The Poincaré-Bendixson theorem plays an important role in the study of the qualitative behavior of dynamical systems on the plane; it describes the structure of limit sets in such systems. We prove a version of the Poincaré-Bendixson Theorem for two dimensional hybrid dynamical systems and describe a method for computing the derivative of the Poincaré return map, a useful object for the stabili...

The choice of a closed regular neighborhood of a finite polyhedron embedded in a manifold enables one to write the ambient space as a union of two manifolds, the neighborhood and its complement, glued along a common boundary component. Poincaré duality embeddings are a homotopy theoretic version of this in which the manifolds are replaced by Poincaré spaces and the gluing is now done with respe...

We call Poincaré time the time associated to the Poincaré (or first return) map of a vector field. In this paper we prove the non-accumulation of isolated critical points of the Poincaré time T on hyperbolic polycycles of polynomial vector fields. The result is obtained by proving that the Poincaré time of a hyperbolic polycycle either has an unbounded principal part or is an almost regular fun...

Recall the notion due to W. Browder of a Poincaré embedding [Br1, Br3, Br2, Br4, Wa, Ra, Wi1, Wi2], which is the homotopy analogue of a smooth embedding of manifolds. Let (M,A) be a simply connected m-dimensional (finite) Poincaré pair. A Poincaré embedding of (M,A) in the sphere S will mean a finite CW-complex W and a map f : A −→ W , such that the homotopy pushout M ∪ι A× [0, 1] ∪f W is homot...

Let M be a Poincaré duality space of dimension d ≥ 4. In this paper we describe a complete obstruction to realizing the diagonal map M → M×M by a Poincaré embedding. The obstruction group depends only on the fundamental group and the parity of d.

It is reported that the characteristic relations of type-II and type-III intermittencies, with respective local Poincaré maps of xn+1 = (1+ǫ)xn+ax 3 n and xn+1 = −(1 + ǫ)xn − ax3n, are both −ln(ǫ) under the assumption of uniform reinjection probability. However, the intermittencies have various characteristic relations such as ǫ−ν (1/2 ≤ ν ≤ 1) depending on the reinjection probability. In this ...

We study numerically the statistics of Poincaré recurrences for the Chirikov standard map and the separatrix map at parameters with a critical golden invariant curve. The properties of recurrences are analyzed with the help of a generalized Ulam method. This method allows us to construct the corresponding Ulam matrix whose spectrum and eigenstates are analyzed by the powerful Arnoldi method. We...

This paper provides algorithms in order to solve an interval implicit function of the Poincaré map generated by a continuous piecewise linear (CPWL) vector field, with the use of interval arithmetic. The algorithms are implemented with the use of MAT LAB and INT LAB. We present an application to verification of canards in two-dimensional CPWL vector field appearing in nonlinear piecewise linear...

We reformulate the integrality property of the Poincaré inner product in the middle dimension, for an arbitrary Poincaré Q-algebra, in classical terms (discriminant and local invariants). When the algebra is 1-connected, we show that this property is the only obstruction to realizing it by a closed manifold, up to dimension 11. We reinterpret a result of Eisenbud and Levine on finite map germs,...