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 show that the stable invariant foliation of codimension 1 near a zero-dimensional hyperbolic set of a $C^{\beta}$ map with $\beta>1$ is $C^{1+\varepsilon}$ with some $\varepsilon>0$. The result is applied to the restriction of higher regularity maps to normally hyperbolic manifolds. An application to the theory of the Newhouse phenomenon is discussed.

We study bifurcations of symmetric elliptic fixed points in the case of $p$:$q$ resonances  with odd $q\geqslant 3$. We consider the case where the initial area-preserving map $\bar z =\lambda z + Q(z,z^*)$ possesses the central symmetry, i.e., is invariant under the change of variables $z\to -z$, $z^*\to  -z^*$. We construct normal forms for such maps in the case $\lambda = e^{i 2\pi \frac{p}{q}}$, where $p$ and $q$ are mutually prime integer numbers, $p\leqslant q$ and $q$ is odd, and study local bifurcations of the fixed point $z=0$ in various settings. We prove the appearance of garlands consisting of four $q$-periodic orbits, two orbits are elliptic and two orbits are saddles, and describe the corresponding bifurcation diagrams for one- and two-parameter families. We also consider the case where the initial map is reversible and find conditions where nonsymmetric periodic orbits of the garlands are nonconservative (contain symmetric pairs of stable and unstable orbits as well as area-contracting and area-expanding saddles).

We examine smooth four-dimensional vector fields reversible under some smooth involution $L$ that has a smooth two-dimensional submanifold of fixed points. Our main interest here is in the orbit structure of such a system near two types of heteroclinic connections involving saddle-foci and heteroclinic orbits connecting them. In both cases we found families of symmetric periodic orbits, multi-round heteroclinic connections and countable families of homoclinic orbits of saddle-foci. All this suggests that the orbit structure near such connections is very complicated. A non-variational version of the stationary Swift – Hohenberg equation is considered, as an example, where such structure has been found numerically.

We study nonconservative quasi-periodic (with $m$ frequencies) perturbations of two-dimensional Hamiltonian systems with nonmonotonic rotation. It is assumed that the perturbation contains the so-called parametric terms. The behavior of solutions in the vicinity of degenerate resonances is described. Conditions for the existence of resonance $(m + 1)$-dimensional invariant tori for which there are no generating ones in the unperturbed system are found. The class of perturbations for which such tori can exist is indicated. The results are applied to the asymmetric Duffing equation under a parametric quasi-periodic perturbation.

We consider a homotopic to the identity family of maps, obtained as a discretization of the Lorenz system, such that the dynamics of the last is recovered as a limit dynamics when the discretization parameter tends to zero. We investigate the structure of the discrete Lorenzlike attractors that the map shows for different values of parameters. In particular, we check the pseudohyperbolicity of the observed discrete attractors and show how to use interpolating vector fields to compute kneading diagrams for near-identity maps. For larger discretization parameter values, the map exhibits what appears to be genuinely-discrete Lorenz-like attractors, that is, discrete chaotic pseudohyperbolic attractors with a negative second Lyapunov exponent. The numerical methods used are general enough to be adapted for arbitrary near-identity discrete systems with similar phase space structure.

This paper studies quasi-periodicity phenomena appearing at the transition from spiking to bursting activities in the Pernarowski model of pancreatic beta cells. Continuing the parameter, we show that the torus bifurcation is responsible for the transition between spiking and bursting. Our investigation involves different torus bifurcations, such as supercritical torus bifurcation, saddle torus canard, resonant torus, self-similar torus fractals, and torus destruction. These bifurcations give rise to complex or multistable dynamics. Despite being a dissipative system, the model still exhibits KAM tori, as we have illustrated. We provide two scenarios for the onset of resonant tori using the Poincaré return map, where global bifurcations happen because of the saddle-node or inverse period-doubling bifurcations. The blue-sky catastrophe takes place at the transition route from bursting to spiking.

The dynamics of two coupled neuron models, the Hindmarsh – Rose systems, are studied. Their interaction is simulated via a chemical coupling that is implemented with a sigmoid function. It is shown that the model may exhibit complex behavior: quasiperiodic, chaotic and hyperchaotic oscillations. A phenomenological scenario for the formation of hyperchaos associated with the appearance of a discrete Shilnikov attractor is described. It is shown that the formation of these attractors leads to the appearance of in-phase bursting oscillations.

We consider the dynamics of a rod on the plane in a flow of non-interacting point particles moving at a fixed speed. When colliding with the rod, the particles are reflected elastically and then leave the plane of motion of the rod and do not interact with it. A thin unbending weightless “knitting needle” is fastened to the massive rod. The needle is attached to an anchor point and can rotate freely about it. The particles do not interact with the needle. The equations of dynamics are obtained, which are piecewise analytic: the phase space is divided into four regions where the analytic formulas are different. There are two fixed points of the system, corresponding to the position of the rod parallel to the flow velocity, with the anchor point at the front and the back. It is found that the former point is topologically a stable focus, and the latter is topologically a saddle. A qualitative description of the phase portrait of the system is obtained.

Let $\mathbb{G}_k^{cod 1}(M^n)$, $k\geqslant 1$, be the set of axiom A diffeomorphisms such that the nonwandering set of any $f\in\mathbb{G}_k^{cod 1}(M^n)$ consists of $k$ orientable connected codimension one expanding attractors and contracting repellers where $M^n$ is a closed orientable $n$-manifold, $n\geqslant 3$. We classify the diffeomorphisms from $\mathbb{G}_k^{cod 1}(M^n)$ up to the global conjugacy on nonwandering sets. In addition, we show that any $f\in\mathbb{G}_k^{cod 1}(M^n)$ is $\Omega$-stable and is not structurally stable. One describes the topological structure of a supporting manifold $M^n$.

The present paper is devoted to a study of orientation-preserving homeomorphisms on three-dimensional manifolds with a non-wandering set consisting of a finite number of surface attractors and repellers. The main results of the paper relate to a class of homeomorphisms for which the restriction of the map to a connected component of the non-wandering set is topologically conjugate to an orientation-preserving pseudo-Anosov homeomorphism. The ambient $\Omega$-conjugacy of a homeomorphism from the class with a locally direct product of a pseudo-Anosov homeomorphism and a rough transformation of the circle is proved. In addition, we prove that the centralizer of a pseudo-Anosov homeomorphisms consists of only pseudo- Anosov and periodic maps.

The focus of the work is the investigation of chaos and closely related dynamic properties of continuous actions of almost open semigroups and $C$-semigroups. The class of dynamical systems $(S, X)$ defined by such semigroups $S$ is denoted by $\mathfrak A$. These semigroups contain, in particular, cascades, semiflows and groups of homeomorphisms. We extend the Devaney definition of chaos to general dynamical systems. For $(S, X)\in\mathfrak A$ on locally compact metric spaces $X$ with a countable base we prove that topological transitivity and density of the set formed by points having closed orbits imply the sensitivity to initial conditions. We assume neither the compactness of metric space nor the compactness of the above-mentioned closed orbits. In the case when the set of points having compact orbits is dense, our proof proceeds without the assumption of local compactness of the phase space $X$. This statement generalizes the well-known result of J. Banks et al. on Devaney's definition of chaos for cascades.The interrelation of sensitivity, transitivity and the property of minimal sets of semigroups is investigated. Various examples are given.

Discharges of different epilepsies are characterized by different signal shape and duration. The authors adhere to the hypothesis that spike-wave discharges are long transient processes rather than attractors. This helps to explain some experimentally observed properties of discharges, including the absence of a special termination mechanism and quasi-regularity. Analytical approaches mostly cannot be applied to studying transient dynamics in large networks. Therefore, to test the observed phenomena for universality one has to show that the same results can be achieved using different model types for nodes and different connectivity terms. Here, we study a class of simple network models of a thalamocortical system and show that for the same connectivity matrices long, but finite in time quasi-regular processes mimicking epileptic spike-wave discharges can be found using nodes described by three neuron models: FitzHugh – Nagumo, Morris – Lecar and Hodgkin – Huxley. This result takes place both for linear and nonlinear sigmoid coupling.

Heteroclinic cycles are widely used in neuroscience in order to mathematically describe different mechanisms of functioning of the brain and nervous system. Heteroclinic cycles and interactions between them can be a source of different types of nontrivial dynamics. For instance, as it was shown earlier, chaotic dynamics can appear as a result of interaction via diffusive couplings between two stable heteroclinic cycles between saddle equilibria. We go beyond these findings by considering two coupled stable heteroclinic cycles rotating in opposite directions between weak chimeras. Such an ensemble can be mathematically described by a system of six phase equations. Using two-parameter bifurcation analysis, we investigate the scenarios of emergence and destruction of chaotic dynamics in the system under study.

We study the local dynamics of chains of coupled nonlinear systems of secondorder ordinary differential equations of diffusion-difference type. The main assumption is that the number of elements of chains is large enough. This condition allows us to pass to the problem with a continuous spatial variable. Critical cases have been considered while studying the stability of the equilibrum state. It is shown that all these cases have infinite dimension. The research technique is based on the development and application of special methods for construction of normal forms. Among the main results of the paper, we include the creation of new nonlinear boundary value problems of parabolic type, whose nonlocal dynamics describes the local behavior of solutions of the original system.