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  • Revisiting the Human and Nature Dynamics Model
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-04-10
    Basil Grammaticos, Ralph Willox, Junkichi Satsuma

    We present a simple model for describing the dynamics of the interaction between a homogeneous population or society, and the natural resources and reserves that the society needs for its survival. The model is formulated in terms of ordinary differential equations, which are subsequently discretised, the discrete system providing a natural integrator for the continuous one. An ultradiscrete, generalised

    更新日期:2020-04-10
  • Simple Flows on Tori with Uncommon Chaos
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-04-10
    Carles Simó

    We consider a family of simple flows in tori that display chaotic behavior in a wide sense. But these flows do not have homoclinic nor heteroclinic orbits. They have only a fixed point which is of parabolic type. However, the dynamics returns infinitely many times near the fixed point due to quasi-periodicity. A preliminary example is given for maps introduced in a paper containing many examples of

    更新日期:2020-04-10
  • Dynamics of Rubber Chaplygin Sphere under Periodic Control
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-04-10
    Ivan S. Mamaev, Evgeny V. Vetchanin

    This paper examines the motion of a balanced spherical robot under the action of periodically changing moments of inertia and gyrostatic momentum. The system of equations of motion is constructed using the model of the rolling of a rubber body (without slipping and twisting) and is nonconservative. It is shown that in the absence of gyrostatic momentum the equations of motion admit three invariant

    更新日期:2020-04-10
  • General Jacobi Coordinates and Herman Resonance for Some Nonheliocentric Celestial N -body Problems
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-04-10
    Chjan C. Lim

    The general Jacobi symplectic variables generated by a combinatorial algorithm from the full binary tree T(N) are used to formulate some nonheliocentric gravitational N-body problems in perturbation form. The resulting uncoupled term HU for (N - 1) independent Keplerian motions and the perturbation term UP are both explicitly dependent on the partial ordering induced by the tree T(N). This leads to

    更新日期:2020-04-10
  • Conservation Laws for Highly Dispersive Optical Solitons in Birefringent Fibers
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-04-10
    Anjan Biswas, Abdul H. Kara, Qin Zhou, Abdullah Kamis Alzahrani, Milivoj R. Belic

    This paper reports conservation laws for highly dispersive optical solitons in birefringent fibers. Three forms of nonlinearities are studied which are Kerr, polynomial and nonlocal laws. Power, linear momentum and Hamiltonian are conserved for these types of nonlinear refractive index.

    更新日期:2020-04-10
  • On the Nonlinear Stability of the Triangular Points in the Circular Spatial Restricted Three-body Problem
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-04-10
    Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas

    The well-known problem of the nonlinear stability of L4 and L5 in the circular spatial restricted three-body problem is revisited. Some new results in the light of the concept of Lie (formal) stability are presented. In particular, we provide stability and asymptotic estimates for three specific values of the mass ratio that remained uncovered. Moreover, in many cases we improve the estimates found

    更新日期:2020-04-10
  • Two Variations on the Periscope Theorem
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-02-20
    Serge Tabachnikov

    A (multidimensional) spherical periscope is a system of two ideal mirrors that reflect every ray of light emanating from some point back to this point. A spherical periscope defines a local diffeomorphism of the space of rays through this point, and we describe such diffeomorphisms. We also solve a similar problem for (multidimensional) reversed periscopes, the systems of two mirrors that reverse the

    更新日期:2020-02-20
  • N -body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-02-20
    Jaime Andrade, Stefanella Boatto, Thierry Combot, Gladston Duarte, Teresinha J. Stuchi

    The formulation of the dynamics of N-bodies on the surface of an infinite cylinder is considered. We have chosen such a surface to be able to study the impact of the surface’s topology in the particle’s dynamics. For this purpose we need to make a choice of how to generalize the notion of gravitational potential on a general manifold. Following Boatto, Dritschel and Schaefer [5], we define a gravitational

    更新日期:2020-02-20
  • On the Nonholonomic Routh Sphere in a Magnetic Field
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-02-20
    Alexey V. Borisov, Andrey V. Tsiganov

    This paper is concerned with the motion of an unbalanced dynamically symmetric sphere rolling without slipping on a plane in the presence of an external magnetic field. It is assumed that the sphere can consist completely or partially of dielectric, ferromagnetic, superconducting and crystalline materials. According to the existing phenomenological theory, the analysis of the sphere’s dynamics requires

    更新日期:2020-02-20
  • On Periodic Poincaré Motions in the Case of Degeneracy of an Unperturbed System
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-02-20
    Anatoly P. Markeev

    This paper is concerned with a one-degree-of-freedom system close to an integrable system. It is assumed that the Hamiltonian function of the system is analytic in all its arguments, its perturbing part is periodic in time, and the unperturbed Hamiltonian function is degenerate. The existence of periodic motions with a period divisible by the period of perturbation is shown by the Poincaré methods

    更新日期:2020-02-20
  • Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-02-20
    Yuri L. Sachkov

    We consider control-linear left-invariant time-optimal problems on step 2 Carnot groups with a strictly convex set of control parameters (in particular, sub-Finsler problems). We describe all Casimirs linear in momenta on the dual of the Lie algebra. In the case of rank 3 Lie groups we describe the symplectic foliation on the dual of the Lie algebra. On this basis we show that extremal controls are

    更新日期:2020-02-20
  • On Dynamics of Jellet’s Egg. Asymptotic Solutions Revisited
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-02-20
    Stefan Rauch-Wojciechowski, Maria Przybylska

    We study here the asymptotic condition \(\dot E = - \mu {g_n}b_A^2 = 0\) for an eccentric rolling and sliding ellipsoid with axes of principal moments of inertia directed along geometric axes of the ellipsoid, a rigid body which we call here Jellett’s egg (JE). It is shown by using dynamic equations expressed in terms of Euler angles that the asymptotic condition is satisfied by stationary solutions

    更新日期:2020-02-20
  • A Map for Systems with Resonant Trappings and Scatterings
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-02-20
    Anton V. Artemyev, Anatoly I. Neishtadt, Alexei A. Vasiliev

    Slow-fast dynamics and resonant phenomena can be found in a wide range of physical systems, including problems of celestial mechanics, fluid mechanics, and charged particle dynamics. Important resonant effects that control transport in the phase space in such systems are resonant scatterings and trappings. For systems with weak diffusive scatterings the transport properties can be described with the

    更新日期:2020-02-20
  • Asymptotic Invariant Surfaces for Non-Autonomous Pendulum-Type Systems
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-02-20
    Alexander A. Burov, Anna D. Guerman, Vasily I. Nikonov

    Invariant surfaces play a crucial role in the dynamics of mechanical systems separating regions filled with chaotic behavior. Cases where such surfaces can be found are rare enough. Perhaps the most famous of these is the so-called Hess case in the mechanics of a heavy rigid body with a fixed point. We consider here the motion of a non-autonomous mechanical pendulum-like system with one degree of freedom

    更新日期:2020-02-20
  • Lax Pairs and Special Polynomials Associated with Self-similar Reductions of Sawada — Kotera and Kupershmidt Equations
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2020-02-20
    Nikolay A. Kudryashov

    Self-similar reductions of the Sawada-Kotera and Kupershmidt equations are studied. Results of Painlevé’s test for these equations are given. Lax pairs for solving the Cauchy problems to these nonlinear ordinary differential equations are found. Special solutions of the Sawada-Kotera and Kupershmidt equations expressed via the first Painlevé equation are presented. Exact solutions of the Sawada-Kotera

    更新日期:2020-02-20
  • Roaming at Constant Kinetic Energy: Chesnavich’s Model and the Hamiltonian Isokinetic Thermostat
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2019-12-10
    Vladimír Krajňák, Gregory S. Ezra, Stephen Wiggins

    We consider the roaming mechanism for chemical reactions under the nonholonomic constraint of constant kinetic energy. Our study is carried out in the context of the Hamiltonian isokinetic thermostat applied to Chesnavich’s model for an ion-molecule reaction. Through an analysis of phase space structures we show that imposing the nonholonomic constraint does not prevent the system from exhibiting roaming

    更新日期:2019-12-10
  • On Resonances in Hamiltonian Systems with Three Degrees of Freedom
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2019-12-10
    Alexander A. Karabanov, Albert D. Morozov

    We address the dynamics of near-integrable Hamiltonian systems with 3 degrees of freedom in extended vicinities of unperturbed resonant invariant Liouville tori. The main attention is paid to the case where the unperturbed torus satisfies two independent resonance conditions. In this case the average dynamics is 4-dimensional, reduced to a generalised motion under a conservative force on the 2-torus

    更新日期:2019-12-10
  • Stability of Periodic Solutions of the N -vortex Problem in General Domains
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2019-12-10
    Björn Gebhard, Rafael Ortega

    We investigate stability properties of a type of periodic solutions of the N-vortex problem on general domains Ω ⊂ ℝ2. The solutions in question bifurcate from rigidly rotating configurations of the whole-plane vortex system and a critical point a0 ∈ Ω of the Robin function associated to the Dirichlet Laplacian of Ω. Under a linear stability condition on the initial rotating configuration, which can

    更新日期:2019-12-10
  • Twisted States in a System of Nonlinearly Coupled Phase Oscillators
    Regul. Chaot. Dyn. (IF 0.933) Pub Date : 2019-12-10
    Dmitry Bolotov, Maxim Bolotov, Lev Smirnov, Grigory Osipov, Arkady Pikovsky

    We study the dynamics of the ring of identical phase oscillators with nonlinear nonlocal coupling. Using the Ott–Antonsen approach, the problem is formulated as a system of partial derivative equations for the local complex order parameter. In this framework, we investigate the existence and stability of twisted states. Both fully coherent and partially coherent stable twisted states were found (the

    更新日期:2019-12-10
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