Abstract
Chaotic oscillators have been designed with embedded systems like field-programmable gate arrays (FPGAs), and applied in different engineering areas. However, the majority of works do not detail the issues when choosing a numerical method and the associated electronic implementation. In this manner, we show the FPGA implementation of chaotic and hyper-chaotic oscillators from the selection of a one-step or multi-step numerical method. We highlight that one challenge is the selection of the time-step h to increase the frequency of operation. The case studies include the application of three one-step and three multi-step numerical methods to simulate three chaotic and two hyper-chaotic oscillators. The numerical methods provide similar chaotic time-series, which are used within a time-series analyzer (TISEAN) to evaluate the Lyapunov exponents and Kaplan–Yorke dimension (DKY) of the (hyper-)chaotic oscillators. The oscillators providing higher exponents and DKY are chosen because higher values mean that the chaotic time series may be more random to find applications in chaotic secure communications. In addition, we choose representative numerical methods to perform their FPGA implementation, which hardware resources are described and counted. It is highlighted that the Forward Euler method requires the lowest hardware resources, but it has lower stability and exactness compared to other one-step and multi-step methods.
Highlights
Chaos is a nonlinear and unpredictable behavior that can be modeled by ordinary differential equations (ODEs)
The evaluation of Equation (6) requires a finite-state machine (FSM) to control the iterative process, a cumulative sum block to process the Runge–Kutta 4 (RK4) method and random access memories (RAMs) to save the past steps f (n), f (n − 1), f (n − 2), f (n − 3), f (n − 4), f (n − 5) that are required for the iteration, and they are controlled by STP (StarT Prediction) and EOP (End Of Prediction)
We have shown the issues with the field-programmable gate arrays (FPGAs) implementation of chaotic and hyperchaotic oscillators from the selection of a one-step and multi-step numerical method
Summary
Chaos is a nonlinear and unpredictable behavior that can be modeled by ordinary differential equations (ODEs). Recent works show the usefulness of chaotic oscillators in different engineering problems [11,12,13], there is no information on the issues related to the implementation of the numerical methods in electronic systems. In this manner, this paper uses three representative chaotic and two hyper-chaotic oscillators as case studies, which are listed, along with their associated name, ODEs and parameter values that are used to generate chaotic behavior.
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