We study the emergence of strong interactions between
dissipative coherent structures (pulses) in spatially extended systems.
Focusing first on the 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 unravelled, 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
L^2_C-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 multi-pulse systems, that can be quantified in
terms of a positive Lyapunov exponent.