# Resonant Tori, Transport Barriers, and Chaos in a Vector Field with a Neimark-Sacker Bifurcation

## TL;DR

This paper numerically investigates a three-dimensional dissipative vector field undergoing a Neimark-Sacker bifurcation, analyzing the invariant torus, manifold embeddings, and their roles in transport barriers and global bifurcations.

## Contribution

It provides a detailed numerical analysis of the invariant torus and manifold structures in a system with a Neimark-Sacker bifurcation, highlighting their influence on global dynamics.

## Key findings

- Identification of the invariant torus and its parameter-dependent behavior.
- Analysis of stable and unstable manifold embeddings as transport barriers.
- Insights into the role of manifolds in global bifurcations.

## Abstract

We make a detailed numerical study of a three dimensional dissipative vector field derived from the normal form for a cusp-Hopf bifurcation. The vector field exhibits a Neimark-Sacker bifurcation giving rise to an attracting invariant torus. Our main goals are to (A) follow the torus via parameter continuation from its appearance to its disappearance, studying its dynamics between these events, and to (B) study the embeddings of the stable/unstable manifolds of the hyperbolic equilibrium solutions over this parameter range, focusing on their role as transport barriers and their participation in global bifurcations. Taken together the results highlight the main features of the global dynamics of the system.