J.M. Muñoz-Pacheco

Frequency limitations in generating multi-scroll chaotic attractors using CFOAs

The current-feedback operational amplifier (CFOA) allows us to implement any kind of circuit useful in analogue signal processing applications. However, it has limited performance in implementing nonlinear circuits. That way, this investigation highlights the experimental results of implementing a multi-scroll chaotic oscillator by using the commercially available CFOA AD844. The chaotic oscillator is based on saturated nonlinear function (SNLF) series, and we show and discuss its frequency limitations to generate 3- to 10-scrolls from 1 kHz to 100 kHz. Finally, we conclude that the frequency limitations are due to the nonideal characteristics of the CFOA-based SNLF block, imposed by the AD844.

Multiscroll floating gate–based integrated chaotic oscillator

In this work, we proposed a voltage-to-current cell based on a Complementary metal-oxide-semiconductor (CMOS) inverter designed by using floating gate transistors. We demonstrate its usefulness for the design of stair-type and sawtooth functions to be used in the implementation of a multiscroll chaotic oscillator. The main advantage of using floating gate transistors to design the nonlinear functions is the elimination of external reference DC sources, as is typically done in most of the nonlinear functions that generate multiscroll attractors. The key guidelines for the design of our proposed voltage-to-current cell are given to provide good performances in the design of an integrated multiscroll chaotic oscillator. HSPICE simulations are presented to demonstrate the usefulness of the proposed cell to generate multiscroll attractors. Finally, simulation results before and after layout are presented to show the good agreement with respect to theoretical results. HSPICE simulations of the post-layout design are in accordance with the system behavior. Copyright

A Novel Chaotic System without Equilibrium

A few special chaotic systems without unstable equilibrium points have been investigated recently. It is worth noting that these special systems are different from normal chaotic ones because the classical Shilnikov criterion cannot be used to prove chaos of such systems. A novel unusual chaotic system without equilibrium is proposed in this work. We discover dynamical properties as well as the synchronization of the new system. Furthermore, a physical realization of the system without equilibrium is also implemented to illustrate its feasibility.

On the trade-off between the number of scrolls and the operating frequency of the chaotic attractors

A trade-off analysis on the number of scrolls and the operating frequency is presented. To do so, nonlinear system parameters, real physical active device parameters and the operating frequency are taken into account to model the behavior of the saturated nonlinear function series approximated by a piecewise-linear description. As a consequence, not only the trade-off between the number of scrolls and the operating frequency is investigated, which depends directly on the behavior at the operating frequency of the nonlinear function, but also chaotic attractors are efficiently synthesized without the application of scaling techniques. Theoretical calculations are in good agreement with numerical simulations and a prediction on the maximum number of scrolls that can be synthesized for a particular frequency is also given.

Frequency scaling simulation of Chua’s circuit by automatic determination and control of step-size

A procedure is introduced to speed-up time simulation of Chua’s circuit by means of determining the eigenvalues of its state matrixes. A multistep method is applied with automatic control of step-size (h), where h is determined by the computation of the eigenvalues of the state matrixes which change according to the slopes of Chua’s diode. It is shown that the proposed simulation procedure enhances the estimation of the values of the circuit elements in Chua’s circuit and the exploration on its frequency scaling behavior. Besides, it is not necessary to give an initial step-size for initializing the simulation.

Analysis of a 4-D Hyperchaotic Fractional-Order Memristive System with Hidden Attractors

In 1695, G. Leibniz laid the foundations of fractional calculus, but mathematicians revived it only 300 years later. In 1971, L.O. Chua postulated the existence of a fourth circuit element, called memristor, but Williams’s group of HP Labs realized it only 37 years later. In recent years, few unusual dynamical systems, such as those with a line of equilibriums, with stable equilibria or without equilibrium, which belong to chaotic systems with hidden attractors, have been reported. By looking at these interdisciplinary and promising research areas, in this chapter, a fractional-order 4-D memristive system with a line of equilibria is introduced. In particular, a hyperchaotic behavior in a simple fractional-order memristor-based system is presented. Systematic studies of the hyperchaotic behavior in the integer and fractional-order form of the system are performed using phase portraits, Poincare maps, bifurcation diagrams and Lyapunov exponents. Simulation results show that both integer-order and fractional-order system exhibit hyperchaotic behavior over a wide range of control parameter. Finally, the electronic circuits for the evaluation of the theoretical model of the proposed integer and fractional-order systems are presented.

Accuracy vs simulation speed trade-off enhancements in the generation of chaotic attractors

In this paper, the trade-off between accuracy and simulation speed in the generation of muti-scroll chaotic attractors at 1-D is analyzed and improved. In a first step, a macromodel based on bipolar transistors and passive elements is used to model the behavior of Opamps, and later on they are used to approach the behavior of all nonlinear system. Hspice simulations are executed to generate chaotic attractors in the phase plane and time-series. CPU-time used during the solution of the chaotic system is computed. In a second stage, a simple and accurate nonlinear model, which includes the most influential performance parameters for Opamps, is coded in C++ and it is used to generate a nonlinear system of equations, which models the behavior of all chaotic system. CPU-time is also computed. Because chaotic waveforms generated have a random behavior, statistical tests are used to measure the similitude/accuracy between two random variables during a long time. Our results indicate that chaotic waveforms can swiftly be generated by using the simple but accurate model for Opamps without the accuracy worsens, independently of the initial conditions of the chaotic system.

A New Chaotic System With Stable Equilibrium: From Theoretical Model to Circuit Implementation

Recent evidences suggest that complex behavior such as chaos can be observed in a nonlinear system with stable equilibria. However, few studies have investigated chaotic systems with only one stable equilibrium. This paper introduces a new 3-D chaotic system having only one stable equilibrium. Dynamics of the new system are discovered by using phase portraits, basin of attraction, bifurcation diagram, and maximal Lyapunov exponents. It is interesting that the system has a state variable related with the freedom of offset boosting. In addition, we have investigated the anti-synchronization of the system via an adaptive control. Furthermore, the feasibility of the system is also discussed through presenting its electronic circuit implementation.

Simulation of Chua’s circuit by automatic control of step-size

The effects on time-simulation by automatic control of step-size is presented to simulate the behavior of Chua’s circuit, which consist of five circuit elements: one linear resistor, one inductor, two capacitors and one nonlinear resistor known as Chua’s diode. The last element is modeled by an I–V piecewise-linear function to formulate a system of state-variables equations, which are solved by applying multistep methods with automatic control of step-size to simulate the generation of chaotic phenomena. It is highlighted the speed-up in time simulation to estimate the values of the circuit elements, so that the synthesis of Chua’s diode can be realized directly using novel electronic circuits.

Synchronization of n-scrolls chaotic systems synthesized from high-level behavioral modeling

The high-level modeling and simulation of a chaotic oscillator based on saturated functions (SFs) is presented for the synthesis of n-scrolls attractors from functional specifications. The applications of the synthesized chaotic circuits in a synchronization approach for secure communications are also described. The synthesis methodology is based on tree hierarchical levels. First, the oscillator is simulated at the electronic system level (ESL) by applying state variables and piecewise-linear approximation. Additionally, design space exploration of n-scroll attractors is performed by procedures to control the scaling of the excursion levels and frequency, R, L and C elements values and the breaking points and slopes of the SF. Second, SFs are synthesized using op-amp blocks. Verilog-A models are used to model the limitations (gain, bandwidth, slew-rate, saturation) of real op-amps. Theoretical results are confirmed by SPICE simulations to show the usefulness of the synthesis approach and the synchronization.