The Lorenz attractor is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its butterfly shape. The map shows how the state of a dynamical system (the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern.
The attractor itself, and the equations from which it is derived, were introduced by Edward Lorenz
, who derived it from the simplified equations of convection
rolls arising in the equations of the atmosphere
techniques are used in complex projects: from spacecrafts to chip design, from robotics to creating large software products to building bridges, Systems engineering uses a host of tools that include modeling & simulation, requirements analysis, and scheduling to manage complexity
The Work Breakdown Structure
(WBS) is a means of organizing system development activities based on system and product decompositions. The systems engineering
process produces system and product descriptions. These product architectures, together with associated services (e.g., program management, systems engineering, etc.) are organized and depicted in a hierarchical tree-like structure that is the WBS.
This image illustrates part of the Mandelbrot set fractal
. The size of the JPEG
file encoding the bitmap of this image is more than 17 kilobytes (approximately 140000 bits). The same file can be generated by a computer program much shorter than 140000 bits, however. Thus, the Kolmogorov complexity
of the JPEG file encoding the bitmap is much less than 140000.
An ecological analysis of CO-2 in an ecosystem
. As systems biology
, systems ecology
seeks a holistic
view of the interactions and transactions within and between biological and ecological systems.
illustrates part of the magnification of the phoenix set.
in the tip vortex
from an airplane
wing. Studies of the critical point
beyond which a system creates turbulence was important for Chaos theory, analyzed for example by the Soviet physicist Lev Landau
who developed the Landau-Hopf theory of turbulence
. David Ruelle
and Floris Takens
later predicted, against Landau, that fluid turbulence
could develop through a strange attractor
, a main concept of chaos theory
An aerial view of a human ecosystem
. Pictured is the city of Chicago
Visualization of a portion of the routes
on the Internet
Planets and dwarf planets of the Solar System
; while the sizes are to scale, the relative distances from the Sun are not.