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

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.