In this paper, we study the nonlinear orbital stability of soliton solutions of the Schamel equation with modular nonlinearity describing solitary waves of different polarities. The proof of the nonlinear stability of these solutions is established within the framework of the general theory for the stability of bound states that decrease rather rapidly on the real axis and correspond to soliton solutions of the translationally invariant infinite-dimensional Hamiltonian system (which is the Schamel equation under consideration). The required stability is established based on verification of the formulated stability conditions. These conditions, being sufficient, imply checking the spectral properties of the operator resulting from linearization of a functional that makes sense of the Lyapunov function. Generally speaking, for translationally invariant differential equations it is impossible to consider the usual stability, when a small perturbation of the solution remains small. The solutions of such equations are subject to investigation by the presence of orbital stability, i. e., stability “up to shear accuracy”. As a result, it is not possible to construct a Lyapunov function (functional) in the problems of stability of boundary states, which would have a local minimum determined by a neighborhood system of the basic functional space of the problem. If the conditions of orbital stability are met, the Lyapunov function has a conditional local minimum, i. e., a local minimum on some nonlinear submanifold of the basic functional space of the system of equations at a point defined by the solution to be investigated for stability. This nonlinear submanifold, as a rule, is determined by the condition of the constancy of the functional, the invariance of which, by virtue of the basic system of equations, is associated with the translational invariance of the problem.
A mathematical billiard consists of a planar region as a table and a ball moving on the table and reflecting from the boundary. By considering the motion as a discrete dynamical system, the billiard map is an area-preserving twist map. In this work, we show the result of nonexistence of invariant curves for two different billiard maps. The first result presents a sufficient condition for the nonexistence of invariant curves. In the second one, we prove the nonexistence of invariant curves near the boundary for Halpern’s billiard, which has a convergent billiard orbit.
In this paper, we provide a complete answer to the question posed in its title. For all closed surfaces for which this question has not yet been resolved, namely, closed nonorientable surfaces of odd genus greater than 5, a series of examples of pseudo-Anosov homeomorphisms are constructed. Each of these homeomorphisms is defined by means of the so-called code, which implies using the construction of a band surface. Combined with previously obtained results, this allows us to fill in the gap and to formulate a general result: pseudo-Anosov homeomorphisms exist on a closed nonorientable surface if and only if its genus is at least 4.
We show that bifurcations of four-dimensional symplectic diffeomorphisms with a quadratic homoclinic tangency to a saddle periodic orbit with real multipliers produce 2-elliptic periodic orbits if the tangency is not partially hyperbolic. We show that a normal form for the rescaled first-return maps near such tangency is given by a four-dimensional symplectic Hénon-like map and study bifurcations of the first-return maps in generic two-parameter families.
Nonholonomic systems are mechanical systems with ideal linear velocity constraints that are not derivable from position constraints and with dynamics identified by the Lagrange – d’Alembert principle. This paper surveys discrete-time nonholonomic mechanics with applications to numerical integration of nonholonomic systems. This includes an exposition of key elements of discrete mechanics, discrete Lagrange – d’Alembert principle, and exact nonholonomic integrators on vector spaces. Exact variational integrators were introduced and exposed in the context of Lagrangian mechanics by Marsden and West. These integrators sample the trajectories of mechanical systems and are useful for developing practical mechanical integrators.
