Publication

Exploring Unstable Periodic Orbits Using Super Stable Ones

2002
Conference Papers

Abstract

This paper deals with Super Stable Orbits (SSOs) of the logistic map, and how they can be used to describe Unstable Periodic Orbits (UPOs).~We will show that Unstable Periodic Orbits have almost the same correlation properties as Super Stable ones from which they were born, and this will be confirmed by simulations. We will present one application, a method to detect UPOs merged in a strange attractor.

Official source

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2002
Conference Papers

Abstract

Official source

Related concepts (32)

Orbital resonance

In celestial mechanics, orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers. Most commonly, this relationship is found between a pair of objects (binary resonance). The physical principle behind orbital resonance is similar in concept to pushing a child on a swing, whereby the orbit and the swing both have a natural frequency, and the body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion.

Attractor

In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space.

Logistic map

The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple nonlinear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation written down by Pierre François Verhulst. Mathematically, the logistic map is written where xn is a number between zero and one, which represents the ratio of existing population to the maximum possible population.

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