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Volume 26, Number 1

Volume 26, Number 1, 2021

Cox C.,  Feres R.,  Zhao B.
Rolling Systems and Their Billiard Limits
Abstract
Billiard systems, broadly speaking, may be regarded as models of mechanical systems in which rigid parts interact through elastic impulsive (collision) forces. When it is desired or necessary to account for linear/angular momentum exchange in collisions involving a spherical body, a type of billiard system often referred to as no-slip has been used. In recent work, it has become apparent that no-slip billiards resemble nonholonomic mechanical systems in a number of ways. Based on an idea by Borisov, Kilin and Mamaev, we show that no-slip billiards very generally arise as limits of nonholonomic (rolling) systems, in a way that is akin to how ordinary billiards arise as limits of geodesic flows through a flattening of the Riemannian manifold.
Keywords: no-slip billiards, nonholonomic systems
Citation: Cox C.,  Feres R.,  Zhao B., Rolling Systems and Their Billiard Limits, Regular and Chaotic Dynamics, 2021, vol. 26, no. 1, pp. 1-21
DOI:10.1134/S1560354721010019
Kaur N.,  Goyal K.
Uncertainty Quantification of Stochastic Epidemic SIR Models Using B-spline Polynomial Chaos
Abstract
Real-life epidemic situations are modeled using systems of differential equations (DEs) by considering deterministic parameters. However, in reality, the transmission parameters involved in such models experience a lot of variations and it is not possible to compute them exactly. In this paper, we apply B-spline wavelet-based generalized polynomial chaos (gPC) to analyze possible stochastic epidemic processes. A sensitivity analysis (SA) has been performed to investigate the behavior of randomness in a simple epidemic model. It has been analyzed that a linear B-spline wavelet basis shows accurate results by involving fewer polynomial chaos expansions (PCE) in comparison to cubic B-spline wavelets. We have carried out our developed method on two real outbreaks of diseases, firstly, influenza which affected the British boarding school for boys in North England in 1978, and secondly, Ebola in Liberia in 2014. Real data from the British Medical Journal (influenza) and World Health Organization (Ebola) has been incorporated into the Susceptible-Infected-Recovered (SIR) model. It has been observed that the numerical results obtained by the proposed method are quite satisfactory.
Keywords: B-spline chaos, Ebola virus outbreak, Galerkin approximation, influenza, SIR model, stochastic ordinary differential equations, uncertainty quantification
Citation: Kaur N.,  Goyal K., Uncertainty Quantification of Stochastic Epidemic SIR Models Using B-spline Polynomial Chaos, Regular and Chaotic Dynamics, 2021, vol. 26, no. 1, pp. 22-38
DOI:10.1134/S1560354721010020
Araujo G. C.,  Cabral H. E.
Parametric Stability of a Charged Pendulum with an Oscillating Suspension Point
Abstract
We consider a planar pendulum with an oscillating suspension point and with the bob carrying an electric charge $q$. The pendulum oscillates above a fixed point with a charge $Q.$ The dynamics is studied as a system in the small parameter $\epsilon$ given by the amplitude of the suspension point. The system depends on two other parameters, $\alpha$ and $\beta,$ the first related to the frequency of the oscillation of the suspension point and the second being the ratio of charges. We study the parametric stability of the linearly stable equilibria and use the Deprit-Hori method to construct the boundary surfaces of the stability/instability regions.
Keywords: charged pendulum, parametric stability, boundary surfaces of stability, Hamiltonian system
Citation: Araujo G. C.,  Cabral H. E., Parametric Stability of a Charged Pendulum with an Oscillating Suspension Point, Regular and Chaotic Dynamics, 2021, vol. 26, no. 1, pp. 39-60
DOI:10.1134/S1560354721010032
Chierchia L.,  Koudjinan C. E.
V. I. Arnold’s ''Global'' KAM Theorem and Geometric Measure Estimates
Abstract
This paper continues the discussion started in [10] concerning Arnold's legacy on classical KAM theory and (some of) its modern developments. We prove a detailed and explicit ''global'' Arnold's KAM theorem, which yields, in particular, the Whitney conjugacy of a non-degenerate, real-analytic, nearly-integrable Hamiltonian system to an integrable system on a closed, nowhere dense, positive measure subset of the phase space. Detailed measure estimates on the Kolmogorov set are provided in case the phase space is: (A) a uniform neighbourhood of an arbitrary (bounded) set times the $d$-torus and (B) a domain with $C^2$ boundary times the $d$-torus. All constants are explicitly given.
Keywords: nearly-integrable Hamiltonian systems, perturbation theory, KAM theory, Arnold’s scheme, Kolmogorov set, primary invariant tori, Lagrangian tori, measure estimates, small divisors, integrability on nowhere dense sets, Diophantine frequencies
Citation: Chierchia L.,  Koudjinan C. E., V. I. Arnold’s ''Global'' KAM Theorem and Geometric Measure Estimates, Regular and Chaotic Dynamics, 2021, vol. 26, no. 1, pp. 61-88
DOI:10.1134/S1560354721010044
Tzemos A. C.,  Contopoulos G.
Integrals of Motion in Time-periodic Hamiltonian Systems: The Case of the Mathieu Equation
Abstract
We present an algorithm for constructing analytically approximate integrals of motion in simple time-periodic Hamiltonians of the form $H=H_0+ \varepsilon H_i$, where $\varepsilon$ is a perturbation parameter. We apply our algorithm in a Hamiltonian system whose dynamics is governed by the Mathieu equation and examine in detail the orbits and their stroboscopic invariant curves for different values of $\varepsilon$. We find the values of $\varepsilon_{crit}$ beyond which the orbits escape to infinity and construct integrals which are expressed as series in the perturbation parameter $\varepsilon$ and converge up to $\varepsilon_{crit}$. In the absence of resonances the invariant curves are concentric ellipses which are approximated very well by our integrals. Finally, we construct an integral of motion which describes the hyperbolic stroboscopic invariant curve of a resonant case.
Keywords: Hamiltonian systems, integrals of motion, Mathieu’s equation
Citation: Tzemos A. C.,  Contopoulos G., Integrals of Motion in Time-periodic Hamiltonian Systems: The Case of the Mathieu Equation, Regular and Chaotic Dynamics, 2021, vol. 26, no. 1, pp. 89-104
DOI:10.1134/S1560354721010056
Hosseini K.,  Matinfar M.,  Mirzazadeh M.
Soliton Solutions of High-order Nonlinear Schrödinger Equations with Different Laws of Nonlinearities
Abstract
In the present paper, high-order nonlinear Schrödinger equations in non-Kerr law media with different laws of nonlinearities are studied. In this respect, after considering a complex envelope and distinguishing the real and imaginary portions of the models, describing the propagation of solitons through nonlinear optical fibers, their soliton solutions are obtained using the well-organized new Kudryashov method. It is believed that the new Kudryashov method provides an effective mathematical tool to look for soliton solutions of high-order nonlinear Schrödinger equations.
Keywords: high-order nonlinear Schrödinger equations, non-Kerr law media, different laws of nonlinearities, new Kudryashov method, soliton solutions
Citation: Hosseini K.,  Matinfar M.,  Mirzazadeh M., Soliton Solutions of High-order Nonlinear Schrödinger Equations with Different Laws of Nonlinearities, Regular and Chaotic Dynamics, 2021, vol. 26, no. 1, pp. 105-112
DOI:10.1134/S1560354721010068

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