Like Chaotic Characteristics Of The Irrational Number Sequence And Its Application In Image Encryption

Chaos is an aperiodic time-asymptotic behavior in a deterministic non-linear system which exhibits sensitive dependence on initial conditions.It is a special state of nonlinear dynamic system.It reveals the common attribute of nonlinear science—unification of order and disorder,unification of certain and stochastic. Irrational number means a real number that can not be expressed as a rational number, and it is similar with chaos because of its non-periodic property.Ting Chen and Feng Li associate pi with chaos in "Chaos-like characteristics of pi series and its application to image encryption".They define pi series and post its chaos-like characteristics by comparison with Logistic chaotic series.At last they apply pi series to image encryption.Irrational number series is proposed by extending the definition and analyses method of pi series to irrational numbers.We get square root of two series and e series, pursuing the similar attributes between them and chaotic series.The paper is mainly composed of two parts.In the first part,we analyze irrational number series of square root of two and e in six aspects involving autocorrelation function,power spectral density(PSD),phase space construction,correlation dimension,principal component analysis(PCA) and maximal Lyapunov exponent. The results verify the chaos-like characteristics of irrational number series.Finally we apply our irrational number series to image encryption and get pleasant encryption effects.In the second part we propose irrational number series based on BBP algorithm, which is a parallel computing algorithm for irrational number.After decimal conversion,we can get the normal irrational number series.Analyses show that chaos-like characteristics also happen to irrational number series based on BBP algorithm.