Volume 29, Number 1
Volume 29, Number 1, 2024
Special Issue: In Honor of Vladimir Belykh and Sergey Gonchenko $\qquad$ Guest Editors: Alexey Kazakov, Vladimir Nekorkin, and Dmitry Turaev
Barabash N., Belykh I., Kazakov A. O., Malkin M. I., Nekorkin V., Turaev D. V.
Abstract
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.

Turaev D. V.
Abstract
We show that the stable invariant foliation of codimension 1 near a zerodimensional 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.

Gonchenko M. S.
Abstract
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 areapreserving 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 twoparameter 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 areacontracting and areaexpanding saddles).

Kulagin N. E., Lerman L. M., Trifonov K. N.
Abstract
We examine smooth fourdimensional vector fields reversible under some smooth
involution $L$ that has a smooth twodimensional submanifold of fixed points. Our main interest
here is in the orbit structure of such a system near two types of heteroclinic connections
involving saddlefoci and heteroclinic orbits connecting them. In both cases we found families
of symmetric periodic orbits, multiround heteroclinic connections and countable families of
homoclinic orbits of saddlefoci. All this suggests that the orbit structure near such connections
is very complicated. A nonvariational version of the stationary Swift – Hohenberg equation is
considered, as an example, where such structure has been found numerically.

Morozov K. E., Morozov A. D.
Abstract
We study nonconservative quasiperiodic (with $m$ frequencies) perturbations of
twodimensional Hamiltonian systems with nonmonotonic rotation. It is assumed that the
perturbation contains the socalled 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 quasiperiodic perturbation.

Kazakov A. O., Murillo A., Vieiro A., Zaichikov K.
Abstract
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 nearidentity maps. For larger discretization parameter
values, the map exhibits what appears to be genuinelydiscrete Lorenzlike 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 nearidentity discrete
systems with similar phase space structure.

Fallah H., Shilnikov A. L.
Abstract
This paper studies quasiperiodicity 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, selfsimilar 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 saddlenode or inverse perioddoubling bifurcations. The
bluesky catastrophe takes place at the transition route from bursting to spiking.

Stankevich N. V., Bobrovsky A. A., Shchegoleva N. A.
Abstract
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 inphase bursting
oscillations.

Davydov A. A., Plakhov A.
Abstract
We consider the dynamics of a rod on the plane in a flow of noninteracting 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.

Grines V. Z., Medvedev V. S., Zhuzhoma E. V.
Abstract
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 $M^n$.

Grines V. Z., Pochinka O. V., Chilina E. E.
Abstract
The present paper is devoted to a study of orientationpreserving homeomorphisms
on threedimensional manifolds with a nonwandering 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 nonwandering set
is topologically conjugate to an orientationpreserving pseudoAnosov homeomorphism. The
ambient $\Omega$conjugacy of a homeomorphism from the class with a locally direct product of a
pseudoAnosov homeomorphism and a rough transformation of the circle is proved. In addition,
we prove that the centralizer of a pseudoAnosov homeomorphisms consists of only pseudo
Anosov and periodic maps.

Zhukova N. I.
Abstract
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 abovementioned 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 wellknown 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.

Kapustnikov A. A., Sysoeva M. V., Sysoev I. V.
Abstract
Discharges of different epilepsies are characterized by different signal shape and
duration. The authors adhere to the hypothesis that spikewave 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 quasiregularity.
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 quasiregular processes mimicking
epileptic spikewave 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.

Emelin A. E., Grines E. A., Levanova T. A.
Abstract
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 twoparameter bifurcation analysis, we investigate the
scenarios of emergence and destruction of chaotic dynamics in the system under study.

Kashchenko S. A.
Abstract
We study the local dynamics of chains of coupled nonlinear systems of secondorder
ordinary differential equations of diffusiondifference 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.
