Chaotic dynamical systems without fixed points are promising in the generation of pseudo-random sequences because orbits will not converge to a fixed point. In this class of systems there are no final fixed points, so the basin of attraction of a fixed point does not exist. The absence of fixed points makes it difficult to analyze the dynamics of the systems because a fixed point gives us a lot of information about the dynamics of the system. In addition, if there is an amplitude control parameter for the generated chaotic signals, the tolerance and adaptability are higher and better in the application process. In view of the aforementioned, in this paper, a novel class of discrete maps is presented and described by a kind of piecewise linear (PWL) maps. Necessary and sufficient conditions are given to guarantee that this class of discrete maps does not have any fixed point. Furthermore, we introduce families of these PWL discrete maps without fixed points that present positive Lyapunov exponents and have chaotic dynamics with amplitude control. From these families, we select one particular map, which is analyzed theoretically and proved to be chaotic in the sense of Devaney.

The employment of chaotic maps in a variety of applications such as cryptosecurity, image encryption schemes, communication schemes, and secure communication has been made possible thanks to their properties of high levels of complexity, ergodicity, and high sensitivity to the initial conditions, mainly. Of considerable interest is the implementation of these dynamical systems in electronic devices such as field programmable gate arrays (FPGAs) with the intention of experimentally reproducing their dynamics, leading to exploiting their chaotic properties in real phenomena. In this work, the implementation of a one-dimensional chaotic map that has no fixed points is performed on an FPGA device with the objective of being able to reproduce its chaotic behavior as well as possible. The chaotic behavior of the introduced system is determined by estimating the Lyapunov exponents and its chaotic behavior is also analyzed using bifurcation diagrams. Simulations of the system are realized via Matlab, as well as in C and the very high-speed integrated circuit (VHSIC) hardware description language (VHDL). Experimental results on FPGA show that they are like those obtained in the simulations; therefore, this chaotic dynamical system could be used as an element in some encryption schemes such as in the generation of cryptographically secure pseudorandom numbers.

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