  Transition state theory is a standard framework for predicting the rate of a
chemical reaction. Although the transition state theory has been successfully
applied to numerous chemical reaction analyses, many experimental and
theoretical studies have reported chemical reactions with a reactivity which
cannot be explained by the transition state theory due to dynamic effects.
Dynamical systems theory provides a theoretical framework for elucidating
dynamical mechanisms of such chemical reactions. In particular, reactive
islands are essential phase space structures revealing dynamical reaction
patterns. However, the numerical computation of reactive islands in a reaction
system of many degrees of freedom involves an intrinsic challenge -- the curse
of dimensionality. In this paper, we propose a dimensionality reduction
algorithm for computing reactive islands in a reaction system of many degrees
of freedom. Using the supervised principal component analysis, the proposed
algorithm projects reactive islands into a low-dimensional phase space with
preserving the dynamical information on reactivity as much as possible. The
effectiveness of the proposed algorithm is examined by numerical experiments
for H\'enon-Heiles systems extended to many degrees of freedom. The numerical
results indicate that our proposed algorithm is effective in terms of the
quality of reactivity prediction and the clearness of the boundaries of
projected reactive islands. The proposed algorithm is a promising elemental
technology for practical applications of dynamical systems analysis to real
chemical systems.
