About the chaos influence on a system with multi-frequency quasi-periodicity and the Landau-Hopf scenario

The interaction of system demonstrating multi-frequency quasi-periodic oscillations and several steps of the Landau-Hopf scenario with chaotic Ro<spacing diaeresis>ssler system is considered. The quasi-periodic subsystem is a network of five non-identical van der Pol oscillators. It is shown that as the coupling parameter between the subsystems decreases, successive quasi-periodic Hopf bifurcations and doublings of high-dimensional invariant tori are observed. The chaos arising in this system can have several (in our case up to five) additional zero Lyapunov exponents. In case of weak coupling parameter between chaotic and quasi-periodic subsystems, when the coupling parameter of van der Pol oscillators changes, the points at which the attractor transformation occurs are observed. This is a new type of bifurcations that are responsible for a consistent increase in the number of additional zero Lyapunov exponents. As the coupling parameter between chaotic and quasi-periodic subsystems increases, the observed stages of the Landau-Hopf scenario turns out to be resistant to interaction with the chaotic system.