Strange attractors generated by a fractional order switching system and its topological horseshoe

Abstract

Chaos generation in a new fractional order unstable dissipative system with only two equilibrium points is reported. Based on the integer version of an unstable dissipative system (UDS) and using the same system’s parameters, chaos behavior is observed with an order less than three, i.e., 2.85. The fractional order can be decreased as low as 2.4 varying the eigenvalues of the fractional UDS in accordance with a switching law that fulfills the asymptotic stability theorem for fractional systems. The largest Lyapunov exponent is computed from the numerical time series in order to prove the chaotic regime. Besides, the presence of chaos is also verified obtaining the topological horseshoe. That topological proof guarantees the chaos generation in the proposed fractional order switching system avoiding the possible numerical bias of Lyapunov exponents. Finally, an electronic circuit is designed to synthesize this fractional order chaotic system.