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# Volume 22, Number 3, 2017

 Barreira L.,  Dragičević D.,  Valls  C. Nonuniform Exponential Dichotomies and Lyapunov Functions Abstract 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 DOI:10.1134/S1560354717030017
 Jalnine A. Y.,  Kuznetsov S. P. Autonomous Strange Nonchaotic Oscillations in a System of Mechanical Rotators Abstract 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 DOI:10.1134/S1560354717030029
 Xue J. Nekhoroshev Estimates for Commuting Nearly Integrable Symplectomorphisms Abstract 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 DOI:10.1134/S1560354717030054
 Kudryashov N. A.,  Gaur I. Y. Weak Nonlinear Asymptotic Solutions for the Fourth Order Analogue of the Second Painlevé Equation Abstract 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 DOI:10.1134/S1560354717030066