For linear systems of the form $Ax = \dot x$, one can find a close solution of the form: $ \dot x(t) = e^{At} x(0) $
Is there a way to show that a nonlinear solution of the form, $A(x) x = \dot x$ has no closed-form solution? You could have a solution in small time steps, for example, $\dot (x+ \tau) = e^{A(x) \tau} x(t)$. But given that $A(x)$ is dependent on x, I guess it is possible to have a closed solution because, at each time step, the x vector needs to be accounted for in A. Is there a way to formally show that?
The reason I ask is people keep seeking this closed for solution for the nonlinear system, and it seems to me impossible, am I wrong?
Thanks!
"Is there a way to show that a nonlinear system has no closed-form solution?" Yes, there is!
The existence of a "closed-form" solution is constrained to an resonance condition. More precisely:
Corollary: If the eigenvalues of $f(x)$ around a fixed point $x_*$, where the Jacobian matrix is semi-simple, are $\Bbb N$-independent, then there is no formal first integral.
This is Corollary 5.2 from Goriely's Integrability and Nonintegrability of Dynamical Systems.