Communications in Nonlinear Science and Numerical Simulation
FPGA realization of multi-scroll chaotic oscillators
Introduction
Chaotic oscillators have been investigated to guarantee chaotic regime by optimizing the maximum Lyapunov exponent (MLE) and ensuring a good distribution of trajectories in the phase-space portraits for generating multi-scroll attractors [1]. Those optimized chaotic oscillators have been realized with different kinds of electronic devices, trying to generate the higher number of scrolls [2]. However, it has been observed the difficulty of programming different values for the parameters involved in the mathematical model of a chaotic oscillator. For example: in [1] the MLE of two multi-scroll chaotic oscillators is optimized; one based on saturated function series and the other on Chua’s circuit. Both chaotic oscillators can have different values of their coefficients as well as their piecewise-linear (PWL) function can change, so that one needs to use precision circuit elements to tune the desired value for the coefficient or PWL function, but the major problem is when tuning values having more than three decimals, i.e. 0.001. Because precision resistors are used to tune those decimal values, and since they have very high variation, then we appeal using field-programmable gate arrays (FPGAs) to program coefficient parameters and PWL functions having three or more decimals.
FPGAs have been used to realize applications on chaotic systems. For example: in [3] a chaotic map with high MLE is implemented into an FPGA, in [4] an FPGA-based 3D chaotic system is realized by an auto-switched numerical resolution of multiple three dimensional continuous chaotic systems, in [5] an FPGA-realization of a self-synchronizing chaotic decoder in the presence of noise is presented, in [6] the FPGA design for a pseudo-random number generator is given, in [7] other FPGA design for a discrete chaotic map is presented, and more recently: in [8] an implementation of an FPGA-based real time novel chaotic oscillator is introduced. Although the MLE and phase portraits are shown, they are not optimized; only 2-scrolls are generated, and the hardware architecture from VHDL (Very High Speed Integrated Circuit Hardware Description Language) simulations is not described. In this manner, this article shows the FPGA realization of two multi-scroll chaotic oscillators having optimal and different MLEs: one based on saturated function series and the other based on Chua’s circuit. In addition, their FPGA realization is described by applying two numerical methods (Forward Euler and Runge Kutta) for solving their models given by three first-order differential equations. We show their hardware description, the computer arithmetic, and their co-simulation for a low-cost FPGA from Xilinx. Both chaotic oscillators are realized for generating from 2- to 6-scrolls. At the end, we show several experimental results for different MLEs, and the FPGA used resources are listed for generating 6-scrolls when applying both numerical methods.
Section snippets
Multi-scroll chaotic oscillators
In [1], two multi-scroll chaotic oscillators are described: one based on saturated function series and the second on Chua’s circuit. They are optimized by maximizing their MLE and minimizing the dispersions of their trajectories in the phase portraits among all scrolls in an attractor.
Hardware realization
The multi-scroll chaotic oscillators described by (1), (6) are simulated herein by applying two numerical integration methods: Forward Euler (FE) and 4th-order Runge Kutta (RK). Those dynamical systems in (1), (6) are associated to solve an initial value problem of the form:
The solution of (8) depends on initial conditions , and includes the PWL functions described by (5), (7). The following subsections show the hardware realization of FE and RK into an FPGA for both chaotic
Co-simulation results in Active-MATLAB
The co-simulation helps to verify the hardware implementation before its physical realization and one can have access to each signal. In this article, the compilation of the chaotic oscillator described by VHDL code is performed by using Active-HDL, which provides an interface for MATLAB-Simulink, and allows to perform the co-simulation of blocks that can be described by mathematical models or VHDL. Active-MATLAB generates a file ∗.m for each entity or component described by VHDL, and herein it
Experimental results
In the experiments we used the low-cost XC3S1000-5FT256 FPGA Spartan-3 from Xilinx. The initial conditions were set to: and . Compared to using electronic devices, as done in [2], the use of an FPGA allows us to program the coefficient values with several decimals, in this work 3 and 4 decimals were used, as listed in Table 1, Table 2. In addition, the generated attractors that are shown in the following subsections are more stable when using FPGAs than those using
Conclusion
It has been shown that by using FPGAs one can realize multi-scroll chaotic oscillators that have better behavior than by using active devices like operational amplifiers. In this article the cases of study were two oscillators: one based on saturated function series and other based on Chua’s circuit. Both chaotic oscillators were realized with FPGAs for generating from 2- to 6-scrolls and characterized by their maximum Lyapunov exponent (MLE), which was optimized to guarantee better
Acknowledgment
This work is partially supported by CONACyT/Mexico under the projects: 237991 and 136056.
References (10)
- et al.
Optimizing the maximum Lyapunov exponent and phase space portraits in multi-scroll chaotic oscillators
Nonlinear Dyn
(2014) - et al.
Frequency limitations in generating multi-scroll chaotic attractors using CFOAs
Int J Electron
(2014) - et al.
Digital chaotic signal generator using robust chaos in compound sinusoidal maps
IEICE Trans Fundam Electron Commun Comput Sci
(2013) - et al.
A new auto-switched chaotic system and its FPGA implementation
Commun Nonlinear Sci Numer Simul
(2013) - et al.
Design and FPGA-realization of a self-synchronizing chaotic decoder in the presence of noise
Commun Nonlinear Sci Numer Simul
(2012)