Volume 28, Number 3

An example of an analytic system of differential equations in $\mathbb{R}^6$ with an equilibrium formally stable and stable for most initial conditions is presented. By means of a divergent formal transformation this system is reduced to a Hamiltonian system with three degrees of freedom. Almost all its phase space is foliated by three-dimensional invariant tori carrying quasi-periodic trajectories. These tori do not fill all phase space. Though the ``gap'' between these tori has zero measure, this set is everywhere dense in $\mathbb{R}^6$ and unbounded phase trajectories are dense in this gap. In particular, the formally stable equilibrium is Lyapunov unstable. This behavior of phase trajectories is quite consistent with the diffusion in nearly integrable systems. The proofs are based on the Poincaré–Dulac theorem, the theory of almost periodic functions, and on some facts from the theory of inhomogeneous Diophantine approximations. Some open problems related to the example are presented.

In this work an exhaustive numerical and analytical investigation of the dynamics of a bi-parametric symplectic map, the so-called rational standard map, at moderate-to-large values of the amplitude parameter is addressed. After reviewing the model, a discussion concerning an analytical determination of the maximum Lyapunov exponent is provided together with thorough numerical experiments. The theoretical results are obtained in the limit of a nearly uniform distribution of the phase values. Correlations among phases lead to departures from the expected estimates. In this direction, a detailed study of the role of stable periodic islands of periods 1, 2 and 4 is included. Finally, an experimental relationship between the Lyapunov and instability times is shown, while an analytical one applies when correlations are irrelevant, which is the case, in general, for large values of the amplitude parameter.

In P.D.McSwiggen’s article, it was proposed Derived from Anosov type construction which leads to a partially hyperbolic diffeomorphism of the 3-torus. The nonwandering set of this diffeomorphism contains a two-dimensional attractor which consists of one-dimensional unstable manifolds of its points. The constructed diffeomorphism admits an invariant onedimensional orientable foliation such that it contains unstable manifolds of points of the attractor as its leaves. Moreover, this foliation has a global cross section (2-torus) and defines on it a Poincar´e map which is a regular Denjoy type homeomorphism. Such homeomorphisms are the most natural generalization of Denjoy homeomorphisms of the circle and play an important role in the description of the dynamics of aforementioned partially hyperbolic diffeomorphisms. In particular, the topological conjugacy of corresponding Poincaré maps provides necessary conditions for the topological conjugacy of the restrictions of such partially hyperbolic diffeomorphisms to their two-dimensional attractors. The nonwandering set of each regular Denjoy type homeomorphism is a Sierpiński set and each such homeomorphism is, by definition, semiconjugate to the minimal translation of the 2-torus. We introduce a complete invariant of topological conjugacy for regular Denjoy type homeomorphisms that is characterized by the minimal translation, which is semiconjugation of the given regular Denjoy type homeomorphism, with a distinguished, no more than countable set of orbits.

For two dimensional surfaces (smooth) Ricci and Yamabe flows are equivalent. In 2003, Chow and Luo developed the theory of combinatorial Ricci flow for circle packing metrics on closed triangulated surfaces. In 2004, Luo developed a theory of discrete Yamabe flow for closed triangulated surfaces. He investigated the formation of singularities and convergence to a metric of constant curvature.
In this note we develop the theory of a naïve discrete Ricci flow and its modification — the so-called weighted Ricci flow. We prove that this flow has a rich family of first integrals and is equivalent to a certain modification of Luo’s discrete Yamabe flow. We investigate the types of singularities of solutions for these flows and discuss convergence to a metric of weighted constant curvature.

In this study, we analyze a planar mathematical pendulum with a suspension point that oscillates harmonically in the vertical direction. The bob of the pendulum is electrically charged and is located between two wires with a uniform distribution of electric charges, both equidistant from the suspension point. The dynamics of this phenomenon is investigated. The system has three parameters, and we analyze the parametric stability of the equilibrium points, determining surfaces that separate the regions of stability and instability in the parameter space. In the case where the parameter associated with the charges is equal to zero, we obtain boundary curves that separate the regions of stability and instability for the Mathieu equation.