To close the system of ODE

Question

I am studying stochastic equations. Lectures use a term "to close a system of ODE" or "the system of ODE can be closed only if coefficients"... What does it mean?

Answer 1

For a system of ODEs:

$$ \frac{dm_1}{dt} = f_1(m_1, m_2 ... m_k, ...m_n) $$ $$ \frac{dm_2}{dt} = f_2(m_1, m_2 ... m_k, ...m_n) $$ $$...$$ $$ \frac{dm_n}{dt} = f_n(m_1, m_2 ... m_k, ...m_n) $$

closing is finding a function H such that all but a few variables are expressed in this form $$ \frac{dm_1}{dt} = f_1(m_1, m_2 ... m_k, H(m_1, m_2 ... m_k)) $$ $$ \frac{dm_2}{dt} = f_2(m_1, m_2 ... m_k, H(m_1, m_2 ... m_k)) $$ $$...$$ $$ \frac{dm_k}{dt} = f_k(m_1, m_2 ... m_k, H(m_1, m_2 ... m_k)) $$

It simplifies handling of systems with infinite many variables