It is well known that the stiff van der Pol equation has a strongly attractive limit cycle. In this paper it is shown that the Euler method applied to the van der Pol equation with small step size, small compared to the perturbation parameter, admits an attractive invariant closed curve close to the limit cycle. To describe closed curves in the vicinity of the limit cycle, 14 charts are introduced. A general graph transform result is derived and applied in these charts. The proof of the main result relies on the contraction principle in a suitable function space. Estimates are given for the distance of the invariant curve to the limit cycle.

We show that the discrete dynamical system obtained by applying the Euler method to the van der Pol relaxation oscillator admits a closed attractive invariant manifold and that this curve is close to the limit cycle of the ODE.

In 1999 Ivanov and Losson [A.F. Ivanov and J. Losson, Stable rapidly oscillating solutions in delay differential equations with negative feedback, Differ. Int. Eqns 12 (1999), pp. 811-832] presented a computer assisted proof that a particular delay differential equation (with negative feedback) admits a stable rapidly oscillating periodic solution (ROPS). In this article the delay equation of Ivanov and Losson is embedded in a five-parametric class of differential equations. Conditions on the parameters are given such that the delay equation admits a stable ROPS. Moreover, it is shown that for odd n 1 the delay equation admits a stable ROPS with n humps per unit time if the parameters satisfy some explicitly given conditions. The delay equation of Ivanov and Losson satisfies all conditions on the five parameters. This gives an analytic proof and a considerable generalization of the result of Ivanov and Losson. The conditions on the parameters are believed to be sharp in a certain sense. The second result proves part of a conjecture in Stoffer [D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dyn. Differ. Eqns 20(1) (2008), pp. 201-238]. For a class of stiff delay differential equations with piecewise constant nonlinearity (positive or negative feedback) and for every n the following holds: if the stiffness parameter is sufficiently large then there are 2a(n) essentially different stable ROPSs with n humps per time unit. a(n) is the number of essentially different binary n-stage shift register sequences.

A computer-assisted proof is presented that the stable and unstable manifolds of the area preserving Henon map intersect transversally.

We prove analytically that there exist delay equations admitting rapidly oscillating stable periodic solutions. Previous results were obtained with the aid of computers, only for particular feedback functions. Our proofs work for stiff equations with several classes of feedback functions. Moreover, we prove that for negative feedback there exists a class of feedback functions such that the larger the stiffness parameter is, the more stable rapidly oscillating periodic solutions there are. There are stable periodic solutions with arbitrarily many zeros per unit time interval if the stiffness parameter is chosen sufficiently large.

We study the delay equation (x)over dot (t) = -ux(t) + f (x(t-1)) with mu > 0 and a nonmonotone C-1-function f obeying xf (x) > 0 (positive feedback) outside a small neighbourhood of zero. By means of a computer-assisted method we prove the existence of asymptotically orbitally stable periodic solutions. The main idea behind our proof is the reduction of the infinite-dimensional dynamics to a finite-dimensional map. In particular, for two classes of nonlinearities f we construct two types of solutions, the dynamics of which is reduced to a one- and a two-dimensional map, respectively.

Using shadowing techniques we prove that the Hénon map $H_{a,b}(x,y)=(a-x^{2}+by,x)$ admits a transversal homoclinic point for a set of parameters which is not small. For the area and orientation preserving Hénon map (corresponding to b=-1) we prove that a transversal homoclinic point exists for a≥0.265625. Applying a computer-assisted version of our scheme we show that the result holds even for a≥-0.866. This supports an old conjecture due to Devaney and Nitecki dating back to 1979, see [4], claiming that the Hénon map in the case b=-1 admits a transversal homoclinic point for a>-1.