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

Area of Science:

• Dynamical Systems Theory
• Nonlinear Dynamics
• Computational Physics

Background:

• Dynamical systems near bifurcations exhibit complex behaviors influenced by disappearing invariant solutions, termed 'ghosts'.
• Previous work focused on the influence of ghosts on temporal dynamics in low-dimensional ordinary differential equations (ODEs).
• The phenomenon of ghosts in spatiotemporal partial differential equations (PDEs) remains less explored.

Purpose of the Study:

• To characterize and define 'ghost states' in the context of spatiotemporal PDEs.
• To develop methods for computing and tracking ghost states of various invariant solutions.
• To demonstrate the relevance of ghost states in diverse nonlinear systems.

Main Methods:

• Defined ghost states as minima of cost functions in state space.
• Employed variational methods for computing and parametrically continuing ghost states.
• Applied methods to equilibria, periodic orbits, and other invariant solutions.

Main Results:

• Successfully computed and continued ghost states for various invariant solutions.
• Demonstrated the relevance of ghost states in explaining observed dynamics.
• Illustrated the phenomenon across diverse systems, including chaotic maps, ODEs, PDEs, and physical models.

Conclusions:

• Ghost states are a significant feature of nonlinear dynamical systems, particularly in spatiotemporal PDEs.
• The developed variational methods provide a powerful tool for analyzing ghost states.
• Understanding ghost states is crucial for comprehending complex dynamics and delayed transitions in various scientific domains.