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# Communications in Mathematical Sciences

## Volume 13 (2015)

### Number 3

### Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

### Sparse time frequency representations and dynamical systems

Pages: 673 – 694

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n3.a4

#### Authors

#### Abstract

In this paper, we establish a connection between the recently developed data-driven time-frequency analysis [T.Y. Hou and Z. Shi, Advances in Adaptive Data Analysis, 3, 1–28, 2011], [T.Y. Hou and Z. Shi, Applied and Comput. Harmonic Analysis, 35, 284–308, 2013] and the classical second order differential equations. The main idea of the data-driven time-frequency analysis is to decompose a multiscale signal into the 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 order of nonlinearity for the associated IMF. This generalizes a concept recently introduced by Prof. N. E. Huang et al. [N.E. Huang, M.-T. Lo, Z. Wu, and Xianyao Chen, US Patent filling number 12/241.565, Sept. 2011]. 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.

#### Keywords

sparse time frequency representations, order of nonlinearity, intrinsic mode function, dynamical system

#### 2010 Mathematics Subject Classification

37M10, 94A12