Coherent Structure Interactions in Spatially Extended Systems Driven by Excited Hidden Modes

Abstract

We study the emergence of strong interactions between dissipative coherent structures (pulses) in spatially extended systems. Focusing first on a prototypical model problem from fluid dynamics, that of liquid film flowing down a vertical plane, we show that under certain conditions, a two-pulse system undergoes a transition from a regime of decaying oscillatory dynamics to one with self-sustained oscillations. Intriguingly, such a transition is not governed by the standard Hopf bifurcation. Instead, a novel governing mechanism for transition to oscillatory dynamics is unraveled via a peculiar and atypical Hopf bifurcation in which a complex-conjugate resonance pair crosses the imaginary axis in the complex plane. Prior to crossing the essential spectrum (including at the bifurcation point), this resonance pair does not appear in the standard LC2-based spectral analysis but reveals itself when appropriate weighted functional spaces are used. We show that such a resonance pair originates from the splitting of a resonance pole of the single-pulse system. While this object is not part of the classical spectrum, it plays a vital role in shaping the system’s dynamics. We further demonstrate that this resonance-pole mechanism extends to a broad range of systems. Specifically, in the generalised Kuramoto-Sivashinsky equation—a model prototype applicable across a wide range of fields from fluid dynamics to geophysics and plasma physics—we observe the same bifurcation and resulting oscillatory pulse interactions. By contrast, in the FitzHugh-Nagumo model—a central model prototype in reaction-diffusion systems—the resonance pole splits into real eigenvalues, and monotonic pulse interactions occur. In addition, we illustrate that the resonance pole may induce oscillatory interactions in three-pulse systems and eventually lead to chaotic dynamics in strongly interacting multipulse systems, which can be quantified in terms of a positive Lyapunov exponent. Published by the American Physical Society 2025