Experiments on chaos synchronization between two BBC micro:bit boards over a radio link

Chaos synchronization between two BBC micro:bit boards connected by their built-in 2.4~GHz radio link is demonstrated experimentally. The drive board (micro:bit V2) integrates in real time a three-dimensional chaotic system with hyperbolic-tangent and absolute-value nonlinearities and transmits its state over the radio. The response board (micro:bit V1.5, which has no hardware floating-point unit) integrates a copy of the system that receives only one variable from the drive. Numerical analysis shows that the classical Pecora--Carroll scheme does not lead to synchronization of this system for any choice of the driving variable: the conditional Lyapunov exponents are zero or positive, because the equation for the third variable, $\dot w=|u|-1$, does not contain $w$ itself. Linear unidirectional coupling through the variable $v$ provides complete synchronization for any coupling strength $k>0$, fastest at $k\approx3$, where the largest transverse Lyapunov exponent is minimal ($\lambda_\perp\approx-0.076$). Coupling through $w$ leads to bistability of synchronous and antisynchronous regimes due to the symmetry of the system. In the experiment, synchronization is reached within $t\approx24$ dimensionless time units; the residual error in $u$ and $v$ is set by the rounding of the transmitted data ($\sim10^{-3}$), and in $w$ it does not exceed $0.011$; no packets were lost out of 8000 and 12\,000. Loss of synchronization when the coupling is switched off and its recovery after the coupling is restored and after an external perturbation of the response system are demonstrated. The mismatch in $w$ is found to decay stepwise: it persists on the parts of the trajectory near $u\approx0$ and decreases during the large loops at transitions between the wings of the attractor. For the multi-wing systems obtained by the Julia fractal transformation $u+iv=K(x+iy)^M$, it is shown numerically that coupling through $v$ synchronizes the variables invariant under rotation by the angle $2\pi/M$, while the coordinates $(x,y)$ of the response system coincide with those of the drive up to such a rotation: $M$ coexisting synchronous regimes arise, the choice between them being determined by the initial conditions. An experiment with six-wing attractors ($M=3$) confirmed this: after perturbations of the response system, all three synchronous states were realized on the boards in succession.