In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813–857, 2012) and Li and Pu...

In this paper we prove existence of global strong-weak two-dimensional solutions to the Navier-Stokes and heat equations coupled by the external force dependent on temperature and the heat dissipation, respectively. The existence is proved in a bounded domain with the Navier boundary conditions for velocity and the Dirichlet boundary condition for temperature. Next, we prove...

A 4-particles chain with different masses represents a natural generalization of the classical Fermi-Pasta-Ulam chain. It is studied by identifying the mass ratios that produce prominent resonances. This is a technically complicated problem as we have to solve an inverse problem for the spectrum of the corresponding linearized equations of motion. In the case of such an...

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...

A superintegrable system has more integrals of motion than degrees d of freedom. The quasi-periodic motions then spin around tori of dimension n<d. Already under integrable perturbations almost all n-tori will break up; in the non-degenerate case the resulting d-tori have n fast and d−n slow frequencies. Such d-parameter families of d-tori do survive Hamiltonian perturbations as...

One of the fundamental problems in control theory is that of controllability, the question of whether one can drive the system from one point to another with a given class of controls. A classical result in geometric control theory of finite-dimensional (nonlinear) systems is Chow–Rashevsky theorem that gives a sufficient condition for controllability on any connected manifold of...

The Korteweg-de Vries (KdV) equation is widely recognized as a simple model for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of fluid. While solutions of the KdV equation describe the shape of the free surface, information about the underlying fluid flow is encoded into the derivation of the equation, and the present article focuses on the...

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...

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 at...

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.

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...

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...