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We take on a Random Matrix theory viewpoint to study the spectrum of certain reversible Markov chains in random environment. As the number of states tends to infinity, we consider the global behavior of the spectrum, and the local behavior…
We consider principal properties of various wave regimes in two selected excitable systems with linear cross-diffusion in one spatial dimension observed at different parameter values. This includes fixed-shape propagating waves, envelope…
Spiral waves are self-repeating waves that can form in excitable media, propagating outward from their center in a spiral pattern. Spiral waves have been observed in different natural phenomena and have been linked to medical conditions…
The random diffusion model is a continuum model for a conserved scalar density field driven by diffusive dynamics where the bare diffusion coefficient is density dependent. We generalize the model from one with a sharp wavenumber cutoff to…
Spiral wave chimeras (SWCs), which combine the features of spiral waves and chimera states, are a new type of dynamical patterns emerged in spatiotemporal systems due to the spontaneous symmetry breaking of the system dynamics. In…
Pattern formation mechanisms of a reaction-diffusion-advection system, with one diffusivity, differential advection, and (Robin) boundary conditions of Danckwerts type, are being studied. Pattern selection requires mapping the domains of…
We consider a reaction-diffusion process with retardation. The particles, immersed in traps initially, remain inactive until another particle is annihilated spontaneously with a rate $\lambda$ at a certain point $\vec x$. In that case the…
A computational linear stability analysis of spiral waves in a reaction-diffusion equation is performed on large disks. As the disk radius R increases, eigenvalue spectra converge to the absolute spectrum predicted by Sandstede and Scheel.…
We follow the trajectories of phase singularities at nulls of intensity in the speckle pattern of waves transmitted through random media as the frequency of the incident radiation is scanned in microwave experiments and numerical…
Rotors of reaction and diffusion, that revolve around a circular singularity, and the corresponding spiral wave activity around it, influence the nature of several excitable media. Their dynamics are known to be affected by target waves and…
The nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter. We show that on a suitable coarse scale this map generates an oscillating…
The dynamics of many natural systems is dominated by non-linear waves propagating through the medium. We show that the dynamics of non-linear wave fronts with positive surface tension can be formulated as a gradient system. The variational…
The diffraction spectrum of coherent waves scattered from fractal supports is calculated exactly. The fractals considered are of the class generated iteratively by successive dilations and translations, and include generalizations of the…
Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological…
We consider a single species reaction diffusion system on a two dimensional lattice where the particles $A$ are biased to move towards their nearest neighbours and annihilate as they meet; $A + A \to \emptyset$. Allowing the bias to take…
We consider general reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. We show that travelling wave solutions to such systems that propagate in rational directions are nonlinearly stable under small…
In many oscillatory or excitable systems, dynamical patterns emerge which are stationary or periodic up in a moving frame of reference. Examples include traveling waves or spiral waves in chemical systems or cardiac tissue. We present a…
In many physical situations involving diverse length scales, waves or rays representing them travel through media characterized by spatially smooth, random, modest refactive index variations. "Primary" diffraction (by individual…
Mathematical modeling is now used commonly in the analysis of signaling networks. With advances in high resolution microscopy, the spatial location of different signaling molecules and the spatio-temporal dynamics of signaling microdomains…
We identify a new type of pattern formation in spatially distributed active systems. We simulate one-dimensional two-component systems with predator-prey local interaction and pursuit-evasion taxis between the components. In a sufficiently…