Smooth dependence of ODE solutions on parameters

Statement

Suppose $F(t,y,\lambda)$ is $C^k$ , $k\ge1$ , near $(t_0,y_0,\lambda_0)$ . The local solution of $y'=F(t,y,\lambda)$ depends $C^k$ on time, initial state, initial time and $\lambda$ , on a common sufficiently small neighborhood. Smooth parameterized initial data can be composed with this solution map.

Variational equation

For one parameter and fixed initial time, $z=\partial_\lambda y$ satisfies

z'=D_yF(t,y,\lambda)z+\partial_\lambda F(t,y,\lambda), \qquad z(t_0)=\partial_\lambda y_0.

The Duhamel formula controls this derivative. Further differentiation gives linear equations in the highest parameter derivative, with sources involving lower derivatives. Uniform estimates over a longer interval require common bounds and a common domain; pointwise existence for each parameter does not supply them.

References

Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, Theorem 2.11.