evidence, is t
hat the second largest Lyapunov exponent passes through zero continuously.
Three examples are presented: a discrete-time map, a continuous-time flow,
and a population model for species dispersal in evolutionary ecology.