Chaos Model Predictions Based On Compressed Sensing

Chaos in a deterministic system is a seemingly random phenomenon. If deterministic nonlinear system solution of intrinsic randomness, we call this solution for the chaos. Such a solution in a short period of time can be predicted, but in the long term is not predictable. Chaos is not a simple disorderly, it had no obvious cyclical and symmetry. Chaos is a new form of existence in nonlinear systems. It has the rich internal hierarchical ordered structures.Predicting chaos is very important in chaos theory application field and research hotspot. It not only can be used to establish model of chaotic system, identification and detection of chaotic, also widely used in such as economy, hydrology, astronomy and stock prediction, and many other fields. Often the model of chaos predicting is divided into two broad categories:based on the nonlinear dynamical method of the mathematical model and the method based on phase space reconstruction. Dynamical method is a common predicting method, based on the people already familiar with the laws of physics, such as movement of the continuous equation, conservation equation, based on the simplification of physical model and numerical solution, so as to achieve the purpose of prediction.In2011, Wang et al. puts forward a method to predict chaotic system based on compressive sensing. In different of the traditional dynamical method, this method is to use time series reconstruction system of equations of the system. The general idea is that if the equation of system can be written in the form of a proper power series expansion, we can accurately calculate the expression of the original system by compressive sensing. In this paper, we introduce this algorithm by the Lorenz model. This approach has two advantages. The first one is that we can accurately calculate the equation of the original system. The second one is that extremely low required measurements. Then we done further research in this method. We found that selecting suitable sampling frequency, can reduce the required measurements. This kind of method applies to a form of power series expansion of system equations. If the system of equations is not entirely in the form of power series expansion, our method can not accurately predict equation of the system. By changing the form of the prediction equation, we still could accurately calculate the equation of the system. In a word, if we can design the equation expression of more general, this method is generalized to more general situations. Finally we predicted the equations of the Rikitake double disk dynamo model by compressive sensing.