We know that a higher-order ode can be converted to dynamical system by replacing each higher-order derivative by a new variable. What about inverse problem? Does a dynamical system convert to a higher-order ode?



Is autonomous dynamical system equivalent to one single higher-order ode?
For example, consider a system such as
$$ \dot{x} = x, \dot{y} = y $$
for which the origin is a proper node. There is no autonomous second-order ODE
$\ddot{z} = f(z, \dot{z})$ that has a proper node: if the linearization at an equilibrium point has a double eigenvalue, it is an improper node. So there 
can be no smooth conversion of the system to an autonomous second-order ODE. 


