How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6

We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order $$O(\epsilon ^2)$$ O ( ϵ 2 ) in the coefficients of the discretization, where $$\epsilon $$ ϵ is the stepsize.

Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game

This paper considers a Bertrand competition between two firms whose decision variables are derived from a quadratic utility function. The first firm produces new products with their own prices while the second firm re-manufactures returned products and sells them in the market at prices that may be less than or equal to the price of the first firm. Dynamically, this competition is constructed on which boundedly rational firms apply a gradient adjustment mechanism to update their prices in each period. According to this mechanism and the nature of the competition, a two-dimensional piecewise smooth discrete dynamic map was constructed in order to study the complex dynamic characteristics of the game. The phase plane of the map was divided into two different regions, separated by border curve. The equilibrium points of the map, in each region on where they are defined, were calculated, and their stability conditions were investigated. Furthermore, we conducted a global analysis to investigate the complex structure of the map, such as closed invariant curves, periodic cycles, and chaotic attractors and their basins, which cause qualitative changes as some parameters are allowed to vary.

Analytic classification of germs of parabolic antiholomorphic diffeomorphisms of codimension k

We investigate the local dynamics of antiholomorphic diffeomorphisms around a parabolic fixed point. We first give a normal form. Then we give a complete classification including a modulus space for antiholomorphic germs with a parabolic fixed point under analytic conjugacy. We then study some geometric applications: existence of real analytic invariant curves, existence of holomorphic and antiholomorphic roots of holomorphic and antiholomorphic parabolic germs, and commuting holomorphic and antiholomorphic parabolic germs.

Chaotic Motion in the Breathing Circle Billiard

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves in the phase space. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map has positive topological entropy. The proof relies on variational techniques based on the Aubry–Mather theory.