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Volume 29, Number 5
Special Issue: Proceedings of RCD Conference 2023
Kozlov V. V.
Abstract
This paper discusses a range of questions concerning the application of solvable
Lie algebras of vector fields to exact integration of systems of ordinary differential equations.
The set of $n$ independent vector fields generating a solvable Lie algebra in $n$-dimensional space
is locally reduced to some “canonical” form. This reduction is performed constructively (using
quadratures), which, in particular, allows a simultaneous integration of $n$ systems of differential
equations that are generated by these fields. Generalized completely integrable systems are
introduced and their properties are investigated. General ideas are applied to integration of the
Hamiltonian systems of differential equations.
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Bizyaev I. A., Mamaev I. S.
Abstract
In this paper we consider the dynamics of a roller bicycle on a horizontal plane.
For this bicycle we derive a nonlinear system of equations of motion in a form that allows us
to take into account the symmetry of the system in a natural form. We analyze in detail the
stability of straight-line motion depending on the parameters of the bicycle. We find numerical
evidence that, in addition to stable straight-line motion, the roller bicycle can exhibit other,
more complex, trajectories for which the bicycle does not fall.
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García-Naranjo L. C., Ortega R., Ureña A. J.
Abstract
We present some results on the absence of a wide class of invariant measures for
dynamical systems possessing attractors. We then consider a generalization of the classical
nonholonomic Suslov problem which shows how previous investigations of existence of invariant
measures for nonholonomic systems should necessarily be extended beyond the class of measures
with strictly positive $C^1$ densities if one wishes to determine dynamical obstructions to the
presence of attractors.
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Bukh A. V., Rybalova E. V., Shepelev I. A., Vadivasova T. E.
Abstract
We study the spike activity of two mutually coupled FitzHugh – Nagumo neurons,
which is influenced by two-frequency signals. The ratio of frequencies in the external signal
corresponds to musical intervals (consonances). It has been discovered that this system can
exhibit selective properties for identifying musical intervals. The mechanism of selectivity is
shown, which is associated with the influence on the spiking frequency of neurons by intensity
of the external signal and nature of the interaction of neurons.
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Karatetskaia E., Koryakin V., Soldatkin K., Kazakov A. O.
Abstract
We provide a detailed bifurcation analysis in a three-dimensional system describing
interaction between tumor cells, healthy tissue cells, and cells of the immune system. As is
well known from previous studies, the most interesting dynamical regimes in this model are
associated with the spiral chaos arising due to the Shilnikov homoclinic loop to a saddle-focus
equilibrium [1–3]. We explain how this equilibrium appears and how it gives rise to Shilnikov
attractors. The main part of this work is devoted to the study of codimension-two bifurcations
which, as we show, are the organizing centers in the system. In particular, we describe bifurcation
unfoldings for an equilibrium state when: (1) it has a pair of zero eigenvalues (Bogdanov – Takens
bifurcation) and (2) zero and a pair of purely imaginary eigenvalues (zero-Hopf bifurcation). It
is shown how these bifurcations are related to the emergence of the observed chaotic attractors.
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Barinova M. K.
Abstract
In this paper we consider an $\Omega$-stable 3-diffeomorphism whose chain-recurrent set consists of isolated periodic points and hyperbolic 2-dimensional nontrivial attractors. Nontrivial attractors in this case can only be expanding, orientable or not. The most known example from the class under consideration is the DA-diffeomorphism obtained from the algebraic Anosov diffeomorphism, given on a 3-torus, by Smale's surgery. Each such attractor has bunches of degree 1 and 2. We estimate the minimum number of isolated periodic points using information about the structure of attractors. Also, we investigate the topological structure of ambient manifolds for diffeomorphisms with k bunches and k isolated periodic points.
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