In this paper a method to rigorously compute several non trivial solutions of the Gray-Scott reaction-diffusion system defined on a 2-dimensional bounded domain is presented. It is proved existence, within rigorous bounds, of non uniform patterns significantly far from being a perturbation of the homogenous states. As a result, a non local diagram of families that bifurcate from ...

In this note we show finite-time blowup of radially symmetric solutions to the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any positive mass. We prove this result by slightly adapting M. Winkler’s method, which he introduced in (Winkler in J. Math. Pures Appl., 10.1016/j.matpur.2013.01.020, 2013) for the semilinear Keller-Segel system in dimensions ...

We prove that every topological dynamical system (X,T) has a zero-dimensional principal extension, i.e. a zero-dimensional extension (Y,S) such that for every S-invariant measure ν on Y the conditional entropy h(ν|X) is zero. This reduces the discussion of many entropy-related properties to the zero-dimensional case which gives access to the various useful tools of symbolic ...

This article describes the work of Ki Hang Kim and Fred Roush on questions of decidability in algebra and number theory.

This is a volume focused on symbolic dynamics, to which Kim and Roush contributed so much. The biographical and review articles are evident from their titles. The research papers reflect both recurring themes and the significant developments in recent years: the study of strong shift equivalence (Boyle-Kim-Roush), a staple since the 1973 Annals paper of Williams; multidimensional ...

We look for singlevalued solutions of the squared modulus M of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using Clunie’s lemma, we first prove that any meromorphic solution M is necessarily elliptic or degenerate elliptic. We then give the two canonical decompositions of the new elliptic solution recently obtained by the subequation method.

This article is mainly historical, except for the discussion of integrability and characteristic exponents in Sect. 2. After summarising the achievements of Henri Poincaré, we discuss his theory of critical exponents. The theory is applied to the case of three degrees-of-freedom Hamiltonian systems in (1:2:n)-resonance (n>4). In addition we discuss Poincaré’s mathematical physics, ...

We consider a class of structured cell population models described by a first order partial differential equation perturbed by a general birth operator which describes in a unified way a wide class of birth phenomena ranging from cell division to the McKendrick model. Using the theory of positive stochastic semigroups we establish new criteria for an asynchronous exponential growth ...

The phenomenon of resonance will be dealt with from the viewpoint of dynamical systems depending on parameters and their bifurcations. Resonance phenomena are associated to open subsets in the parameter space, while their complement corresponds to quasi-periodicity and chaos. The latter phenomena occur for parameter values in fractal sets of positive measure. We describe a ...