Linear Ordinary Differential Equations
The most simple example of dynamical system is a system of linear, constant-coefficients ordinary differential equations, i.e. let and :
whose direct closed-form solution involves computation of the matrix exponential:
Another way, provided the solution is restricted to the local Lebesgue space of -dimensional vector fields, is to use its Laplace transform . In this case
The matrix function is called the resolvent matrix of the differential operator . It is meromorphic with respect to the complex parameter since its matrix elements are rational functions whose denominator is equal for all to . Its polar singularities are the eigenvalues of, whose order equals their index for it, i.e. .
Read more about this topic: Jordan Matrix
Famous quotes containing the words ordinary and/or differential:
“To an ordinary human being, love means nothing if it does not mean loving some people more than others.”
—George Orwell (1903–1950)
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)