The bifurcation structure of a constraint Duffing van der Pol oscillator with a diode is analyzed and an objective bifurcation diagram is illustrated in detail in this work. An idealized case, where the diode is assumed to operate as a switch, is considered.In this case, the Poincaré map is constructed as a one dimensional map: a circle map. The parameter boundary between a torusgenerating regi...

In the domain of analysis situs Poincaré has recently brought us an abundance of new results, but at the same time he has raised an abundance of new questions that still await settlement. Thus while we have known, for a long time, a set of necessary and sufficient conditions for the existence of a one-toone continuous map between two two-dimensional manifolds, at present such a system of condit...

This work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit. We show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit. When the regular orbit is a limit cycle, we can determine its associated Poincaré return map in terms of the in...

Let M,N and B ⊂ N be compact smooth manifolds of dimensions n + k, n and ` , respectively. Given a map f : M → N , we give homological conditions under which g−1(B) has nontrivial cohomology (with local coefficients) for any map g homotopic to f . We also show that a certain cohomology class in Hj(N,N−B) is Poincaré dual (with local coefficients) under f ∗ to the image of a corresponding class ...

Many natural phenomena found in various areas, such as orbital mechanics, fluid dynamics or quantum mechanics, can be described in terms of Hamiltonian dynamics. Multi-dimensional Hamiltonian systems often exhibit chaotic behaviour, which makes their analysis difficult. The use of surfaces of sections, such as a Poincaré return map, is commonly employed to get insight into the phase space of dy...

Is the one-to-one continuous image of an open set in R still open? This problem is the so called invariance of domain and, as H. Freudenthal pointed out in his essay on the history of topology [6] (see also [12]), it has been considered of a great importance at the beginning of this (20) century “for it was essential for the justification of the Poincaré continuation method in the theory of aut...

We study the linearization of the Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scattering matrix for an elliptic operator of 0-type, modified by some differential...

In order to study the geometries of a hyperbolic plane, it is necessary to understand the set of transformations that map from the space to itself. This paper shows that in the Poincaré half-plane model, this group of transformations is the general Möbius group. This is done by showing that the general Möbius group is a group of homeomorphisms that map circles to circles on the extended complex...

For every hyperbolic group Γ with Gromov boundary ∂Γ, one can form the cross product C∗-algebra C(∂Γ)⋊Γ. For each such algebra we construct a canonical K-homology class, which induces a Poincaré duality map K∗(C(∂Γ)⋊Γ) → K (C(∂Γ)⋊Γ). We show that this map is an isomorphism in the case of Γ = F2 the free group on two generators. We point out a direct connection between our constructions and the ...