  In this paper, we establish a connection between the recently developed
data-driven time-frequency analysis \cite{HS11,HS13-1} and the classical second
order differential equations. The main idea of the data-driven time-frequency
analysis is to decompose a multiscale signal into a sparsest collection of
Intrinsic Mode Functions (IMFs) over the largest possible dictionary via
nonlinear optimization. These IMFs are of the form $a(t) \cos(\theta(t))$ where
the amplitude $a(t)$ is positive and slowly varying. The non-decreasing phase
function $\theta(t)$ is determined by the data and in general depends on the
signal in a nonlinear fashion. One of the main results of this paper is that we
show that each IMF can be associated with a solution of a second order ordinary
differential equation of the form $\ddot{x}+p(x,t)\dot{x}+q(x,t)=0$. Further,
we propose a localized variational formulation for this problem and develop an
effective $l^1$-based optimization method to recover $p(x,t)$ and $q(x,t)$ by
looking for a sparse representation of $p$ and $q$ in terms of the polynomial
basis. Depending on the form of nonlinearity in $p(x,t)$ and $q(x,t)$, we can
define the degree of nonlinearity for the associated IMF. %and the
corresponding coefficients for the associated highest order nonlinear terms.
This generalizes a concept recently introduced by Prof. N. E. Huang et al.
\cite{Huang11}. Numerical examples will be provided to illustrate the
robustness and stability of the proposed method for data with or without noise.
This manuscript should be considered as a proof of concept.
