Construction of Dynamical Systems
The concept of evolution in time is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of classical mechanical systems, that is the study of the initial value problems for their describing systems of ordinary differential equations.
where
- represents the velocity of the material point x
- v: T × M → M is a vector field in Rn or Cn and represents the change of velocity induced by the known forces acting on the given material point. Depending on the properties of this vector field, the mechanical system is called
-
- autonomous, when v(t, x) = v(x)
- homogeneous when v(t, 0) = 0 for all t
The solution is the evolution function already introduced in above
Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy
where is a functional from the set of evolution functions to the field of the complex numbers.
Read more about this topic: Dynamical System (definition)
Famous quotes containing the words construction of, construction and/or systems:
“The construction of life is at present in the power of facts far more than convictions.”
—Walter Benjamin (1892–1940)
“Striving toward a goal puts a more pleasing construction on our advance toward death.”
—Mason Cooley (b. 1927)
“The geometry of landscape and situation seems to create its own systems of time, the sense of a dynamic element which is cinematising the events of the canvas, translating a posture or ceremony into dynamic terms. The greatest movie of the 20th century is the Mona Lisa, just as the greatest novel is Gray’s Anatomy.”
—J.G. (James Graham)