Research On The Applications Of Chaotic Systems Reconstruction Based On Compressive Sensing

The prediction of chaotic dynamical systems based on time series has been an outstanding problem in nonlinear dynamics. Despite the previous efforts in using the standard delay-coordinate embedding method to decode the topological properties of the undelying system, how to accurately infer the chaotic system equations remains largely an unsolved problem. In recent years, compressive sensing theory has received lots of attention, whose core idea is reconstruct the sparse signal measured from the linear projections of the original signal. The number of measurements demanded for accurate reconstruction is far less than the traditional signal processing methods especially when the sufficient mesurements are not allowed. Thus, it is of great significance to combine the compressive sensing scheme with chaotic systems prediction.In this thesis, a generalized method is demonstrated, which is chaotic systems reconstruction based on compressive sensing scheme providing that only time series of variables are available. This method has a wide range of applications involving nonlinear structure. Some reserchers have applied it to predict catastrophic bifurcation, detect hidden nodes and forecast the adiabatically time-varying system and so on. In this paper, the proposed method is applied to two different fields which include optical security technique and topology prediction of complex network.(1) In our proposed optical image cryptosystem, two pairs of phase-amplitude masks are generated from the chaotic web map for image encryption in the 4f double-random phase-amplitude encoding (DRPAE) system. Instead of transmitting the real keys and the enormous masks codes, only few observed points intermittently chosen from the masks are delivered. Based on compressive sensing paradigm, we suitably refine the series expansions of web map equations to better reconstruct the underlying system. Then secret keys of the encryption system can be successfully calculated and the chaotic equations are also obtained to regenerate correct decrypting masks. Numerical simulations show that this cryptosystem has the advantages of sufficiently low occupation of the transmitted key codes and security improvement of information transmission without sending the real keys.(2) Chaotic system reconstruction is extended to Duffing oscillator to detect the weak sinusodol signal. Due to the strict demands of traditional detecting method, we aim to solve the problem of low efficiency for unknown frequency signals. Also, the relative frequency error between the inner driving force and the weak signal should be under a constant, which is an issue to be solved. Based on compressive sensing reconstruction for Duffing system, these problems are gone in some sense.