In this paper, the emergence of multistable behavior through the use of fractional-order-derivatives in a Piece-Wise Linear (PWL) multi-scroll generator is presented. Using the integration-order as a bifurcation parameter, the stability in the system is modified in such a form that produces a basin of attraction segmentation, creating many stable states as scrolls are generated in the integer-order system. The results here presented reproduce the same phenomenon reported in systems with integer-order derivatives, where the multistable regimen is obtained through a parameter variation. The multistable behavior reported is also validated through electronic simulation. The presented results are not only applicable in engineering fields, but they also enrich the analysis and the understanding of the implications of using fractional integration orders, boosting the development of further and better studies.

In this work, we present a simple algorithm to design n × n-bits substitution boxes (S-boxes) based on chaotic time series of the logistic map for different carrying capacities. e use of different carrying capacities in the chaotic map leads to low computational complexity, which is desirable to get high-speed communication systems. We generate a main sequence by means of two auxiliary sequences with uniform distribution via the logistic map for different carrying capacities. e elements of the main sequence are useful for generating the elements of an S-box. e auxiliary sequences are generated by considering lag time chaotic series; this helps to hide the chaotic map used. e U-shape distribution of logistic chaotic map is also avoided, in contrast with common chaos-based schemes without considering lag time chaotic series, and uncorrelated S-box elements are obtained. e proposed algorithm guarantees the generation of strong S-boxes that fulfill the following criteria: bijection, nonlinearity, strict avalanche criterion, output bits independence criterion, criterion of equiprobable input/output XOR distribution, and maximum expected linear probability. Finally, an application premised on polyalphabetic ciphers principle is developed to obtain a uniform distribution of the plaintext via dynamical S-boxes.

In this paper, we investigate the dynamical behavior of a one-dimensional piecewise map based on the logistic map, where generalized multistability can be observed. The proposed system has the unique property that the function is symmetric with respect to the origin but not its behavior, furthermore this system can display three types of multistability, and chaos for both, monostable and bistable behaviors. The stability analysis of the proposed system is presented. We describe the structure of bistable regions in the bifurcation diagram. Particular attention is paid to the chaotic regions. Corresponding to coexisting attractors, three scenarios of coexisting attractors, namely fixed points, periodic orbits, and chaotic attractors, can be found, which are unreported behaviors in discrete chaotic systems. The mechanism that leads to multistability phenomenon including pitchfork bifurcation, period-halving bifurcations, and the coexisting invariant sets is demonstrated. Furthermore, the Lyapunov exponent is analyzed with the type of multistability distinguished for a given set of parameters.

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