M. Malkin

This issue of RCD is dedicated to the outstanding Russian mathematician Leonid Pavlovich Shilnikov, a leading figure in the theory of dynamical systems, one of the founders of the mathematical theory of dynamical chaos.
The impact of the Shilnikov work on nonlinear dynamics and its applications is indeed enormous. We mention only a few of his groundbreaking scientific achievements: the theory of global bifurcations of multidimensional dynamical systems, the discovery of spiral chaos, the theory of Lorenz-like attractors, the mathematical theory of transition from synchronization to chaos, the theory of homoclinic chaos, among many others.

This paper deals with the problem of smoothness of the stable invariant foliation for a homoclinic bifurcation with a neutral saddle in symmetric systems of differential equations. We give an improved sufficient condition for the existence of an invariant smooth foliation on a cross-section transversal to the stable manifold of the saddle. It is shown that the smoothness of the invariant foliation depends on the gap between the leading stable eigenvalue of the saddle and other stable eigenvalues. We also obtain an equation to describe the one-dimensional factor map, and we study the renormalization properties of this map. The improved information on the smoothness of the foliation and the factor map allows one to extend Shilnikov’s results on the birth of Lorenz attractors under the bifurcation considered.

This special issue is dedicated to the anniversaries of two famous Russian mathematicians, Sergey V.Gonchenko and Vladimir N.Belykh. Over the years, they have made a lasting impact in the theory of dynamical systems and applications. In this issue we have collected a series of papers by their friends and colleagues devoted to modern aspects and trends of the theory of dynamical chaos.

We consider piecewise monotone (not necessarily, strictly) piecewise $C^2$ maps on the interval with positive topological entropy. For such a map $f$ we prove that its topological entropy $h_{top}(f)$ can be approximated (with any required accuracy) by restriction on a compact strictly $f$-invariant hyperbolic set disjoint from some neighborhood of prescribed set consisting of periodic attractors, nonhyperbolic intervals and endpoints of monotonicity intervals. By using this result we are able to generalize main theorem from [1] on chaotic behavior of multidimensional perturbations of solutions for difference equations which depend on two variables at nonperturbed value of parameter.

In this paper, we study the family of Arneodo–Coullet–Tresser maps $F(x,y,z)=(ax-b(y-z)$, $bx+a(y-z)$, $cx-dxk+e z)$ where $a$, $b$, $c$, $d$, $e$ are real parameters with $bd \ne 0$ and $k>1$ is an integer. We find regions of parameters near anti-integrable limits and near singularities for which there exist hyperbolic invariant sets such that the restriction of $F$ to these sets is conjugate to the full shift on two or three symbols.