Ghost states underlying spatial and temporal patterns: How nonexistent invariant solutions control nonlinear dynamics

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 ordinary differential equations (ODEs), the influence of such "ghosts" on the temporal behavior of the system, namely delayed transitions, has been studied previously. We consider spatiotemporal partial differential equations (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 spatiotemporally chaotic Kuramoto-Sivashinsky PDE, the buckling of an elastic arc, and 3D Rayleigh-Bénard convection.