Chaos and bifurcations in chaotic maps with parameter q: Numerical and analytical studies

In this paper, a class of chaotic maps with parameter q are introduced and bifurcations and chaos in proposed maps are numerical and analytical studied. Euler method is employed to get the continuous systems corresponding to chaotic maps and the fractional styles in Caputo's definition. Based on that, we finally infer a class of chaotic maps with the Adams–Bashforth–Moulton predictor-corrector method. In the simulation and analysis, we discuss the Logistic map with q and Hénon map with q, observe the route from period to chaos and do tests to analyze properties of maps with parameter q

Fractional Dynamical Systems

In this paper the author presents the results of the preliminary investigation of fractional dynamical systems based on the results of numerical simulations of fractional maps. Fractional maps are equivalent to fractional differential equations describing systems experiencing periodic kicks. Their properties depend on the value of two parameters: the non-linearity parameter, which arises from the corresponding regular dynamical systems; and the memory parameter which is the order of the fractional derivative in the corresponding non-linear fractional differential equations. The examples of the fractional Standard and Logistic maps demonstrate that phase space of non-linear fractional dynamical systems may contain periodic sinks, attracting slow diverging trajectories, attracting accelerator mode trajectories, chaotic attractors, and cascade of bifurcations type trajectories whose properties are different from properties of attractors in regular dynamical systems. The author argues that discovered properties should be evident in the natural (biological, psychological, physical, etc.) and engineering systems with power-law memory.Comment: 6 pages, 4 figure

A chaotic spread spectrum system for underwater acoustic communication

The work is supported in part by NSFC (Grant no. 61172070), IRT of Shaanxi Province (2013KCT-04), EPSRC (Grant no.Ep/1032606/1).Peer reviewedPostprin

Effect of Random Parameter Switching on Commensurate Fractional Order Chaotic Systems

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.The paper explores the effect of random parameter switching in a fractional order (FO) unified chaotic system which captures the dynamics of three popular sub-classes of chaotic systems i.e. Lorenz, Lu and Chen's family of attractors. The disappearance of chaos in such systems which rapidly switch from one family to the other has been investigated here for the commensurate FO scenario. Our simulation study show that a noise-like random variation in the key parameter of the unified chaotic system along with a gradual decrease in the commensurate FO is capable of suppressing the chaotic fluctuations much earlier than that with the fixed parameter one. The chaotic time series produced by such random parameter switching in nonlinear dynamical systems have been characterized using the largest Lyapunov exponent (LLE) and Shannon entropy. The effect of choosing different simulation techniques for random parameter FO switched chaotic systems have also been explored through two frequency domain and three time domain methods. Such a noise-like random switching mechanism could be useful for stabilization and control of chaotic oscillation in many real-world applications

Chaos In New Polynomial Discrete Logistic Maps With Fractional Derivative And Applications For Text Encryption

In this paper, we propose new polynomial discrete logistic equations based on the classical logistic map, which exhibit chaotic behavior as control parameters vary. We also explore versions with fractional derivatives. Using the chaotic sequence generated by these equations, we develop an encryption scheme for text. The scheme relies on initial conditions, control parameters, and a transformation of text characters into values between 0 and 1, followed by a transformation to discrete chaotic values for transmissio

Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue