TL;DR
A normal-form reduction at critical spectra corresponding to Turing and Hopf bifurcations is developed and a general framework for bifurcation analysis in reaction-diffusion equations with memory and nonlocality is provided and how these effects interact to produce spatiotemporal complexity is clarified.
We study a general class of nonlinear reaction-diffusion equations that model pattern-forming systems. The class includes the Gierer-Meinhardt PDE, Brusselator model, and van der Pol PDE, as well as the Gray-Scott, Klausmeier, Lengyel-Epstein, Schnakenberg PDEs and others of activator-inhibitor type. In the limit in wh...
The model produces the so-called "Turing-before-Tipping"phenomenon demonstrating that the arrangement into spatial patterns is an effective resilience mechanism against harsh conditions, and there is a kernel-dependent dichotomy in pattern behavior.
A delayed reaction–diffusion extension of a restrained modified Brusselator model is studied under homogeneous Neumann boundary conditions. The delay is inserted into the restrained autocatalytic channel and is interpreted as an effective memory generated by intermediate-complex formation, catalytic activation, or resi...
We study linear instabilities and pattern formation in a spatially extended Selkov model for glycolysis with diffusion and chemotactic interactions between two species. Linear stability analysis reveals two distinct bifurcations: a zero-wavevector, finite-frequency Hopf bifurcation leading to spatially homogeneous osci...
PatternFormer is developed, a large language model-based framework for learning the multiple solutions of nonlinear partial differential equations that produces structured solution sets in a single autoregressive pass, terminating automatically for finite families and enforcing physical residual constraints for unbound...
Whereas the effect of monotonically decaying memory kernels on reaction dynamics has been studied extensively, the effect of oscillatory memory, observed for widely different systems, is much less explored. We analyze non-Markovian barrier-crossing dynamics in the presence of a single exponentially decaying oscillatory...
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