Fractional Zaslavsky and Henon Discrete Maps

This paper is devoted to the memory of Professor George M. Zaslavsky passed away on November 25, 2008. In the field of discrete maps, George M. Zaslavsky introduced a dissipative standard map which is called now the Zaslavsky map. G. Zaslavsky initialized many fundamental concepts and ideas in the fractional dynamics and kinetics. In this paper, starting from kicked damped equations with derivatives of non-integer orders we derive a fractional generalization of discrete maps. These fractional maps are generalizations of the Zaslavsky map and the Henon map. The main property of the fractional differential equations and the correspondent fractional maps is a long-term memory and dissipation. The memory is realized by the fact that their present state evolution depends on all past states with special forms of weights.Comment: 26 pages, LaTe

Dissipative quantum chaos: transition from wave packet collapse to explosion

Using the quantum trajectories approach we study the quantum dynamics of a dissipative chaotic system described by the Zaslavsky map. For strong dissipation the quantum wave function in the phase space collapses onto a compact packet which follows classical chaotic dynamics and whose area is proportional to the Planck constant. At weak dissipation the exponential instability of quantum dynamics on the Ehrenfest time scale dominates and leads to wave packet explosion. The transition from collapse to explosion takes place when the dissipation time scale exceeds the Ehrenfest time. For integrable nonlinear dynamics the explosion practically disappears leaving place to collapse.Comment: 4 pages, 4 figure

Quantum synchronization

Using the methods of quantum trajectories we study numerically the phenomenon of quantum synchronization in a quantum dissipative system with periodic driving. Our results show that at small values of Planck constant $\hbar$ the classical devil's staircase remains robust with respect to quantum fluctuations while at large $\hbar$ values synchronization plateaus are destroyed. Quantum synchronization in our model has close similarities with Shapiro steps in Josephson junctions and it can be also realized in experiments with cold atoms.Comment: 5 pages, 5 figs, 1 fig added, research at http://www.quantware.ups-tlse.f

Hiding text in speech signal using K-means, LSB techniques and chaotic maps

In this paper, a new technique that hides a secret text inside a speech signal without any apparent noise is presented. The technique for encoding the secret text is through first scrambling the text using Chaotic Map, then encoding the scraped text using the Zaslavsky map, and finally hiding the text by breaking the speech signal into blocks and using only half of each block with the LSB, K-means algorithms. The measures (SNR, PSNR, Correlation, SSIM, and MSE) are used on various speech files (“.WAV”), and various secret texts. We observed that the suggested technique offers high security (SNR, PSNR, Correlation, and SSIM) of an encrypted text with low error (MSE). This indicates that the noise level in the speech signal is very low and the speech purity is high, so the suggested method is effective for embedding encrypted text into speech files

The ratchet effect and the transporting islands in the chaotic sea

We study directed transport in a classical deterministic dissipative system. We consider the generic case of mixed phase space and show that large ratchet currents can be generated thanks to the presence, in the Hamiltonian limit, of transporting stability islands embedded in the chaotic sea. Due to the simultaneous presence of chaos and dissipation the stationary value of the current is independent of initial conditions, except for initial states with very small measure.Comment: 5 pages, 6 figure

Quantum ratchets in dissipative chaotic systems

Using the method of quantum trajectories we study a quantum chaotic dissipative ratchet appearing for particles in a pulsed asymmetric potential in the presence of a dissipative environment. The system is characterized by directed transport emerging from a quantum strange attractor. This model exhibits, in the limit of small effective Planck constant, a transition from quantum to classical behavior, in agreement with the correspondence principle. We also discuss parameter values suitable for implementation of the quantum ratchet effect with cold atoms in optical lattices.Comment: Significant changes: Several text improvements and new results. Figure 2 modified. Figure 4 adde

Adaptive Chaotic Maps in Cryptography Applications

Chaotic cryptography is a promising area for the safe and fast transmission, processing, and storage of data. However, many developed chaos-based cryptographic primitives do not meet the size and composition of the keyspace and computational complexity. Another common problem of such algorithms is dynamic degradation caused by computer simulation with finite data representation and rounding of results of arithmetic operations. The known approaches to solving these problems are not universal, and it is difficult to extend them to many chaotic systems. This chapter describes discrete maps with adaptive symmetry, making it possible to overcome several disadvantages of existing chaos-based cryptographic algorithms simultaneously. The property of adaptive symmetry allows stretching, compressing, and rotating the phase space of such maps without significantly changing the bifurcation properties. Therefore, the synthesis of one-way piecewise functions based on adaptive maps with different symmetry coefficients supposes flexible control of the keyspace size and avoidance of dynamic degradation due to the embedded technique of perturbing the chaotic trajectory