Spiral organization of quasi-periodic shrimp-shaped domains in a discrete predator-prey system

In this paper, we report the discovery of some novel dynamical scenarios for quasi-periodic shrimp-shaped structures embedded within chaotic phases in bi-parameter space of a discrete predator-prey system. By constructing high-resolution, two-dimensional stability diagrams based on Lyapunov exponents, we observe the abundance of both periodic and quasi-periodic shrimp-shaped organized domains in a certain parameter space of the system. A comprehensive comparative analysis is conducted to elucidate the similarities and differences between these two types of shrimps. Our analysis reveals that, unlike periodic shrimp, quasi-periodic shrimp induces (i) torus bubbling transition to chaos and (ii) multistability with multi-tori, torus-chaotic, and multi-chaotic coexisting attractors, resulting from the crossing of its two inner antennae. The basin sets of the coexisting attractors are analyzed, and we observe the presence of intriguing basin boundaries. We also verify that, akin to periodic shrimp structures, quasi-periodic shrimps also maintain the three-times self-similarity scaling. Furthermore, we encounter the occurrence of spiral organization for the self-distribution of quasi-periodic shrimps within a large chaotic domain. We believe that these novel findings will significantly enhance our understanding of shrimp-shaped structures and the intricate dynamics exhibited by their distribution in chaotic regimes.