In spite of the fact that the article was published in a meteorological journal, it is regarded as the beginning of a new discipline of mathematics known as the "chaos theory". This discipline is devoted to considerations of the problem of sensitive dependence of some systems on minor changes in the initial conditions. In another paper Lorentz described this phenomenon in a very picturesque way: One meteorologist remarked that if the theory were correct, one flap of a sea gull's wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls. This observation is often known as the "butterfly effect". Task Design a code that solves the Lorentz system. Apply the values of constants given above or play with them to find some other values for which the solution is chaotic. Analyze two cases with slightly different initial conditions. For each case present the results in two forms: Time evolution of each variable x(t), y(t), z(t), 3-dimensional curve in phase space (xy2) Code (main program): Code (function called by ODE procedure): Screenshot (case 1): Screenshot (case 2): Comments:

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
icon
Related questions
Question

Solve this on MATLAB

In 1963 an American metheorologist- Edward Lorentz was trying to develop a simplified
mathematical model for atmospheric convection. He considered an isolated fluid volume
that was heated from the bottom and cooled from above. He applied method known as the
Galerkin approximation and obtained a simplified model of three differential equations:
dx
dt
= o(y -x)
dy
=rx - y - xz
dt
dz
= xy – bz
dt
where the variable x is proportional to the intensity of a convective motion, y is proportional
to the temperature difference between ascending and descending currents and z accounts
for the distortion of temperature profile from linearity. a >1 is referred to as Prandti
number and can be interpreted as the ratio of viscous diffusion over thermal diffusion. r
describes the temperature difference on a heated layer, while b depends on the geometry of
a fluid cell.
Lorentz observed that for a =
way and gives different results each time he ran the simulation. He finally concluded that
this resulted from the fact that the accuracy of the computer was reaching only 6 decimal
places while the system was sensitive to much smaller changes. He published a research
paper, where this matter was described in the following way:
10, r = 28, b = the system behaves in a quite unpredicted
Two states differing by imperceptible amounts may eventually evolve into two
considerably different states . If, then, there is any error whatever in observing the
and in any real system such errors seem inevitable-an acceptable
present state-
prediction of an instantaneous state in the distant future may well be impossible.In
view of the inevitable inaccuracy and incompleteness of weather observations,
precise very-long-range forecasting would seem to be nonexistent.
Transcribed Image Text:In 1963 an American metheorologist- Edward Lorentz was trying to develop a simplified mathematical model for atmospheric convection. He considered an isolated fluid volume that was heated from the bottom and cooled from above. He applied method known as the Galerkin approximation and obtained a simplified model of three differential equations: dx dt = o(y -x) dy =rx - y - xz dt dz = xy – bz dt where the variable x is proportional to the intensity of a convective motion, y is proportional to the temperature difference between ascending and descending currents and z accounts for the distortion of temperature profile from linearity. a >1 is referred to as Prandti number and can be interpreted as the ratio of viscous diffusion over thermal diffusion. r describes the temperature difference on a heated layer, while b depends on the geometry of a fluid cell. Lorentz observed that for a = way and gives different results each time he ran the simulation. He finally concluded that this resulted from the fact that the accuracy of the computer was reaching only 6 decimal places while the system was sensitive to much smaller changes. He published a research paper, where this matter was described in the following way: 10, r = 28, b = the system behaves in a quite unpredicted Two states differing by imperceptible amounts may eventually evolve into two considerably different states . If, then, there is any error whatever in observing the and in any real system such errors seem inevitable-an acceptable present state- prediction of an instantaneous state in the distant future may well be impossible.In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-long-range forecasting would seem to be nonexistent.
In spite of the fact that the article was published in a meteorological journal, it is regarded as
the beginning of a new discipline of mathematics known as the "chaos theory". This
discipline is devoted to considerations of the problem of sensitive dependence of some
systems on minor changes in the initial conditions. In another paper Lorentz described this
phenomenon in a very picturesque way:
One meteorologist remarked that if the theory were correct, one flap of a sea gull's
wings would be enough to alter the course of the weather forever. The controversy
has not yet beenm settled, but the most recent evidence seems to favor the sea gulls.
This observation is often known as the "butterfly effect".
Task
Design a code that solves the Lorentz system. Apply the values of constants given above or
play with them to find some other values for which the solution is chaotic. Analyze two cases
with slightly different initial conditions. For each case present the results in two forms:
Time evolution of each variable x(t), y(t), z(t),
3-dimensional curve in phase space (xy.2)
Code (main program):
Code (function called by ODE procedure):
Screenshot (case 1):
Screenshot (case 2):
Comments:
Transcribed Image Text:In spite of the fact that the article was published in a meteorological journal, it is regarded as the beginning of a new discipline of mathematics known as the "chaos theory". This discipline is devoted to considerations of the problem of sensitive dependence of some systems on minor changes in the initial conditions. In another paper Lorentz described this phenomenon in a very picturesque way: One meteorologist remarked that if the theory were correct, one flap of a sea gull's wings would be enough to alter the course of the weather forever. The controversy has not yet beenm settled, but the most recent evidence seems to favor the sea gulls. This observation is often known as the "butterfly effect". Task Design a code that solves the Lorentz system. Apply the values of constants given above or play with them to find some other values for which the solution is chaotic. Analyze two cases with slightly different initial conditions. For each case present the results in two forms: Time evolution of each variable x(t), y(t), z(t), 3-dimensional curve in phase space (xy.2) Code (main program): Code (function called by ODE procedure): Screenshot (case 1): Screenshot (case 2): Comments:
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Knowledge Booster
Computational Systems
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Database System Concepts
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
C How to Program (8th Edition)
C How to Program (8th Edition)
Computer Science
ISBN:
9780133976892
Author:
Paul J. Deitel, Harvey Deitel
Publisher:
PEARSON
Database Systems: Design, Implementation, & Manag…
Database Systems: Design, Implementation, & Manag…
Computer Science
ISBN:
9781337627900
Author:
Carlos Coronel, Steven Morris
Publisher:
Cengage Learning
Programmable Logic Controllers
Programmable Logic Controllers
Computer Science
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education