Joanna Janczewska

The shadowing chain lemma for singular Hamiltonian systems involving strong forces

We consider a planar autonomous Hamiltonian system :q+∇V(q) = 0, where the potential V: ℝ2 \{ζ}→ ℝ has a single well of infinite depth at some point ζ and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ζ we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits — the shadowing chain lemma — via minimization of action integrals and using simple geometrical arguments.

Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point

In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×Rk→Y is a C2-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R1 that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ0)∈X×Rk and we describe the solution set of the studied equation in a small neighbourhood of this point.

Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3

We consider a conservative second order Hamiltonian system

On Von Kármán Equations and the Buckling of a Thin Circular Elastic Plate

\ddot q + \nabla V(q) = 0

Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential

in ℝ3 with a potential V having a global maximum at the origin and a line l ∩ {0} = ϑ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.

Heteroclinic solutions for a class of the second order Hamiltonian systems

The study of existence and multiplicity of solutions of differential equations possessing a variational nature is a problem of great meaning since most of them derives from mechanics and physics. In particular, this relates to Hamiltonian systems including Newtonian ones. During the past 30 years there has been a great deal of progress in the use of variational methods to find periodic, homoclinic and heteroclinic solutions of Hamiltonian systems. Hamiltonian systems with singular potentials, i.e., potentials that become infinite at a point or a larger subset of \(\mathbb{R}^{n}\), are among those of the greatest interest. Let us remark that such potentials arise in celestial mechanics. For example, the Kepler problem with

Bifurcation of equilibrium forms of an elastic rod on a two-parameter Winkler foundation

\displaystyle{V (q) = - \frac{1} {\vert q -\xi \vert }}