# Parameter space of experimental chaotic circuits with high-precision control parameters

## Summary (2 min read)

### Introduction

- These factors are behind their motivation to propose an approach to obtain experimental parameter spaces that are not only reliable, autonomous, and reproducible but that can also reproduce nominal parameter values considered in numerical experiments.
- With these potentiometers and a set of resistors, switches, and relays, the authors were able to autonomously obtain a high resolution parameter space of the Chua’s circuit (with resolutions of 400 562 and 1023 126 points, varying one resistor with step sizes of 0.200001 X and another with 0.10022 X), which could remarkably reproduce the numerically obtained parameter spaces.
- In particular, the authors showed by calculating the Lyapunov exponents numerically and experimentally that this parameter space presents self-similar periodic structures, the shrimps, embedded in a domain of chaos.

### II. EXPERIMENTAL AND NUMERICAL ASPECTS

- Let us start describing the digital potentiometer.
- The 10 pin left connector stands for the digital data coming from input/ output (I/O) digital ports of the DAQ board.
- The five-fold piecewise linear element that provides the nonlinear character of the Chua’s circuit consists of two operational amplifiers and the resistances R1 to R6.
- For each time series, the potentiometers R and rL were switched by a LabView routine with values previously determined and calibrated to give precise equivalent steps.
- By starting at parameters leading to such attractors provides results as if the system was never switched off.

### III. RESULTS AND DISCUSSION

- The parameter space in Fig. 4(a) shows by colors the values of the largest Lyapunov exponent k, calculated by the method of Sano and Sawada25 from the 400 562 experimental time series with R and rL as the control parameters.
- The simulated parameter space in Fig. 4(b) considered also the values of k obtained from time series generated by 1600 values of R and 562 values of rL in the same range of the experimental data.
- The authors estimated from the measured time series that this noise is between 1 mV and 2 mV.
- This result led us to conclude that their experimentally and numerically obtained periodic windows have a complex structure as expected in Refs. 1, 2, 9, and 27.
- Fitting results in Fig. 7 indicate that the decay exponents, b ¼ 0:3060:04 (experimental results) and b ¼ 0:3960:09 (numerical results), are in the same order of magnitude of the largest positive Lyapunov exponent of the chaotic attractor in the chaotic regions surrounding the shrimps, as expected.

### IV. CONCLUSIONS

- The authors have successfully built autonomous, reliable, and reproducible digital potentiometers that allowed precise measurements of the set of experimental physical parameters of electronic circuits.
- As an application of the power of their component, the authors obtained high-resolution experimental parameter spaces of Chua’s circuit that is remarkably similar to the one obtained by simulations using the same set of physical parameters values.
- To the best of their knowledge, this work provides, for the first time, experimental Lyapunov exponent parameter spaces of a spiral cascade structure of several shrimps of electronic circuits.
- Also, with respect to simulations, the authors show that considering features such as 5-fold idðxÞ and careful consideration of its equations drawn directly from measurements, it was performed simulations with results close to the ones measured experimentally with regard to the shape of the periodic structures, its exponential decay law, as well as the overall range of Lyapunov exponent values.
- The very high precision and stability on the resistance steps improved the definition of periodic structure borders when compared with other methods of parameter variation.

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...In some systems these structures organize themselves with specific bifurcation cascades, for example, period-adding cascades, and along preferred directions on the parameter-space [Gallas, 2015; de Sousa et al., 2016; Gallas, 2016]....

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...We have chosen vary bias [Rocha & Medrano-T., 2009; Medrano-T & Rocha, 2014] instead of resistors [de Sousa et al., 2016; Tahir et al., 2016; Viana et al., 2010, 2012], which are commonly used in circuitry implementations of dynamical systems, for a better fine tuning that we can obtain in the…...

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...As far as we know, commercial or home-made (as the digital [de Sousa et al., 2016]) potentiometers, are noise sources in circuitry implementations, and in nonlinear dynamical systems, noise signal with a large enough strength may perturb the periodic attractors [Viana et al., 2010, 2012], leading…...

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