Meminductor is a novel nonlinear inductor with memory. A meminductor-based chaotic oscillating circuit that has only two linear resistors, two linear capacitors and a meminductor, is designed based on a mathematical model of the flux-controlled meminductor to study its characteristics in nonlinear circuit. Through the analysis of bifurcations, dynamic map and Lyapunov exponents, it is found that the system can exhibit some complex characteristics, such as an infinite number of equilibrium points and burst chaos. Especially, bifurcation without parameters and coexisting attractors appear under a fixed set of parameter values but different initial conditions. Moreover, random characteristics of the PN sequences generated from the chaotic circuit are tested via the test suit of National Institute of Standards and Technology (NIST), and the tested randomness definitely reaches the standards of NIST. Finally, a scheme for digitally realising this oscillating circuit is provided using the digital signal processor (DSP).