The Fractional Order Qi(Hyper)Chaotic System And Its Application In Image Encryption

Due to its complex characteristics, such as inherent randomness, unpredictability, broadband characteristics, ergodicity, extremely sensitive to initial conditions and so forth, chaotic system has great development prospect in image encryption. And the study of chaotic system and its application has caused many scholars’enthusiasm. Compared to the general chaos system, hyperchaotic system has more complex dynamical behaviors. And compared with the integer order chaotic system, the fractional order chaotic system has more complex dynamics characteristics and larger key space. Therefore, the study of fractional (hyper)chaotic system and its application in image encryption is of universal and great importance in practice.Based on the integer order Qi system, the fractional order Qi system(hyperchaotic Qi system),its circuit design and implementation and the application in the image encryption are investigated in this paper.At first,With the predictor-corrector method and the frequency domain approximation method, we investigate the dynamics characteristics of a fractional order Qi chaotic system and a fractional order Qi hyperchaotic system by analyzing their phase portraits, Poincare sections, bifurcation diagrams and Lyapunov exponent spectra, and we find chaos phenomena occur in the fractional order Qi chaotic system when fractional order q>0.868 and the system present chaos in a large range of system parameters. Moreover, the system will present chaotic attractors, periodic attractors and quasi-periodic attractors which indicates that there are abundant dynamics behaviors in the chaotic system. We find that the fractional order Qi hyperchaotic system will present chaos when fractional order q>0.761. By drawing its Lyapunov exponent spectra, we can see that the system has two positive Lyapunov exponents in a large range of parameters, which indicate the system presents hyperchaos and its hyperchaotic attractor is four-wing fan shaped when system parameters change in a certain range.Secondly, The improved chaotic circuit modular design method is adopted to design and implement the circuits of the two fractional order system.Through circuit simulation and circuit experiment, we find that the circuit simulation result, circuit experiment result are in accordance with the numerical simulation result, verifying the existence of the chaotic attractor of fractional-order Qi chaotic system and the hyper-chaotic attractor of fractional-order Qi hyper-chaotic system on the physical level.At last, study on the application of the fractional order (hyper)chaotic system in image encryption.The image encryption schemes based on fractional-order Qi chaotic system and fractional-order Qi hyper-chaotic system are designed, and the color digital image is encrypted and decrypted by using these schemes. On this base, the histogram, adjacent pixels correlation, key sensitivity of ciphertext are analyzed, and the analys is results show that the algorithm of image encryption based on fractional Qi (hyper)chaotic system has a large key space and high security.