Completely degenerate lower-dimensional invariant tori for Hamiltonian system

We study the persistence of lower-dimensional invariant tori for a nearly integrable completely degenerate Hamiltonian system. It is shown that the majority of unperturbed invariant tori can survive from the perturbations which are only assumed the smallness and smoothness.     

Existence of weak solutions for a Bingham fluid-rigid body system

We consider the motion of a rigid body in a viscoplastic material. This material is modeled by the 3D Bingham equations, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is an inequality (due to the plasticity of the fluid), and it involves a free boundary (due to the motion of the rigid body). We approximate it by regularizing the convex terms in the Bingham fluid and by using a penalty method to take into account the presence of the rigid body.     

An integrable coupling hierarchy of the Mkdv_integrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy

A four-by-four matrix spectral problem is introduced, locality of solution of the related stationary zero curvature equation is proved. An integrable coupling hierarchy of the Mkdv_integrable systems is presented. The Hamiltonian structure of the resulting integrable coupling hierarchy is established by means of the variational identity. It is shown that the resulting integrable couplings are all Liouville integrable Hamiltonian systems. Ultimately, through the nonisospectral zero curvature representation, a nonisospectral integrable hierarchy associated with the resulting integrable couplings is constructed.     

The hierarchy of an integrable coupling and its Hamiltonian structure

A type of higher dimensional loop algebra is constructed from which an isospectral problem is established. It follows that an integrable coupling, actually an extended integrable model of the existed solitary hierarchy of equations, is obtained by taking use of the zero curvature equation, whose Hamiltonian structure is worked out by employing the constructed quadratic identity.     

An affine model for the actions in an integrable system with monodromy

We give a standard model for the flat affine geometry defined by the local action variables of a completely integrable system. We are primarily interested in the affine structure in the neighborhood of a critical value with nontrivial monodromy.      

On degenerate resonances and “vortex pairs”

Hamiltonian systems with 3/2 degrees of freedom close to non-linear autonomous are studied. For unperturbed equations with a nonlinearity in the form of a polynomial of the fourth or fifth degree, their coefficients are specified for which the period on closed phase curves is not a monotone function of the energy and has extreme values of the maximal order. When the perturbation is periodic in time, this non-monotonicity leads to the existence of degenerate resonances. The numerical study of the Poincaré map was carried out and bifurcations related to the formation of the vortex pairs within the resonance zones were found. For systems of a general form at arbitrarily small perturbations the absence of vortex pairs is proved. An explanation of the appearance of these structures for the Poincaré map is presented.      

Local and global existence of solutions to a quasilinear degenerate chemotaxis system with unbounded initial data

This paper is concerned with local and global existence of solutions to the parabolic‐elliptic chemotaxis system . Marinoschi (J. Math. Anal. Appl. 2013; 402:415–439) established an abstract approach using nonlinear m‐accretive operators to giving existence of local solutions to this system when 0 < D₀≤D′(r)≤D_∞<∞ and (r₁,r₂)?K(r₁,r₂)r₁ is Lipschitz continuous on , provided that the initial data is assumed to be small. The smallness assumption on the initial data was recently removed (J. Math. Anal. Appl. 2014; 419:756–774). However the case of non‐Lipschitz and degenerate diffusion, such as D(r) = r^m(m > 1), is left incomplete. This paper presents the local and global solvability of the system with non‐Lipschitz and degenerate diffusion by applying (J. Math. Anal. Appl. 2013; 402:415–439) and (J. Math. Anal. Appl. 2014; 419:756–774) to an approximate system. In particular, the result in the present paper does not require any properties of boundedness, smoothness and radial symmetry of initial data. This makes it difficult to deal with nonlinearity. Copyright © 2015 John Wiley & Sons, Ltd.     

The Serret-Andoyer formalism in rigid-body dynamics: II. Geometry,stabilization, and control

This paper continues the review of the Serret-Andoyer (SA) canonical formalism in rigid-body dynamics, commenced by [1], and presents some new result. We discuss the applications of the SA formalism to control theory. Considerable attention is devoted to the geometry of the Andoyer variables and to the modeling of control torques. We develop a new approach to Stabilization of rigid-body dynamics, an approach wherein the state-space model is formulated through sets of canonical elements that partially or completely reduce the unperturbed Euler-Poinsot problem. The controllability of the system model is examined using the notion of accessibility, and is shown to be accessible. Based on the accessibility proof, a Hamiltonian controller is derived by using the Hamiltonian as a natural Lyapunov function for the closed-loop dynamics. It is shown that the Hamiltonian controller is both passive and inverse optimal with respect to a meaningful performance-index. Finally, we point out the possibility to apply methods of structure-preserving control using the canonical Andoyer variables, and we illustrate this approach on rigid bodies containing internal rotors.      

A molecular dynamics model for symplectic intergrators

Mechanical systems in the very large scale like in celestial mechanics or in the very small scale like in the molecular dynamics can be modelled without dissipation. The resulting Hamiltonian systems possess conservation properties, which are characterized with the term symplecticness, Numerical integration schemes should preserve the symplecticness. Different methods are introduced and their performance is studied for constant and variable step size. As test examples two systems from molecular dynamics are introduced.     

A Lagrangian camel

We prove the Lagrangian analogue of the symplectic camel theorem: there are compact Lagrangian submanifolds of that cannot be moved through a small hole by a global Hamiltonian isotopy with compact support. Received: November 9, 1998.     

The influence of the kinetic energy in equilibrium of Hamiltonian systems

We provide a simple and explicit example of the influence of the kinetic energy in the stability of the equilibrium of classical Hamiltonian systems of the type . We construct a potential energy π of class C^k with a critical point at 0 and two different positive defined matrices B₁andB₂, both independent of q, and show that the equilibrium (0,0) is stable according to Lyapunov for the Hamiltonian , while for the equilibrium is unstable. Moreover, we give another example showing that even in the analytical situation the kinetic energy has influence in the stability, in the sense that there is an analytic potential energy π and two kinetic energies, also analytic, T₁ and T₂ such that the attractive basin of (0,0) is a two-dimensional manifold in the system of Hamiltonian π+T₁ and a one-dimensional manifold in the system of Hamiltonian π+T₂.