Exploring chaotic attractors in nonlinear dynamical system under fractal theory

Abstract

This paper introduces a new method to exploit chaotic attractors of nonlinear dynamics. The method decreases both the correlation and the noise of samples while preserves chaotic characteristics of samples in real time applications. The fractal prediction method is used to compressive sensing method in order to concentrate the sparse data on a trajectory. The proposed method can be applied to chaotic noise reduction, signal compression, and object’s movement synthesis in video. The experimental results indicate that the proposed method outperforms other state-of-the-art methods. Moreover, the results demonstrate that the chaotic extraction method is most effective to represent a chaotic dynamics of nonlinear time series for signal processing.

Appendix

There have been many methods to reduce the noise of chaotic time series such as 4D variational method (4DVAR) and Pseudo-orbit Data Assimilation (PDA). The 4DVAR algorithm, the initial condition of system is found based on observations. In the algorithm minimizes mismatch between trajectory and observations as

$$\begin{aligned} C_{4DVAR}= & {} \frac{1}{2}\left( {x_{-n+1} -x_{-n+1}^b } \right) ^{T}B^{-1}\left( {x_{-n+1} -x_{-n+1}^b } \right) \\&+\,\frac{1}{2}\mathop \sum \limits _{t=-n+1}^0 [h\left( {x_t-s_t} \right] ^{T}{\Gamma }^{-1}\left[ {h\left( {x_t } \right) -s_t } \right] \end{aligned}$$

where the background or initial value is ${x_{-n+1}^{b}}$ and the covariance matrix is B. The observations in time t is ${s_{t}}$.

PDA is applied to an initial condition ensemble to identify a reference trajectory with the projected observation sequence into the state space. The method minimizes the mismatches of points, which are not trajectories of model, using the gradient descent method based on Ikeda map. The gradient descent (GD) algorithm is used to minimize the mismatch error ${e_{n}}=|F({u}_{n})-{u_{n+1}}|,n=-m+1,\ldots ,-1$ with the cost function ${C(U)=\sum {e_{n}^{2}}}$. The pseudo orbit U is obtained using the gradient-based method

$$\begin{aligned} \frac{\partial C\left( U \right) }{\partial u_n }=2\times \left\{ {{\begin{array}{ll} -\left[ {u_{n+1} -F\left( {u_n } \right) } \right] d_n F\left( {u_n } \right) &{}\quad \quad n=-m+1 \\ \left[ {u_n -F\left( {u_{n+1} } \right) } \right] -\left[ {u_{n+1} -F\left( {u_n } \right) } \right] d_n F\left( {u_n } \right) &{} -m+1<n<0 \\ u_n -F\left( {u_{n-1} } \right) &{}\quad \quad \quad \quad \quad \quad n=0 \\ \end{array} }} \right. \end{aligned}$$

where $d_n F\left( {u_n } \right)$ is the Jacobian matrix of F. The pseudo orbit U is updated as

$$\begin{aligned} U=U+\frac{\partial C\left( U \right) }{\partial u_n } \end{aligned}$$

to generate a trajectory in the state space. To select ensemble members using a likelihood function, the candidate trajectories can be created by sampling a local state around the reference trajectory. A reference trajectory can be generated based on middle component $y_{-m/2}$ from pseudo trajectory $y_{-n+1} ,\ldots ,y_{-1} ,y_0$ and equation $z_{n+1} =F\left( {z_n } \right)$ with $y_{-m/2} =z_{-m/2}$. The starting point $z_{-m/2}$ is perturbed to generate some candidate trajectories using random variable $\zeta$, which is Gaussian with zero mean and standard deviation of the difference between the truth and $z_{-m/2}$. Then, ensemble initial conditions are selected from the candidate trajectories using a likelihood function

$$\begin{aligned} L\left( {z^{*}} \right) =\frac{1}{2}\mathop \sum \limits _{n=-m/2}^0 \left[ {h\left( {z_n^*} \right) -S_n } \right] ^{T}{\Gamma }^{-1}\left[ {h\left( {z_n^*} \right) -S_n } \right] \end{aligned}$$

where ${\Gamma }^{-1}$ is the inverse of the covariance matrix, $z_n^*$ is candidate trajectory and the end component of selected candidate trajectory is considered as ensemble member. The observation is $S_n =h\left( {X_n } \right) +\eta _n$, where $h\left( . \right)$ is the observation operator and the observational noise is $\eta _n \in R^{2}$ where $X_n =\left( {x_n ,y_n } \right)$.