相关论文: Variational method for finding periodic orbits in …

One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg…

In laboratory studies and numerical simulations, we observe clear signatures of unstable time-periodic solutions in a moderately turbulent quasi-two-dimensional flow. We validate the dynamical relevance of such solutions by demonstrating…

We address several concerns related to the derivation of drift-ordered fluid equations. Starting from a fully Galilean invariant fluid system, we show how consistent sets of perturbative drift-fluid equations in the case of a isothermal…

Using the damped pendulum system we introduce the averaging method to study the periodic solutions of a dynamical system with small perturbation. We provide sufficient conditions for the existence of periodic solutions with small amplitude…

The dynamics of periodic swimming is studied for two models of a deformable sphere, the dipole-quadrupole model and the quadrupole-octupole model. For the two models the solution of the Navier-Stokes equations can be found exactly to second…

We propose a simple method to identify unstable parameter regions in general inviscid unidirectional shear flow stability problems. The theory is applicable to a wide range of basic flows, including those that are non-monotonic. We…

We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic…

Computer-aided engineering techniques are indispensable in modern engineering developments. In particular, partial differential equations are commonly used to simulate the dynamics of physical phenomena, but very large systems are often…

In this article, we investigate the existence and properties of time-periodic solutions for damped evolutionary partial differential equations subject to periodic forcing. Particular emphasis is placed on configurations where the energy…

An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework…

Evolution equations for the orientation distribution of axisymmetric particles in periodic flows are derived in the regime of small but non-zero Brownian rotations. The equations are based on a multiple time scale approach that allows fast…

Hamiltonian variational principles provided, since 60s, the means of developing very successful wave theories for nonlinear free-surface flows, under the assumption of irrotationality. This success, in conjunction with the recognition that…

All complex fluid motions, such as transition and turbulence, obeying the Navier-Stokes equations are non-linear phenomena. Some aspects of the non-linear terms of these equations are not well understood and are, in fact, misunderstood. The…

This paper develops a computational method for studying stable/unstable manifolds attached to periodic orbits of differential equations. The method uses high order Chebyshev-Taylor series approximations in conjunction with the…

We use geometric methods of equivariant dynamical systems to address a long-standing open problem in the theory of nematic liquid crystals, namely a proof of the existence and asymptotic stability of kayaking periodic orbits for which the…

We consider a linear differential system of Mathieu equations with periodic coefficients over periodic closed orbits and we prove that, arbitrarily close to this system, there is a linear differential system of Hamiltonian damped Mathieu…

Wilton ripples are a type of periodic traveling wave solution of the full water wave problem incorporating the effects of surface tension. They are characterized by a resonance phenomenon that alters the order at which the resonant harmonic…

This paper proves the existence of variational rotating solutions to the compressible non-isentropic Euler-Poisson equations with prescribed total mass. This extends the result of the isentropic case [Auchmuty and Beals, Arch. Ration. Mech.…

We shall deal with the periodic problem for nonlinear perturbations of abstract hyperbolic evolution equations generating an evolution system of contractions. We prove an averaging principle for the translation along trajectories operator…

In this short exposition, we describe equilibria and periodic orbits in terms of the flow map, {\Phi}, and discuss the essentials of the Jacobian-free Newton-Krylov (JFNK) method that can be used to find them. This method requires little…