O. G. Félix-Beltrán

Determining the Lyapunov Spectrum of Continuous-Time 1D and 2D Multiscroll Chaotic Oscillators via the Solution of -PWL Variational Equations

An algorithm to compute the Lyapunov exponents of piecewise linear function-based multidirectional multiscroll chaotic oscillators is reported. Based on the regions in the piecewise linear functions, the suggested algorithm determines the individual expansion rate of Lyapunov exponents from -piecewise linear variational equations and their associated -Jacobian matrices whose entries remain constant during all computation cycles. Additionally, by considering OpAmp-based chaotic oscillators, we study the impact of two analog design procedures on the magnitude of Lyapunov exponents. We focus on analyzing variations of both frequency bandwidth and voltage/current dynamic range of the chaotic signals at electronic system level. As a function of the design parameters, a renormalization factor is proposed to estimate correctly the Lyapunov spectrum. Numerical simulation results in a double-scroll type chaotic oscillator and complex chaotic oscillators generating multidirectional multiscroll chaotic attractors on phase space confirm the usefulness of the reported algorithm.

Synchronization of multi-directional multi-scroll chaos generators: A Hamiltonian approach

In this paper, a chaotic synchronization scheme for multi-directional multi-scroll chaos generators is presented. We use Generalized Hamiltonian forms approach to determine the synchronization conditions for two unidirectionally coupled multi-directional multi-scroll chaotic attractors. First, two state-variables of the master system are used to control the two nonlinear functions in the slave system and consequently, reach the synchronization of two 2D-4-scroll chaotic systems. Similarly, the synchronization of two 3D-4-scroll chaotic systems is achieved by using all three state-variables in order to control the three nonlinear functions for the slave system. Finally, theoretical calculations are in good agreement with numerical simulations and a prediction on the values for observer gain is also given.

Experimental Verification of Optimized Multiscroll Chaotic Oscillators Based on Irregular Saturated Functions

Multiscroll chaotic attractors generated by irregular saturated nonlinear functions with optimized positive Lyapunov exponent are designed and implemented. The saturated nonlinear functions are designed in an irregular way by modifying their parameters such as slopes, delays between slopes, and breakpoints. Then, the positive Lyapunov exponent is optimized using the differential evolution algorithm to obtain chaotic attractors with 2 to 5 scrolls. We observed that the resulting chaotic attractors present more complex dynamics when different patterns of irregular saturated nonlinear functions are considered. After that, the optimized chaotic oscillators are physically implemented with an analog discrete circuit to validate the use of proposed irregular saturated functions. Experimental results are consistent with MATLAB™ and SPICE circuit simulator. Finally, the synchronization between optimized and nonoptimized chaotic oscillators is demonstrated.

Fault conditions of a simple chaotic circuit under capacitor nonlinear effects

In this paper a tolerance analysis in the electronic design of a simple chaos generator is reported. This simple chaotic oscillator is composed by four resistors, three capacitors and two opamps. A Verilog-A model for the opamps and capacitors is used herein. For the opamp, the model contains input impedance, finite bandwidth with a dominant pole and voltage saturation effects. In case of capacitor, a nonlinear model based on a varactor is considered, which includes the charge-dependence with the voltage. By using H-Spice simulator, the sensitivity of the chaos generation in the simple chaotic oscillator as a function of the varactor is analyzed. Several H-Spice simulations are given.

Determining the number of scrolls in a multi-scroll chaotic oscillator under uncertainties

A trade-off analysis on the chaotic regime in a multi-scroll chaotic system and the uncertainties of its parameters is presented. The chaotic oscillator is modeled by using the state-variables approximation and its nonlinear function is represented as a piecewise linear (PWL) function. In particular, a 4-scroll chaotic system with four parameters is studied. Due to the chaotic behavior depends on the system parameters; linear and random sweeps are performed to obtain an interval for each parameter. This trust interval is used to predict the number of scrolls that can be synthesized under uncertainties and also, the implementation with an electronic circuit based on operational amplifiers. MATLAB simulations confirm the usefulness of the proposed approach.

Chaotic Planning Paths Generators by Using Performance Surfaces

Chaotic systems have been widely used as path planning generators in autonomous mobile robots due to the unpredictability of the generated trajectories and the coverage rate of the robots workplace. In order to obtain a chaotic mobile robot, the chaotic signals are used to generate True RNGs (TRNGs), which, as is known, exploit the nondeterministic nature of chaotic controllers. Then, the bits obtained from TRNGs can be continuously mapped to coordinates (\(x_n, y_n\)) for positioning the robot on the terrain. A frequent technique to obtain a chaotic bitstream is to sample analog chaotic signals by using thresholds. However, the performance of chaotic path planning is a function of optimal values for those levels. In this framework, several chaotic systems which are used to obtain TRNGs but by computing a quasi-optimal performance surface for the thresholds is presented. The proposed study is based on sweeping the Poincare sections to find quasi-optimal values for thresholds where the coverage rate is higher than those obtained by using the equilibrium points as reference values. Various scenarios are evaluated. First, two scroll chaotic systems such as Chua’s circuit, saturated function, and Lorenz are used as entropy sources to obtain TRNGS by using its computed performance surface. Afterwards, n-scrolls chaotic systems are evaluated to get chaotic bitstreams with the analyzed performance surface. Another scenario is dedicated to find the performance surface of hybrid chaotic systems, which are composed by three chaotic systems where one chaotic system determines which one of the remaining chaotic signals will be used to obtain the chaotic bitstream. Additionally, TRNGs from two chaotic systems with optimized Lyapunov exponents are studied. Several numerical simulations to compute diverse metrics such as coverage rate against planned points, robot’s trajectory evolution, covered terrain, and color map are carried out to analyze the resulting TRNGs. This investigation will enable to increase several applications of TRNGs by considering the proposed performance surfaces.

PWL function-based model of a beta cell

In this paper, a new model based on piecewise linear (PWL) functions is proposed and analyzed by considering the well known Pernarowskis mathematical model for an isolated beta cell. Contrary to Pernarowskis model, we replace the original cubic functions with PWL functions with multi-segments in order to obtain bursting electrical activity (BEA) of beta cell. The number of segments depends on the required accuracy to generate bursting. Additionally, an electronic design of the proposed model using voltage operational amplifiers is realized. Several SPICE circuit simulations are carried out to validate the PWL function-based model.

Quasi-optimal values in the Hamiltonian-based synchronization of chaotic systems

In this paper a quasi-optimal surface for the observer gain in a Hamiltonian-based controller with applications in chaos synchronization is reported. The synchronization scheme is based on a master-slave topology composed of two chaotic oscillators with identical parameters but by using different initial conditions. Therefore, a trade-off analysis on the synchronization regime and the observer gains (K) in an n-scroll chaotic system is obtained. Lyapunov exponents are not required to prove the stability of the synchronization error, which could expand the study to many others chaotic systems. The synchronization error can be obtained as lower than 0.0001 for certain types of permutations of K. Numerical simulations validate the theoretical background and the usefulness of the proposed approach.

Sensitivity analysis of multi-scroll chaotic oscillators at circuit level

In this paper a sensitivity analysis in the electronic design of a chaos generator with n-scrolls in one direction (1-D) is reported. The chaotic oscillator is modeled by using state variables and a piecewise linear function. From this model, a circuit synthesis is carried out to get an electronic circuit based on voltage operational amplifiers. This circuit is analyzed by applying the Monte Carlo method embedded in the T-Spice simulator. Two cases are studied, the first one is related with the sensitivity of the number of scrolls in the chaotic oscillator as a function of the variations in the passive elements. Later, a non ideal macro-model is introduced to analyze the impact of the second order effects of the operational amplifiers against the number of scrolls. Several Spice simulations are given to demonstrate the usefulness of this approach.

On the Synchronization of 1D and 2D Multi-scroll Chaotic Oscillators

In this chapter, the guidelines to synchronize one-directional (1D) and two-directional (2D) multi-scroll chaos generators by means of Generalized Hamiltonian forms are presented. First, the multi-scroll chaotic oscillator is simulated at the electronic system level by applying state-variables and piecewise-linear approaches. Besides, we apply scaling procedures to modify the breaking points, slopes and frequency of the chaotic signals in order to reduce their excursion levels within practical values for electronic devices. Second, a chaotic synchronization scheme for multi-directional multi-scroll chaos generators is introduced. We use Generalized Hamiltonian forms approach to determine the synchronization conditions when one and two state-variables of the master system are sent to control the nonlinear functions in the slave system. Additionally, two schemes are set-up to transmit encrypted binary and analog signals by applying chaotic switching technique and additive chaotic masking, respectively. Both schemes are implemented by using traditional operational amplifiers. Finally, theoretical results are confirmed by performing numerical and SPICE simulation results.