This book presents a new theory on the transition to dynamical chaos for two-dimensional nonautonomous, and three-dimensional, many-dimensional and infinitely-dimensional autonomous nonlinear dissipative systems of differential equations including nonlinear partial differential equations and differential equations with delay arguments. The transition is described from the Feigenbaum cascade of period doubting bifurcations of the original singular cycle to the complete or incomplete Sharkovskii subharmonic cascade of bifurcations of stable limit cycles with arbitrary period and finally to the...