Crocheting the Lorenz Manifold

  title={Crocheting the Lorenz Manifold},
  author={Hinke M. Osinga and Bernd Krauskopf},
  journal={The Mathematical Intelligencer},
You have probably seen a picture of the famous butterfly-shaped Lorenz attractor — on a book cover, a conference poster, a coffee mug or a friend’s T-shirt. The Lorenz attractor is the best known image of a chaotic or strange attractor. We are concerned here with its close cousin, the two-dimensional stable manifold of the origin of the Lorenz system, which we call the Lorenz manifold for short. This surface organizes the dynamics in the three-dimensional phase space of the Lorenz system. It is… 
Visualizing curvature on the Lorenz manifold
The Lorenz manifold is an intriguing two-dimensional surface that illustrates chaotic dynamics in the well-known Lorenz system. While it is not possible to find an explicit analytic expression for
Global bifurcations of the Lorenz manifold
In this paper we consider the interaction of the Lorenz manifold—the two-dimensional stable manifold of the origin of the Lorenz equations—with the two-dimensional unstable manifolds of the secondary
How to Crochet a Space-Filling Pancake: the Math, the Art and What Next
The chaotic behavior of the famous Lorenz system is organized by an amazing surface: the Lorenz manifold. Initial conditions on different sides of this surface will behave differently after some
The Lorenz manifold: Crochet and curvature
We present a crocheted model of an intriguing two-dimensional surface — known as the Lorenz manifold — which illustrates chaotic dynamics in the well-known Lorenz system. The crochet instructions are
Visualizing the transition to chaos in the Lorenz system
The Lorenz system still fascinates many people because of the simplicity of the equations that generate such complicated dynamics on the famous butterfly attractor. This paper addresses the role of
Visualizing global manifolds during the transition to chaos in the Lorenz system
This work discusses an algorithm for computing global manifolds of vector fields that is decidedly geometric in nature, and allows to visualize the resulting surface by making use of the geodesic parametrization.
The Sculpture Manifold: A Band from a Surface, a Surface from a Band
The steel sculpture Manifold consists of an 8 cm wide closed band of stainless steel that winds around in an intricate way, curving and coming very close to itself. It is based on a complicated
Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields
We consider a homoclinic bifurcation of a vector field in $\R^3$, where a one-dimensional unstable manifold of an equilibrium is contained in the two-dimensional stable manifold of this same
Computation and visualization of invariant manifolds
This thesis starts with the basic concepts of dynamical systems, then introduces the general types of problems that the well-known software package AUTO solves, and discusses the basic bifurcation and stability analysis of general ODE systems.


Visualizing the structure of chaos in the Lorenz system
The Lorenz manifold as a collection of geodesic level sets
We demonstrate a method to compute a two-dimensional global stable or unstable manifold of a vector field as a sequence of approximate geodesic level sets. Specifically, we compute the Lorenz
Numerical approximations of strong (un)stable manifolds
The method of computing global one-dimensional stable or unstable manifolds of a hyperbolic equilibrium of a smooth vector field is well known. Such manifolds consist only of two trajectories and
The Lorenz attractor exists
Two-dimensional global manifolds of vector fields.
An efficient algorithm for computing two-dimensional stable and unstable manifolds of three-dimensional vector fields that allows one to study manifolds geometrically and obtain important features of dynamical behavior.
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.-
Computing Geodesic Level Sets on Global (Un)stable Manifolds of Vector Fields
Many applications give rise to dynamical systems in the form of a vector field with a phase space of moderate dimension. Examples are the Lorenz equations, mechanical and other oscillators, and mod...
Nonlinear Dynamics and Chaos
Nonlinear dynamics deals with more-or-less regular fluctuations in system variables caused by feedback intrinsic to the system (as opposed to external forces). Chaos is the most exotic form of
COVER ILLUSTRATION: The Lorenz manifold as a collection of geodesic level sets
A tone source is located at each telephone set of a private branch exchange (PBX) or a key telephone system. The tone source provides call progress tones and warble ring tone, as well as a feedback
Deterministic nonperiodic flow
Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with