Diversity of chaos with additional zero Lyapunov exponents in heterogeneous ensemble of quasi-periodic and chaotic systems

We analyze the interaction of a quasi-periodic subsystem, in which two-, three-, and four-frequency quasi-periodic oscillations are possible, with a chaotic subsystem. A discrete version is obtained via the Euler scheme from the corresponding flow model. The dynamics of individual subsystems are discussed, including quasi-periodic bifurcations of invariant tori and the occurrence of two-frequency quasi-periodic structures known as “shrimps”. For the coupled system, chaos with several zero Lyapunov exponents is observed, arising from the doubling of high-frequency tori. Bifurcations of chaotic attractors associated with an increase in the number of zero exponents have been discovered. A two-parameter analysis in the plane of coupling parameters reveals new codimension-2 bifurcations, associated with different bifurcation lines of tori, namely, saddle-node bifurcations, quasi-periodic Hopf bifurcations, and doublings of tori. In the vicinity of such points, quasi-periodic regimes with different numbers of incommensurate frequencies and chaos with different numbers of zero Lyapunov exponents are possible. In the parameter plane of the chaotic subsystem, a quasi-periodic shrimp based on a five-frequency regime is found, and its internal structure is discussed. In the parameter plane of the quasi-periodic subsystem, structures similar in configuration to those of the individual subsystem are observed, but they correspond to chaos with a different number of zero Lyapunov exponents.