ON THE DYNAMIC OF HENON MAP. HOW TO CONTROL IT!
Ylldrita SALIHI, Krutan RASIMI
Abstract
Most of the dynamics displayed by highly complicated nonlinear systems also appear in simple nonlinear systems. The purpose of the two-dimensional map with a strange attractor was for it to be a simple mapping that possesses similar properties to the Lorenz system and its Poincare map. Henon map is investigated, periodic points are found, and chaotic attractors are produced.
In this paper, we will demonstrate an orbit of the Henon map with 10000 points, which vary on the initial conditions of the orbit and the values of the two parameters of the system. Known that the chaotic attractors in the Henon map are neither area filling (dimension 2) nor a simple curve (of dimension 1), the dimensions of these complicated geometries must be non-integer values between 1 and 2, and the chaotic attractors are then called fractals or strange attractors. Thee capacity or box-counting dimension dbox is the simplest possible way to measure such pathologies. We use The OGY (Ott, Grebogi, and Yorke) Method with the idea to make small time-dependent linear perturbations to the control parameter p in order to nudge the state towards the stable manifold of the desired fixed point.
Acting on 500 equally spaced initial points (x0,y0) on a circle or a square, represent an numerical experiment which may give us some hints about why Jupiter's red spot and Saturn's hexagon-shaped hurricane seem to exist forever without contracting.
Pages:
398 - 406