On Multiscroll Chaos Generation Based on Tchebytchev Polynomials: Theoretical Study and Arduino Microcontroller-Based Experiment

This paper proposes a new approach for obtaining multi-scroll signals based on the Tchebytchev polynomials-type nonlinearity as well as the Chua system. This specific nonlinearity induces in the Chua system several equilibrium points whose number depends on the order of the polynomial considered. To shed light on the mechanism that results in the creation of multi-scroll chaos in the system, we use appropriate numerical (i.e., bifurcation diagrams, attraction basins and phase space trajectory plots) and analytical techniques (i.e., Hopf bifurcation theorems, Routh–Hurtwitz criterion). The experimental results reveal that the scroll number of the chaotic attractor exhibits a monotonic increase with respect to the polynomial order. For example, 3, 4, 5, 6, 7 and 8 scrolls are obtained for a polynomial of order 5, 7, 9, 11, 13 and 15, respectively. In addition, the parametric analysis highlights parallel bifurcation branches in several areas where two or more attractors coexist for certain combinations of parameters. To verify these theoretical predictions, an experimental study is carried out using an Arduino card. It should be pointed out that the new approach introduced in this work stands out from traditional techniques in that it is based on an infinitely differentiable function with simple electronic implementation.