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Important Notice: Change of Publisher
Dear authors, please be advised that, as of 2027, the journal Regular and Chaotic Dynamics will have a new publisher. You can find more detailed information about this at the link.Call for papers
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Call for Papers: Special Issue Dedicated to the 80th Anniversary of Carles Simó
The deadline for manuscript submissions has been extended to December 15, 2026. The issue is provisionally scheduled for publication in March 2027. |
Volume 31, Number 5
Theory of Dynamical and Control Systems and its Applications: On the 125th Anniversary of the Birth of Academician A. A. Andronov (Issue Editors: Dmitry Balandin, Vladimir Belykh, Sergey Gonchenko, and Alexey Kazakov)
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Karatetskaia E., Kazakov A. O., Koryakin V., Safonov K. A.
Abstract
We study the chaotic dynamics of the Suslov top, a classical nonholonomic
system. This system possesses a rich bifurcation structure, which we explore numerically
via Lyapunov diagrams and bifurcation analysis in a two-parameter plane. This structure is
organized by a codimension-2 point where an equilibrium state has eigenvalues $(0, i\omega,−i\omega)$ and
a homoclinic loop. From this point, two important bifurcation curves originate: the Andronov – Hopf bifurcation curve, corresponding to the birth of a stable periodic orbit from an equilibrium
state, and the Andronov – Vitt – Shilnikov bifurcation curve, corresponding to the birth of
a stable periodic orbit from a homoclinic loop of a saddle-node equilibrium. We show how
chaotic attractors and repellers emerge from equilibrium states, grow, and merge into mixed
dynamics. We also discuss the connection of this organizing codimension-2 point with the blue
sky catastrophe phenomenon: the birth of a stable periodic orbit of infinite period and length.
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Barabash N. V., Suntsova M. V., Belykh V. N.
Abstract
In this paper, we constructed a new model of the neuron membrane potential in the form of a piecewise-smooth (PWS) slow-fast system. Its fast part is a two-dimensional PWS subsystem specially “sewn” from the normal forms of a stable focus and a saddle-node bifurcation, and the slow one-dimensional equation serves to periodically force the fast one. Our approach to constructing PWS systems in this way allows them to qualitatively reproduce the phase space structure of their smooth counterpart and also provide explicit solutions. By obtaining an exact Poincaré map as a composition of solutions and examining it, we proved
the complete bifurcation structure of the fast subsystem. By selecting three bifurcation routes and applying a time-periodic change of the subsystem parameter along them using the third slow equation, we obtained three different types of bursting patterns. Finally, we showed that, when coupled into a network, our PWS models form a synthetic central pattern generator that manifests a traveling bursting wave.
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Treschev D. V.
Abstract
We discuss algebraic and combinatorial aspects of the Hamiltonian normal form
theory. The main objective is to describe the normal form near a singular point by an explicit
formula in terms of the original Hamiltonian, avoiding the normalization procedure. In the
case of one degree of freedom we compute the normal form as an explicit nonlinear functional,
applied to the original Hamiltonian. In arbitrary dimension the corresponding formulas are
more complicated but still explicit.
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Medvedev V. S., Zhuzhoma E. V.
Abstract
We study Axiom A diffeomorphisms (in the sense of Smale) on a closed orientable $n$-manifold $M^n$, $n\geqslant 4$, whose nonwandering sets consist of codimension one orientable attractors, repellers, and isolated saddle periodic orbits with the same Morse index. Such systems combine chaotic dynamics on basic sets with regular dynamics near trivial periodic orbits. We prove that the supporting manifold $M^n$ is homeomorphic to a connected sum of several copies of the $n$-dimensional torus. Moreover, we obtain a complete classification of such systems up to global conjugacy on their nonwandering sets. The invariant of conjugacy is a finite collection of Anosov automorphisms of the torus, called a tube, defined up to equivalence under linear automorphisms with determinants of the same sign. This extends the classical Plykin – Grines classification of individual codimension one basic sets to systems with multiple coexisting attractors and repellers.
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Bolotin S. V.
Abstract
We consider the plane three-body problem with two of the masses much smaller
than the third. Periodic solutions with near collisions of the small bodies were named by
Poincaré second species solutions. Poincaré gave an idea of how to prove their existence, but
didn’t provide an actual proof. Complete proofs appeared much later and only for the restricted
three-body problem. We reduce the problem of second species solutions of the general threebody
problem to the study of dynamics of a billiard type problem and show how to transform
the arguments of Poincaré into a rigorous proof.
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Balandin D. V., Tuzikov A. M., Biryukov R. S.
Abstract
This paper extends the modeling and control of a spherical robot with electromagnetic drive (SRED) from planar to fully three-dimensional motion on uneven surfaces. The robot consists of two concentric spherical shells: an outer shell carrying permanent magnets (PMs) and an inner shell with solenoid-type electromagnets (EMs); rolling is produced by the controlled electromagnetic interaction between the two magnet arrays. To enable omnidirectional rolling, the magnet placement is generalized to vertex-transitive polyhedra with cubic or icosahedral symmetry, which preserves the scalarity of the inertia tensors and allows the existing optimal control framework to be applied without modification. Within this class, $M=60$ PMs on a rhombicosidodecahedron and $S=24$ EMs on a truncated octahedron are selected; numerical evaluation over $10^{4}$ relative orientations confirms that a nonzero torque can be generated about an arbitrary axis. The drive model is reformulated as a three-dimensional torque map matrix, and coil currents and terminal voltages are recovered from a reference torque via the generalized-$H_{2}$ optimal controller, a Tikhonov-regularized pseudoinverse, and Kirchhoff's circuit equations with back-EMF terms. Simulations of straight-line rolling, lateral-offset correction, and three-dimensional path tracking on a Gaussian bell-shaped surface confirm convergence of the tracking error to zero, with currents and voltages within the range of standard low-voltage motor drivers.
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Stankevich N. V., Gonchenko A. S., Popova E. S., Bagautdinova E. R.
Abstract
We study bifurcation scenarios leading to the so-called Shilnikov singular attractors
as a result of breakdown of closed invariant curves in the Chialvo map, which is a twodimensional
endomorphism demonstrating neuron-like dynamics. We show that two different
routes, soft and hard, of the emergence of such closed invariant curves can be traced here: the
soft one corresponds to the birth of an invariant curve as a result of a supercritical Neimark – Sacker bifurcation, and the hard one relates to the immediately occurring big invariant curve
after disappearance of a stable fixed point at the saddle-node bifurcation. We study both
these mechanisms and trace subsequent scenarios of breakdown of the invariant curve and
chaos development leading to the emergence of Shilnikov singular attractors. Additionally, we
study geometrical peculiarities of these attractors, such as structures of rotating patterns of
orbits inside the Shilnikov singular funnel and present a two-parametric analysis with Lyapunov
exponents and minimal a distance between the chaotic attractor and the unstable focus.
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Pyatko L., Aksenov S.
Abstract
In the circular restricted three-body problem, families of periodic and quasi-periodic
orbits exist in the vicinity of the libration points and are of significant interest for practical
applications. This work presents a neural network-based algorithm for orbit construction and
station-keeping, applicable to both periodic and quasi-periodic orbits near collinear libration
points. The method employs a fully connected neural network trained on labeled trajectory data
and iteratively refined using trajectories generated during previous stages of the algorithm. The
proposed approach was applied to orbit generation near the Sun-Earth L1 libration point and
demonstrated the ability to generalize to a broader class of trajectories than those included in
the training set.
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Lubchenko D.
Abstract
Strange nonchaotic attractors (SNA) are fractal invariant sets characterized by
aperiodic dynamics and nonpositive largest Lyapunov exponents. Due to the fractal geometry
and the absence of exponential sensitivity to initial conditions, SNA are often difficult to
distinguish from chaotic attractors using time-series data. This study examines the applicability
of wavelet-based analysis for the detection of strange nonchaotic regimes in dynamical systems.
The proposed approach involves the analysis of wavelet scalograms and global wavelet energy
spectra. It is shown that quasiperiodic and strange regimes can be distinguished by the structure
of their scalograms. To distinguish between strange nonchaotic and chaotic regimes, global
wavelet energy spectra are employed. Numerical results demonstrate the applicability of this
method for identifying quasiperiodic, strange nonchaotic, and chaotic regimes from time-series
data.
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Sumenkov O. Y., Gusev S. V.
Abstract
This paper treats the problem of finding feasible periodic motions and that of
their robust orbital stabilization for a spherical robot. The first problem is tackled using the
servoconstraint approach based on the mathematical model of a pendulum-type spherical robot
moving on a plane under the no-slip and no-loss-of-contact assumptions. The second problem
is addressed through a generalization of the well-known Andronov – Vitt theorem, specifically
by introducing the transverse coordinates of co-dimension one that are zero on a desired orbit.
Linearizing the dynamics in the transverse coordinates results in a linear system with periodic
coefficients (LSPC). A rigorous procedure is suggested for stabilizing the LSPC in the presence
of structured uncertainties and matched disturbances using vector relay control. This control
strategy forces the LSPC solutions onto a predefined time-periodic family of subspaces, where
the exponential stability is ensured. The effectiveness of the overall framework is demonstrated
through simulations.
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