| Abstract: | The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations,
including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential
equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional
unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional
unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge
between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows
the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples
and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations. |
