Volume 22, Number 3

Volume 22, Number 3, 2017

Barreira L.,  Dragičević D.,  Valls  C.
For the nonautonomous dynamics defined by a sequence of bounded linear operators acting on an arbitrary Hilbert space, we obtain a characterization of the notion of a nonuniform exponential dichotomy in terms of quadratic Lyapunov sequences. We emphasize that, in sharp contrast with previous results, we consider the general case of possibly noninvertible linear operators, thus requiring only the invertibility along the unstable direction. As an application, we give a simple proof of the robustness of a nonuniform exponential dichotomy under sufficiently small linear perturbations.
Keywords: nonuniform exponential dichotomies, Lyapunov functions
Citation: Barreira L.,  Dragičević D.,  Valls  C., Nonuniform Exponential Dichotomies and Lyapunov Functions, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 197-209
Jalnine A. Y.,  Kuznetsov S. P.
We investigate strange nonchaotic self-oscillations in a dissipative system consisting of three mechanical rotators driven by a constant torque applied to one of them. The external driving is nonoscillatory; the incommensurable frequency ratio in vibrational-rotational dynamics arises due to an irrational ratio of diameters of the rotating elements involved. It is shown that, when losing stable equilibrium, the system can demonstrate two- or three-frequency quasi-periodic, chaotic and strange nonchaotic self-oscillations. The conclusions of the work are confirmed by numerical calculations of Lyapunov exponents, fractal dimensions, spectral analysis, and by special methods of detection of a strange nonchaotic attractor (SNA): phase sensitivity and analysis using rational approximation for the frequency ratio. In particular, SNA possesses a zero value of the largest Lyapunov exponent (and negative values of the other exponents), a capacitive dimension close to 2 and a singular continuous power spectrum. In general, the results of this work shed a new light on the occurrence of strange nonchaotic dynamics.
Keywords: autonomous dynamical system, mechanical rotators, quasi-periodic oscillations, strange nonchaotic attractor, chaos
Citation: Jalnine A. Y.,  Kuznetsov S. P., Autonomous Strange Nonchaotic Oscillations in a System of Mechanical Rotators, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 210-225
Roozegar  M.,  Mahjoob M. J.,  Ayati M.
This paper deals with adaptive estimation of the unknown parameters and states of a pendulum-driven spherical robot (PDSR), which is a nonlinear in parameters (NLP) chaotic system with parametric uncertainties. Firstly, the mathematical model of the robot is deduced by applying the Newton–Euler methodology for a system of rigid bodies. Then, based on the speed gradient (SG) algorithm, the states and unknown parameters of the robot are estimated online for different step length gains and initial conditions. The estimated parameters are updated adaptively according to the error between estimated and true state values. Since the errors of the estimated states and parameters as well as the convergence rates depend significantly on the value of step length gain, this gain should be chosen optimally. Hence, a heuristic fuzzy logic controller is employed to adjust the gain adaptively. Simulation results indicate that the proposed approach is highly encouraging for identification of this NLP chaotic system even if the initial conditions change and the uncertainties increase; therefore, it is reliable to be implemented on a real robot.
Keywords: nonholonomic spherical robot, adaptive estimation, nonlinear in parameters, speed gradient method; fuzzy logic controller, Newton–Euler strategy
Citation: Roozegar  M.,  Mahjoob M. J.,  Ayati M., Adaptive Estimation of Nonlinear Parameters of a Nonholonomic Spherical Robot Using a Modified Fuzzy-based Speed Gradient Algorithm, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 226-238
Bizyaev I. A.
This paper addresses the problem of the inertial motion of a roller racer, which reduces to investigating a dynamical system on a (two-dimensional) torus and to classifying singular points on it. It is shown that the motion of the roller racer in absolute space is asymptotic. A restriction on the system parameters in which this motion is bounded (compact) is presented.
Keywords: roller racer, invariant measure, nonholonomic mechanics, scattering map
Citation: Bizyaev I. A., The Inertial Motion of a Roller Racer, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 239-247
Xue J.
In this paper, we prove the Nekhoroshev estimates for commuting nearly integrable symplectomorphisms. We show quantitatively how $\mathbb{Z}^m$ symmetry improves the stability time. This result can be considered as a counterpart of Moser’s theorem [11] on simultaneous
conjugation of commuting circle maps in the context of Nekhoroshev stability. We also discuss the possibility of Tits’ alternative for nearly integrable symplectomorphisms.
Keywords: Nekhoroshev estimates, commuting symplectomorphisms, generating functions, resonances
Citation: Xue J., Nekhoroshev Estimates for Commuting Nearly Integrable Symplectomorphisms, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 248-265
Kudryashov N. A.,  Gaur I. Y.
The fourth-order analogue of the second Painlevé equation is considered. The monodromy manifold for a Lax pair associated with the $P_2^2$ equation is constructed. The direct monodromy problem for the Lax pair is solved. Asymptotic solutions expressed via trigonometric functions in the Boutroux variables along the rays $\phi = \frac{2}{5}\pi(2n+1)$ on the complex plane have been found by the isomonodromy deformations technique.
Keywords: $P^2_2$ equation, isomonodromy deformations technique, special functions, Painlevé transcendents
Citation: Kudryashov N. A.,  Gaur I. Y., Weak Nonlinear Asymptotic Solutions for the Fourth Order Analogue of the Second Painlevé Equation, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 266-271
Naik S.,  Lekien F.,  Ross S. D.
Lobe dynamics and escape from a potential well are general frameworks introduced to study phase space transport in chaotic dynamical systems.While the former approach studies how regions of phase space get transported by reducing the flow to a two-dimensional map, the latter approach studies the phase space structures that lead to critical events by crossing certain barriers. Lobe dynamics describes global transport in terms of lobes, parcels of phase space bounded by stable and unstable invariant manifolds associated to hyperbolic fixed points of the system. Escape from a potential well describes how the critical events occur and quantifies the rate of escape using the flux across the barriers. Both of these frameworks require computation of curves, intersection points, and the area bounded by the curves. We present a theory for classification of intersection points to compute the area bounded between the segments of the curves. This involves the partition of the intersection points into equivalence classes to apply the discrete form of Green’s theorem. We present numerical implementation of the theory, and an alternate method for curves with nontransverse intersections is also presented along with a method to insert points in the curve for densification.
Keywords: chaotic dynamical systems, numerical integration, phase space transport, lobe dynamics
Citation: Naik S.,  Lekien F.,  Ross S. D., Computational Method for Phase Space Transport with Applications to Lobe Dynamics and Rate of Escape, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 272-297
Kilin A. A.,  Pivovarova E. N.
This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous right-hand side. Equations describing the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded.
Keywords: integrable system, system with discontinuity, nonholonomic constraint, bifurcation diagram, absolute dynamics
Citation: Kilin A. A.,  Pivovarova E. N., The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 298-317

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