Ghost states underlying spatial and temporal patterns: how non-existing invariant solutions control nonlinear dynamics

TL;DR

This paper introduces the concept of ghost states in spatio-temporal PDEs, showing how they influence nonlinear dynamics near bifurcations through variational methods and applications to various complex systems.

Contribution

It defines and computes ghost states in PDEs, extending the understanding of invariant solutions' influence on nonlinear dynamics.

Findings

Ghost states can be characterized as minima of cost functions.

Variational methods enable computation and continuation of ghost states.

Ghost states significantly influence the dynamics of complex nonlinear systems.

Abstract

Close to a saddle-node bifurcation, when two invariant solutions collide and disappear, the behavior of a dynamical system can closely resemble that of a solution which is no longer present at the chosen parameter value. For bifurcating equilibria in low-dimensional ODEs, the influence of such 'ghosts' on the temporal behavior of the system, namely delayed transitions, has been studied previously. We consider spatio-temporal PDEs and characterize the phenomenon of ghosts by defining representative state-space structures, which we term 'ghost states,' as minima of appropriately chosen cost functions. Using recently developed variational methods, we can compute and parametrically continue ghost states of equilibria, periodic orbits, and other invariant solutions. We demonstrate the relevance of ghost states to the observed dynamics in various nonlinear systems including chaotic maps, the Lorenz ODE system, the spatio-temporally chaotic Kuramoto-Sivashinsky PDE, the buckling of an elastic arc, and 3D Rayleigh-B\'enard convection.