Moving equilibrium theorem
Encyclopedia
Consider a dynamical system
(1)..........
(2)..........
with the state variables
and 
. Assume that 
is fast and 
is slow. Assume that the system (1) gives, for any fixed 
, an asymptotically stable solution 
. Substituting this for 
in (2) yields
(3)..........
Here
has been replaced by 
to indicate that the solution 
to (3) differs from the solution for 
obtainable from the system (1), (2).
The Moving Equilibrium Theorem suggested by Lotka
states that the solutions
obtainable from (3) approximate the solutions 
obtainable from (1), (2) provided the partial system (1) is asymptotically stable in 
for any given 
and heavily damped (fast).
The theorem has been proved for linear systems comprising real vectors
and 
. It permits reducing high-dimensional dynamical problems to lower dimensions and underlies Alfred Marshall
's temporary equilibrium method
.
(1)..........
(2)..........
with the state variables
and . Assume that is fast and is slow. Assume that the system (1) gives, for any fixed , an asymptotically stable solution . Substituting this for in (2) yields(3)..........
Here
has been replaced by to indicate that the solution to (3) differs from the solution for obtainable from the system (1), (2).The Moving Equilibrium Theorem suggested by Lotka
Alfred J. Lotka
Alfred James Lotka was a US mathematician, physical chemist, and statistician, famous for his work in population dynamics and energetics. An American biophysicist best known for his proposal of the predator-prey model, developed simultaneously but independently of Vito Volterra...
states that the solutions
obtainable from (3) approximate the solutions obtainable from (1), (2) provided the partial system (1) is asymptotically stable in for any given and heavily damped (fast).The theorem has been proved for linear systems comprising real vectors
and . It permits reducing high-dimensional dynamical problems to lower dimensions and underlies Alfred MarshallAlfred Marshall
Alfred Marshall was an Englishman and one of the most influential economists of his time. His book, Principles of Economics , was the dominant economic textbook in England for many years...
's temporary equilibrium method
Temporary equilibrium method
The temporary equilibrium method has been devised by Alfred Marshall for analyzing economic systems that comprise interdependent variables of different speed....
.