Volume 28, Number 2

Volume 28, Number 2, 2023

Medvedev V. S.,  Zhuzhoma E. V.
We introduce Smale A-homeomorphisms that include regular, semichaotic, chaotic, and superchaotic homeomorphisms of a topological $n$-manifold $M^n$, $n\geqslant 2$. Smale A-homeomorphisms contain axiom A diffeomorphisms (in short, A-diffeomorphisms) provided that $M^n$ admits a smooth structure. Regular A-homeomorphisms contain all Morse–Smale diffeomorphisms, while semichaotic and chaotic A-homeomorphisms contain A-diffeomorphisms with trivial and nontrivial basic sets. Superchaotic A-homeomorphisms contain A-diffeomorphisms whose basic sets are nontrivial. The reason to consider Smale A-homeomorphisms instead of A-diffeomorphisms may be attributed to the fact that it is a good weakening of nonuniform hyperbolicity and pseudo-hyperbolicity, a subject which has already seen an immense number of applications.
We describe invariant sets that determine completely the dynamics of regular, semichaotic, and chaotic Smale A-homeomorphisms. This allows us to get necessary and sufficient conditions of conjugacy for these Smale A-homeomorphisms (in particular, for all Morse–Smale diffeomorphisms). We apply these necessary and sufficient conditions for structurally stable surface diffeomorphisms with an arbitrary number of expanding attractors. We also use these conditions to obtain a complete classification of Morse–Smale diffeomorphisms on projectivelike manifolds.
Keywords: conjugacy, topological classification, Smale homeomorphism
Citation: Medvedev V. S.,  Zhuzhoma E. V., Smale Regular and Chaotic A-Homeomorphisms and A-Diffeomorphisms, Regular and Chaotic Dynamics, 2023, vol. 28, no. 2, pp. 131-147
Ronzhina M. I.,  Manita L. A.
In this paper, we consider the minimum time problem for a space rocket whose dynamics is given by a control-affine system with drift. The admissible control set is a disc. We study extremals in the neighbourhood of singular points of the second order. Our approach is based on applying the method of a descending system of Poisson brackets and the Zelikin–Borisov method for resolution of singularities to the Hamiltonian system of Pontryagin’s maximum principle. We show that in the neighbourhood of any singular point there is a family of spiral-like solutions of the Hamiltonian system that enter the singular point in a finite time, while the control performs an infinite number of rotations around the circle.
Keywords: Hamiltonian system of Pontryagin’s maximum principle, singular extremal, controlaffine system with drift, descending system of Poisson brackets, resolution of singularity, blowup, coupled attitude orbit problem
Citation: Ronzhina M. I.,  Manita L. A., Spiral-Like Extremals near a Singular Surface in a Rocket Control Problem, Regular and Chaotic Dynamics, 2023, vol. 28, no. 2, pp. 148-161
Zhong X.,  Zhao J.,  Yu K.,  Xu M.
This paper presents the stability of resonant rotation of a symmetric gyrostat under third- and fourth-order resonances, whose center of mass moves in an elliptic orbit in a central Newtonian gravitational field. The resonant rotation is a special planar periodic motion of the gyrostat about its center of mass, i. e., the body performs one rotation in absolute space during two orbital revolutions of its center of mass. The equations of motion of the gyrostat are derived as a periodic Hamiltonian system with three degrees of freedom and a constructive algorithm based on a symplectic map is used to calculate the coefficients of the normalized Hamiltonian. By analyzing the Floquet multipliers of the linearized equations of perturbed motion, the unstable region of the resonant rotation and the region of stability in the first-order approximation are determined in the dimensionless parameter plane. In addition, the thirdand fourth-order resonances are obtained in the linear stability region and further nonlinear stability analysis is performed in the third- and fourth-order resonant cases.
Keywords: Hamiltonian systems, normal form, gyrostat, stability analysis, symplectic maps, periodic motion
Citation: Zhong X.,  Zhao J.,  Yu K.,  Xu M., Stability Analysis of Resonant Rotation of a Gyrostat in an Elliptic Orbit Under Third- and Fourth-Order Resonances, Regular and Chaotic Dynamics, 2023, vol. 28, no. 2, pp. 162-190
Lopes L. M.,  Grácio C.,  Fernandes S.,  Fournier-Prunaret D.
The chaotic behaviour of dynamical systems can be suppressed if we couple them in some way. In order to do that, the coupling strengths must assume particular values. We illustrate it for the situation that leads to a fixed point behaviour, using two types of couplings corresponding either to a diffusive interaction or a migrative one. For both of them, we present strategies that easily calculate coupling strengths that suppress the chaotic behaviour. We analyse the particular situation of these couplings that consists in a symmetric one and we propose a strategy that provides the suppression of the chaotic evolution of a population.
Keywords: suppression of chaos, diffusive coupling, migrative coupling, symmetric coupling, stabilisation of a population
Citation: Lopes L. M.,  Grácio C.,  Fernandes S.,  Fournier-Prunaret D., Using Couplings to Suppress Chaos and Produce a Population Stabilisation Strategy, Regular and Chaotic Dynamics, 2023, vol. 28, no. 2, pp. 191-206
Ivanov A. V.
We consider a skew product $F_{A} = (\sigma_{\omega}, A)$ over irrational rotation $\sigma_{\omega}(x) = x + \omega$ of a circle $\mathbb{T}^{1}$. It is supposed that the transformation $A: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ which is a $C^{1}$-map has the form $A(x) = R\big(\varphi(x)\big) Z\big(\lambda(x)\big)$, where $R(\varphi)$ is a rotation in $\mathbb{R}^{2}$ through the angle $\varphi$ and $Z(\lambda)= \text{diag}\{\lambda, \lambda^{-1}\}$ is a diagonal matrix. Assuming that $\lambda(x) \geqslant \lambda_{0} > 1$ with a sufficiently large constant $\lambda_{0}$ and the function $\varphi$ is such that $\cos \varphi(x)$ possesses only simple zeroes, we study hyperbolic properties of the cocycle generated by $F_{A}$. We apply the critical set method to show that, under some additional requirements on the derivative of the function $\varphi$, the secondary collisions compensate weakening of the hyperbolicity due to primary collisions and the cocycle generated by $F_{A}$ becomes uniformly hyperbolic in contrast to the case where secondary collisions can be partially eliminated.
Keywords: linear cocycle, hyperbolicity, Lyapunov exponent, critical set
Citation: Ivanov A. V., On $SL(2,\mathbb{R})$-Cocycles over Irrational Rotations with Secondary Collisions, Regular and Chaotic Dynamics, 2023, vol. 28, no. 2, pp. 207-226
Takano H.
This paper investigates the dynamics of a toy known as the chatter ring. Specifically, it examines the mechanism by which the small ring rotates around the large ring, the mechanism by which the force from the large ring provides torque to the small ring, and whether the motion of the small ring is the same as that of a hula hoop. The dynamics of a chatter ring has been investigated in previous work [13–15]; however, a detailed analysis has not yet been performed. Thus, to understand the mechanisms described above, the equations of motion and constraint conditions are obtained, and an analysis of the motion is performed. To simplify the problem, a model consisting of a straight rod and a washer ring is analyzed under the no-slip condition. The motion of a washer has two modes: the one point of contact (1PC) mode and two points of contact (2PC) mode. The motion of the small ring of the chatter ring is similar to that of a washer in the 2PC mode, whereas the motion of a hula hoop is similar to that of a washer in the 1PC mode. The analysis indicates that the motion of a washer with two points of contact is equivalent to free fall motion. However, in practice, the velocity reaches a constant value through energy dissipation. The washer rotates around an axis that passes through the two points of contact. The components of the forces exerted by the rod at the points of contact that are normal to the plane of the washer provide rotational torque acting at the center of mass. The components of the forces parallel to the horizontal plane are centripetal forces, which induce the circular motion of the center of mass.
Keywords: rigid body, chatterring, jitter ring, hula hoop, washer, rod, numerical simulation
Citation: Takano H., Analyzing the Motion of a Washer on a Rod, Regular and Chaotic Dynamics, 2023, vol. 28, no. 2, pp. 227-250

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