Hardware implementation of a Julia-fractal multi-wing chaotic generator on an ESP32 microcontroller with a wireless oscilloscope, and analysis of mobile robot control

We present a hardware implementation of a multi-wing chaotic generator obtained by applying the Julia fractal process u + iv = (x + iy)^m + Z₀ to our recently proposed three-dimensional chaotic system with the rational saturating nonlinearity f(u) = u/(1 + k|u|). The generator runs in real time on an ESP32-CAM module and streams its signals over Wi-Fi to a web browser that acts as a phase-plane oscilloscope and data logger. We propose an embedded algorithm in which the base system is integrated and the variables (x, y) are recovered as an m-th root with continuous branch selection. The algorithm is equivalent to integrating the system in the (x, y, z) coordinates but contains no singular factor |x + iy|^−(2m−2). We show that the scaling coefficient α of the modified fractal reduces to an exact similarity transformation: the attractor in the (x, y) plane shrinks as α^−1/m, while its shape, the variable z and the Lyapunov exponents remain unchanged. We establish that, under a natural ergodicity assumption, the angular spectrum of the multi-wing attractor contains only harmonics n = 2jm, and that the amplitude of the n = 2m harmonic does not depend on m; this provides an objective measure of the number of wings. The mean time between sign changes of x grows linearly, T± ≈ πm/Ω̄, where Ω̄ ≈ 0.93 is the mean angular velocity of the base-system phase point. Records acquired from the microcontroller for m = 2,…,6 and α = 1, 30, 100, 200 agree with a reference computation: the median attractor radius deviates by no more than 2% (7% for α = 30), and the dominant angular harmonic equals 2m in all cases. The generator is mounted on a differential-drive mobile robot chassis, and wheel control following the scheme of our previous work is implemented in the firmware. We show numerically that the area covered under this control is determined by the fraction of time the base system spends near the direction Φ* = m(45° − β) that corresponds to straight-line motion. This explains the small coverage for m = 4, and the exploration rate can be changed by about an order of magnitude by rotating the coupling by an angle β. Supplementary material: Arduino firmware for the ESP32-CAM (julia_chaos_robot.ino).