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For a class of Hamiltonian systems in R 4 the set of homoclinic and hete-roclinic orbits which connect saddle-focus equilibria is studied using a vari-ational approach. The oscillatory properties of a saddle-focus equilibrium and the variational nature of the problem give rise to connections in many homotopy classes of the connguration plane punctured at… (More)
The extended Fisher Kolmogorov equation, u t = ?u xxxx + u xx +u?u 3 , > 0, models a binary system near the Lifshitz critical point and is known to exhibit a stationary heteroclinic solution joining the equilibria 1. For the classical case, = 0, the heteroclinic is u(x) = tanh(x= p 2) and is unique up to the obvious symmetries. We prove the conjecture that… (More)
We provide upper estimates on the spectral radius of a directed graph. In particular we prove that the spectral radius is bounded by the maximum of the geometric mean of in-degree and out-degree taken over all vertices.
We construct a diffeomorphism of the two-dimensional torus which is isotopic to the identity and whose rotation set is not a polygon.
We describe homotopy classes of self-homeomorphisms of solenoids and Knaster continua. In particular, we demonstrate that homeomorphisms within one homotopy class have the same (explicitly given) topological en-tropy and that they are actually semi-conjugeted to an algebraic homeomor-phism in the case when the entropy is positive.
In generalizing the classical theory of circle maps, we study the rotation set for maps of the real line x 7 ! f(x) with almost periodic displacement f(x) ? x. Such maps are in one-to-one correspondence with maps of compact abelian topological groups that have a dense 1-parameter subgroup preserved by the dynamics. For homeomorphisms, we show existence of… (More)
We give a geometric proof of stability for spatially nonho-mogeneous equilibria in the singular perturbation problem u t = 2 u xx + f(x; u); t 2 R + ; ?1 u 1, with the Neumann boundary conditions on x 2 0; 1]. The nonlinearity is of the form f(x; u) := (1 ?u 2)(u ?c(x)) where c(x) is merely continuous with a nite number of zeros. The strength of the method… (More)
Motivated by the computations in the theory of cohomological Conley index, cocyclic subshifts are the supports of locally constant matrix cocycles on the full shift over a nite alphabet. They properly generalize sooc systems and topological Markov chains; and, via the Wedderburn-Artin theory of nite-dimensional algebras, admit a similar structure theory… (More)