Torus and hyperchaos in 3D Lotka-Volterra map

In this study, we investigate the occurrence of a three-frequency quasiperiodic torus in a three-dimensional Lotka-Volterra map. Our analysis extends to the observation of a doubling bifurcation of a closed invariant curve, leading to a subsequent transition into a state of hyperchaos. The absorption of various saddle periodic orbits into the hyperchaotic attractor is demonstrated through distance computation, and we explore the dimensionality of both stable and unstable manifolds. Various routes to cyclic and disjoint quasiperiodic structures are presented. Specifically we showcase the transition from a saddle-node connection to a saddle-focus connection, leading to the formation of quasiperiodic closed cyclic disjoint curves, as revealed by the computation of one-dimensional unstable manifold. Additionally, we show an unusual transition from a period-two orbit to a period-six orbit and uncover the mechanism related to two subsequent bifurcations: a) subcritical Neimark-Sacker bifurcation, and (b) saddle-node bifurcation. Our approach involves the use of computational methods for constructing one-dimensional manifolds, extending saddle periodic orbits through a one-parameter continuation, and employing a multi-dimensional Newton-Raphson approach for pinpointing the saddle periodic orbits in the three-dimensional map.