Resonant tori, transport barriers, and chaos in a vector field with a Neimark–Sacker bifurcation

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.

Periodic forcing of a heteroclinic network

“…In the context of dissipative vortex dynamics Fleurantin and James have studied in [9] the Langford system, a one-parameter family of three-dimensional vector fields. The flow of this model exhibits a sink, two saddle-foci of different Morse indices and a non-trivial periodic solution with a complex conjugate pair of Floquet exponents.…”

“…Our route to chaos from an attracting network is different from the routes described by [14] and [5], where the authors use coupled oscillators. Another different itinerary has been described by [9] in the context of the Langford system. These works are discussed in Section 7.…”

Computer assisted proofs of two-dimensional attracting invariant tori for ODEs

“…For all α ∈ [0, 0.95] there exists a fixed point in Σ of P 2 which corresponds to a periodic orbit τ of the ODE. The periodic orbit has complex conjugate Floquet multipliers which are stable for small α but which later cross the unit circle, loosing stability in a Neimark-Sacker bifurcation [57], which occurs at α ≈ 0.69714 and gives birth to a C k torus. We give a plot of such torus for one of the parameters in Figure 8.…”