FPGA-based implementation of chaotic oscillators by applying the numerical method based on trigonometric polynomials

E. Tlelo-Cuautle,A. D. Pano-Azucena,G. Rodriguez-Gomez,L. G. de la Fraga

doi:10.1063/1.5038583

E. Tlelo-Cuautle, A. D. Pano-Azucena + Show 2 more

Open Access

https://doi.org/10.1063/1.5038583

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Abstract

Chaotic systems are integrated via numerical methods but the main challenge is determining the correct time-step. For instance, traditional numerical methods like Forward Euler (FE) and 4th-order Runge-Kutta (RK), have been applied to simulate and to implement chaotic oscillators into embedded systems like the field-programmable gate array (FPGA). However, if one does not choose the correct time-step, numerical methods may induce artificial chaos suppression or can engender the appearance of spurious solutions. To cope with these issues when solving chaotic systems, one can apply numerical methods for problems having oscillatory characteristics. In this manner, we show that methods like the one based on trigonometric polynomials are ad hoc in simulating chaotic oscillators because provide better accuracy than FE, and as also shown herein requires lower FPGA resources compared to 4th-order RK. To demonstrate the usefulness of the method based on trigonometric polynomials, five chaotic oscillators are simulated and compared to the traditional FE, 4th-order RK and ODE45 (available into MatlabTM). The comparison considers time-execution and number of calls for evaluating the mathematical models of the oscillators. The experimental results when implementing the methods within an FPGA demonstrate that the method based on trigonometric polynomials has similar accuracy than ODE45, similar time-execution compared to FE, and its FPGA implementation requires lower hardware resources than RK. Therefore, we conclude that trigonometric polynomials is much better than FE and RK when one knows a priori that the problem has oscillatory characteristics.

Highlights

Numerical methods are the key to solve dynamical systems as in the case of chaotic oscillators
Those chaotic oscillators are implemented in an field-programmable gate array (FPGA) and at the end we show that the FPGA-based implementation of the trigonometric polynomials method requires lower hardware resources than RK, and it is quite useful to implement chaos generators in FPGAs
This work shows the application of the numerical method based on trigonometric polynomials, which is a special method to solve the ordinary differential equations (ODEs) modeling a dynamical system with oscillatory characteristics, as the chaotic oscillators are.[22]

Summary

INTRODUCTION

Numerical methods are the key to solve dynamical systems as in the case of chaotic oscillators. FE has been applied to synthesize chaotic oscillators into an embedded system as the field-programmable gate array (FPGA).[2] On the other hand, if the step-size is not the good one, chaotic behavior can be suppressed in the very short time Is for this reason that we show the usefulness of the trigonometric polynomial method to simulate and to implement chaotic oscillators into an FPGA for engineering applications. In the current literature one can find numerical methods to solve special problems, as when the solution exhibits exponential characteristics,[21] or oscillatory behavior.[17] In this manner, this work shows the application of the numerical method based on trigonometric polynomials, which is a special method to solve the ODEs modeling a dynamical system with oscillatory characteristics, as the chaotic oscillators are.[22] In addition, the chaotic systems are implemented with FPGAs, which are modern reconfigurable architectures.[23].

TRIGONOMETRIC POLYNOMIALS TO SIMULATE CHAOTIC OSCILLATORS

NUMERICAL SIMULATION OF CHAOTIC OSCILLATORS BY TRIGONOMETRIC POLYNOMIALS

FPGA-BASED IMPLEMENTATION BY APPLYING TRIGONOMETRIC POLYNOMIALS

CONCLUSIONS