This paper presents the application of three optimization algorithms to increase the chaotic behavior of the fractional order chaotic Chen system. This is achieved by optimizing the maximum Lyapunov exponent (MLE). The applied optimization techniques are evolutionary algorithms (EAs), namely: differential evolution (DE), particle swarm optimization (PSO), and invasive weed optimization (IWO). In each algorithm, the optimization process is performed using 100 individuals and generations from 50 to 500, with a step of 50, which makes a total of ten independent runs. The results show that the optimized fractional order chaotic Chen systems have higher maximum Lyapunov exponents than the non-optimized system, with the DE giving the highest MLE. Additionally, the results indicate that the chaotic behavior of the fractional order Chen system is multifaceted with respect to the parameter and fractional order values. The dynamical behavior and complexity of the optimized systems are verified using properties, such as bifurcation, LE spectrum, equilibrium point, eigenvalue, and sample entropy. Moreover, the optimized systems are compared with a hyper-chaotic Chen system on the basis of their prediction times. The results show that the optimized systems have a shorter prediction time than the hyper-chaotic system. The optimized results are suitable for developing a secure communication system and a random number generator. Finally, the Halstead parameters measure the complexity of the three optimization algorithms that were implemented in MATLAB. The results reveal that the invasive weed optimization has the simplest implementation.

Fractional-order chaotic oscillators (FOCOs) have shown more complexity than integer-order chaotic ones. However, the majority of electronic implementations were performed using embedded systems; compared to analog implementations, they require huge hardware resources to approximate the solution of the fractional-order derivatives. In this manner, we propose the design of FOCOs using fractional-order integrators based on operational transconductance amplifiers (OTAs). The case study shows the implementation of FOCOs by cascading first-order OTA-based filters designed with complementary metal-oxide-semiconductor (CMOS) technology. The OTAs have programmable transconductance, and the robustness of the fractional-order integrator is verified by performing process, voltage and temperature variations as well as Monte Carlo analyses for a CMOS technology of 180 nm from the United Microelectronics Corporation. Finally, it is highlighted that post-layout simulations are in good agreement with the simulations of the mathematical model of the FOCO.

This paper considers a three-dimensional nonlinear dynamical system capable of generating spherical attractors. The main activity is the realization of a spherical chaotic attractor on Intel and Xilinx FPGA boards, with a focus on implementation of a secure communication system. The first major contribution is the successful synchronization of two chaotic spherical systems, in VHDL program, in a master-slave topology using Hamiltonian forms. The synchronization errors show that the two spherical chaotic systems synchronize in a very short time after which the error signals become zero. The second major contribution is the FPGA realization of a spherical chaotic attractor-based secure communication system, which involves encrypting both grayscale and RGB images with chaos and diffusion key at the transmitting system, sending the encrypted image via the state variables, and reconstructing the encrypted image at the receiving system. The Intel Stratix III and Xilinx Artix-7 AC701 results are the same as those of MATLAB. The statistical analyses of the encrypted and received images show that the implemented system is very effective, as it reveals high degree of randomness in the encrypted images with the entropy test, and the obtained correlation coefficient, which is zero, removes relativity between the original and encrypted images. Finally, the transmission system fully recovers the original grayscale and RGB images without loss of information.

In this paper, we present an adaptive modeling and linearization algorithm using the weighted memory polynomial model (W-MPM) implemented in a chain involving the indirect learning approach (ILA) as a linearization technique. The main aim of this paper is to offer an alternative to correcting the undesirable effect of spectral regrowth based on modeling and linearization stages, where the 1-dB compression point (P1dB) of a nonlinear device caused by memory effects within a short time is considered. The obtained accuracy is tested for a highly nonlinear behavior power amplifier (PA) properly measured using a field-programmable gate array (FPGA) system. The adaptive modeling stage shows, for the two PAs under test, performances with accuracies of −32.72 dB normalized mean square error (NMSE) using the memory polynomial model (MPM) compared with −38.03 dB NMSE using the W-MPM for the (i) 10 W gallium nitride (GaN) high-electron-mobility transistor (HEMT) radio frequency power amplifier (RF-PA) and of −44.34 dB NMSE based on the MPM and −44.90 dB NMSE using the W-MPM for (ii) a ZHL-42W+ at 2000 MHz. The modeling stage and algorithm are suitably implemented in an FPGA testbed. Furthermore, the methodology for measuring the RF-PA under test is discussed. The whole algorithm is able to adapt both stages due to the flexibility of the W-MPM model. The results prove that the W-MPM requires less coefficients compared with a static model. The error vector magnitude (EVM) is estimated for both the static and adaptive schemes, obtaining a considerable reduction in the transmitter chain. The development of an adaptive stage such as the W-MPM is ideal for digital predistortion (DPD) systems where the devices under test vary their electrical characteristics due to use or aging degradation.

The phase portrait for dynamic systems is a tool used to graphically determine the instantaneous behavior of its trajectories for a set of initial conditions. Classic phase portraits are limited to two dimensions and occasionally snapshots of 3D phase portraits are presented; unfortunately, a single point of view of a third or higher order system usually implies information losses. To solve that limitation, some authors used an additional degree of freedom to represent phase portraits in three dimensions, for example color graphics. Other authors perform states combinations, empirically, to represent higher dimensions, but the question remains whether it is possible to extend the two-dimensional phase portraits to higher order and their mathematical basis. In this paper, it is reported that the combinations of states to generate a set of phase portraits is enough to determine without loss of information the complete behavior of the immediate system dynamics for a set of initial conditions in an n-dimensional state space. Further, new graphical tools are provided capable to represent methodically the phase portrait for higher order systems.