Global dynamics of the general second order difference equation in the first quadrant

We investigate the global behavior of a general polynomial second order difference equation with non-negative parameters and initial conditions. We establish the relations for local stability of the equilibrium solutions and existence of the period-two solutions. We then use this result to give global behavior results for special ranges of parameters and determine the basins of attraction of all equilibrium and periodic points. We give a class of examples of second order difference equations for which the Julia set can be found explicitly and is represented by a planar curve

Asymptotic approximations of a stable and unstable manifolds of a two-dimensional quadratic map

We find the asymptotic approximations of the stable and unstable manifolds of the saddle equilibrium solutions and the saddle period-two solutions of the following difference equation xn+1= cx2n-1+dxn+1; where the parameters c and d are positive numbers and initial conditions x-1and x0are arbitrary nonnegative numbers. These manifolds determine completely global dynamics of this equation

Global dynamics of quadratic second order difference equation in the first quadrant

We investigate the global behavior of a quadratic second order difference equationxn+1=Axn2+Bxnxn-1+Cxn-12+ Dxn+Exn-1+F,n=0,1,.with non-negative parameters and initial conditions. We find the global behavior for all ranges of parameters and determine the basins of attraction of all equilibrium points. © 2013 Elsevier Inc. All rights reserved

The DECam Ecliptic Exploration Project (DEEP). VI. First Multiyear Observations of Trans-Neptunian Objects

We present the first set of trans-Neptunian objects (TNOs) observed on multiple nights in data taken from the DECam Ecliptic Exploration Project. Of these 110 TNOs, 105 do not coincide with previously known TNOs and appear to be new discoveries. Each individual detection for our objects resulted from a digital tracking search at TNO rates of motion, using two-to-four-hour exposure sets, and the detections were subsequently linked across multiple observing seasons. This procedure allows us to find objects with magnitudes m _VR ≈ 26. The object discovery processing also included a comprehensive population of objects injected into the images, with a recovery and linking rate of at least 94%. The final orbits were obtained using a specialized orbit-fitting procedure that accounts for the positional errors derived from the digital tracking procedure. Our results include robust orbits and magnitudes for classical TNOs with absolute magnitudes H ∼ 10, as well as a dynamically detached object found at 76 au (semimajor axis a ≈ 77 au). We find a disagreement between our population of classical TNOs and the CFEPS-L7 three-component model for the Kuiper Belt