Effect of Random Parameter Switching on Commensurate Fractional Order Chaotic Systems

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.Comment: 31 pages, 17 figures, 5 Table

When an attacker meets a cipher-image in 2018: A Year in Review

This paper aims to review the encountered technical contradictions when an attacker meets the cipher-images encrypted by the image encryption schemes (algorithms) proposed in 2018 from the viewpoint of an image cryptanalyst. The most representative works among them are selected and classified according to their essential structures. Almost all image cryptanalysis works published in 2018 are surveyed due to their small number. The challenging problems on design and analysis of image encryption schemes are summarized to receive the attentions of both designers and attackers (cryptanalysts) of image encryption schemes, which may promote solving scenario-oriented image security problems with new technologies.Comment: 12 page

Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions

Using the Mellin transform approach, it is shown that, in contrast with integer-order derivatives, the fractional-order derivative of a periodic function cannot be a function with the same period. The three most widely used definitions of fractional-order derivatives are taken into account, namely, the Caputo, Riemann-Liouville and Grunwald-Letnikov definitions. As a consequence, the non-existence of exact periodic solutions in a wide class of fractional-order dynamical systems is obtained. As an application, it is emphasized that the limit cycle observed in numerical simulations of a simple fractional-order neural network cannot be an exact periodic solution of the system.Comment: 15 pages, 2 figure, submitted for publication: April 27th 2011; accepted: November 24, 201

Unusual dynamics and hidden attractors of the Rabinovich-Fabrikant system

This paper presents some unusual dynamics of the Rabinovich-Fabrikant system, such as "virtual" saddles, "tornado"-like stable cycles and hidden chaotic attractors. Due to the strong nonlinearity and high complexity, the results are obtained numerically with some insightful descriptions and discussions.Comment: replace

Synchronization of piece-wise continuous systems of fractional order

The aim of this study is to prove analytically that synchronization of a piece-wise continuous class of systems of fractional order can be achieved. Based on our knowledge, there are no numerical methods to integrate differential equations with discontinuous right hand side of fractional order which model these systems. Therefore, via Filippov's regularization [1] and Cellina's Theorem [2,3], we prove that the initial value problem can be converted into a continuous problem of fractional-order, to which numerical methods for fractional orders apply. In this way, the synchronization problem transforms into a standard problem for continuous systems of fractional order. Three examples of fractional-order piece-wise systems are considered: Sprott system, Chen and Shimizu-Morioka system.Comment: Examples revise

Asymptotic Methods in Non Linear Dynamics

This paper features and elaborates recent developments and modifications in asymptotic techniques in solving differential equation in non linear dynamics. These methods are proved to be powerful to solve weakly as well as strongly non linear cases. Obtained approximate analytical solutions are valid for the whole solution domain. In this paper, limitations of traditional perturbation methods are illustrated with various modified techniques. Mathematical tools such as variational approach, homotopy and iteration technique are discussed to solve various problems efficiently. Asymptotic methods such as Variational Method, modified Lindstedt-Poincare method, Linearized perturbation method, Parameter Expansion method, Homotopy Perturbation method and Perturbation-Iteration methods(singular and non singular cases) have been discussed in various situations. Main emphasis is given on Singular perturbation method and WKB method in various numerical problems.Comment: submit, to appear in Journal of Non Linear Science and Applications, 201

Spatiotemporal correlation uncovers fractional scaling in cardiac tissue

Complex spatiotemporal patterns of action potential duration have been shown to occur in many mammalian hearts due to a period-doubling bifurcation that develops with increasing frequency of stimulation. Here, through high-resolution optical mapping and numerical simulations, we quantify voltage length scales in canine ventricles via spatiotemporal correlation analysis as a function of stimulation frequency and during fibrillation. We show that i) length scales can vary from 40 to 20 cm during one to one responses, ii) a critical decay length for the onset of the period-doubling bifurcation is present and decreases to less than 3 cm before the transition to fibrillation occurs, iii) fibrillation is characterized by a decay length of about 1 cm. On this evidence, we provide a novel theoretical description of cardiac decay lengths introducing an experimental-based conduction velocity dispersion relation that fits the measured wavelengths with a fractional diffusion exponent of 1.5. We show that an accurate phenomenological mathematical model of the cardiac action potential, fine-tuned upon classical restitution protocols, can provide the correct decay lengths during periodic stimulations but that a domain size scaling via the fractional diffusion exponent of 1.5 is necessary to reproduce experimental fibrillation dynamics. Our study supports the need of generalized reaction-diffusion approaches in characterizing the multiscale features of action potential propagation in cardiac tissue. We propose such an approach as the underlying common basis of synchronization in excitable biological media.Comment: 8 pages, 6 figure

Note on a parameter switching method for nonlinear ODEs

In this paper we study analytically a parameter switching (PS) algorithm applied to a class of systems of ODE, depending on a single real parameter. The algorithm allows the numerical approximation of any solution of the underlying system by simple periodical switches of the control parameter. Near a general approach of the convergence of the PS algorithm, some dissipative properties are investigated and the dynamical behavior of solutions is investigated with the Lyapunov function method. A numerical example is presentedComment: Mathematica Slovaca, 66, 439-448 (2016

HPC optimal parallel communication algorithm for the simulation of fractional-order systems

A parallel numerical simulation algorithm is presented for fractional-order systems involving Caputo-type derivatives, based on the Adams-Bashforth-Moulton (ABM) predictor-corrector scheme. The parallel algorithm is implemented using several different approaches: a pure MPI version, a combination of MPI with OpenMP optimization and a memory saving speedup approach. All tests run on a BlueGene/P cluster, and comparative improvement results for the running time are provided. As an applied experiment, the solutions of a fractional-order version of a system describing a forced series LCR circuit are numerically computed, depicting cascades of period-doubling bifurcations which lead to the onset of chaotic behavior

Delayed feedback control of fractional-order chaotic systems

We study the possibility to stabilize unstable steady states and unstable periodic orbits in chaotic fractional-order dynamical systems by the time-delayed feedback method. By performing a linear stability analysis, we establish the parameter ranges for successful stabilization of unstable equilibria in the plane parametrizad by the feedback gain and the time delay. An insight into the control mechanism is gained by analyzing the characteristic equation of the controlled system, showing that the control scheme fails to control unstable equilibria having an odd number of positive real eigenvalues. We demonstrate that the method can also stabilize unstable periodic orbits for a suitable choice of the feedback gain, providing that the time delay is chosen to coincide with the period of the target orbit. In addition, it is shown numerically that delayed feedback control with a sinusoidally modulated time delay significantly enlarges the stability region of the steady states in comparison to the classical time-delayed feedback scheme with a constant delay.Comment: 9 figures, 17 pages, RevTeX, title changed, additional section on control of unstable periodic orbits included, version published in Journal of Physics A: Mathematical and Theoretica