Geometry of complex instability and escape in four-dimensional symplectic maps

TL;DR

This paper explores the geometry and dynamics of complex instability in four-dimensional symplectic maps, revealing how escape mechanisms are governed by invariant manifolds and transport across phase space.

Contribution

It introduces a detailed analysis of complex unstable fixed points in 4D symplectic maps, including visualization techniques and normal-form descriptions of transport mechanisms.

Findings

Escape is governed by transport along unstable manifolds.

Complex instability leads to long-time confinement near fixed points.

Visualization of phase-space slices reveals geometric changes during transition.

Abstract

In four-dimensional symplectic maps complex instability of periodic orbits is possible, which cannot occur in the two-dimensional case. We investigate the transition from stable to complex unstable dynamics of a fixed point under parameter variation. The change in the geometry of regular structures is visualized using 3D phase-space slices and in frequency space using the example of two coupled standard maps. The chaotic dynamics is studied using escape time plots and by computations of the 2D invariant manifolds associated with the complex unstable fixed point. Based on a normal-form description, we investigate the underlying transport mechanism by visualizing the escape paths and the long-time confinement in the surrounding of the complex unstable fixed point. We find that the escape is governed by the transport along the unstable manifold across invariant planes of the normal-form.