On the amplitude of External Perturbation and the Chaos via Devil's Staircase -Stability of Attractors -

Abstract

We made the chaotic circuit proposed by Chua and the memristic circuit proposed by Muthuswamy and Chua, and analyzed the behavior of the voltage of the capacitor, electric current in the inductor and the voltage of the memristor by adding an external sinusoidal oscillation \({\dot y}(t)\simeq {\dot i_L}(t)\) of a type \(\gamma\omega \cos\omega t\), while the \({\dot x}(t)\simeq {\dot v_C}(t)\) is given by \(y(t)/C\), and studied the Devil's staircase route to chaos. We compared the frequency of the driving oscillation \(f_s\) and the frequency of the response \(f_d\) in the window and assigned \(W=f_s/f_d\) to each window. When capacitor \(C=1.0\), we observe stable attractors of Farey sequences \(\displaystyle W=\{\frac{1}{2}, \frac{2}{3},\frac{3}{4},\frac{4}{5}, \cdots ,\frac{14}{15},\frac{1}{1}\}\), which can be interpreted as hidden attractors, while when \(C=1.2\), we observe unstable attractors. Possible role of octonions in quantum mechanics and Cartan's supersymmetry is discussed.