Regular and Chaotic Dynamics

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Exact Solutions to the Beltrami Equation with a Non-constant $$\alpha({\bf x})$$
Regular and Chaotic Dynamics - Tập 26 - Trang 692-699 - 2021
Oleg Bogoyavlenskij, Yuyang Peng
Infinite families of new exact solutions to the Beltrami equation with a non-constant $$\alpha({\bf x})$$ are derived. Differential operators connecting the steady axisymmetric Klein – Gordon equation and a special case of the Grad – Shafranov equation are constructed. A Lie semi-group of nonlinear transformations of the Grad – Shafranov equation is found.... hiện toàn bộ
Heteroclinic and Homoclinic Structures in the System of Four Identical Globally Coupled Phase Oscillators with Nonpairwise Interactions
Regular and Chaotic Dynamics - - 2019
Evgeny A. Grines, Grigory V. Osipov
Systems of N identical globally coupled phase oscillators can demonstrate a multitude of complex behaviors. Such systems can have chaotic dynamics for N > 4 when a coupling function is biharmonic. The case N = 4 does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies have shown that some of ch... hiện toàn bộ
Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations
Regular and Chaotic Dynamics - Tập 16 - Trang 562-576 - 2011
Maria V. Demina, Nikolai A. Kudryashov
Rational solutions and special polynomials associated with the generalized K 2 hierarchy are studied. This hierarchy is related to the Sawada-Kotera and Kaup-Kupershmidt equations and some other integrable partial differential equations including the Fordy-Gibbons equation. Differential-difference relations and differential equations satisfied by the polynomials are derived. The relationship betwe... hiện toàn bộ
Poisson structures for geometric curve flows in semi-simple homogeneous spaces
Regular and Chaotic Dynamics - Tập 15 - Trang 532-550 - 2010
G. Marí Beffa, P. J. Olver
We apply the equivariant method of moving frames to investigate the existence of Poisson structures for geometric curve flows in semi-simple homogeneous spaces. We derive explicit compatibility conditions that ensure that a geometric flow induces a Hamiltonian evolution of the associated differential invariants. Our results are illustrated by several examples of geometric interest.
The Lagrange-D’Alembert-Poincaré equations and integrability for the Euler’s disk
Regular and Chaotic Dynamics - Tập 12 - Trang 56-67 - 2007
H. Cendra, V. A. Díaz
Nonholonomic systems are described by the Lagrange-D’Alembert’s principle. The presence of symmetry leads, upon the choice of an arbitrary principal connection, to a reduced D’Alembert’s principle and to the Lagrange-D’Alembert-Poincaré reduced equations. The case of rolling constraints has a long history and it has been the purpose of many works in recent times. In this paper we find reduced equa... hiện toàn bộ
Dynamics of the finite-dimensional Kuramoto model: Global and cluster synchronization
Regular and Chaotic Dynamics - Tập 20 Số 1 - Trang 37-48 - 2015
Belykh, Vladimir N., Petrov, Valentin S., Osipov, Grigory V.
Synchronization phenomena in networks of globally coupled non-identical oscillators have been one of the key problems in nonlinear dynamics over the years. The main model used within this framework is the Kuramoto model. This model shows three main types of behavior: global synchronization, cluster synchronization including chimera states and totally incoherent behavior. We present new sufficient ... hiện toàn bộ
Projective dynamics and classical gravitation
Regular and Chaotic Dynamics - - 2008
Alain Albouy
The Centroid-Deformation Decomposition for Buoyant Vortex Patch Motion
Regular and Chaotic Dynamics - Tập 26 - Trang 577-599 - 2021
Banavara N. Shashikanth, Rangachari Kidambi
The motion of a two-dimensional buoyant vortex patch, i. e., a vortex patch with a uniform density different from the uniform density of the surrounding fluid, is analyzed in terms of evolution equations for the motion of its centroid, deformation of its boundary and the strength distribution of a vortex sheet which is essential to enforce pressure continuity across the boundary. The equations for... hiện toàn bộ
Normal forms, stability and splitting of invariant manifolds II. Finitely differentiable Hamiltonians
Regular and Chaotic Dynamics - Tập 18 - Trang 261-276 - 2013
Abed Bounemoura
This paper is a sequel to “Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians”, in which we gave a new construction of resonant normal forms with an exponentially small remainder for near-integrableGevrey Hamiltonians at a quasiperiodic frequency, using a method of periodic approximations. In this second part we focus on finitely differentiable Hamiltonians, and we... hiện toàn bộ
On Periodic Poincaré Motions in the Case of Degeneracy of an Unperturbed System
Regular and Chaotic Dynamics - Tập 25 - Trang 111-120 - 2020
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... hiện toàn bộ
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