Evolution from regular movement patterns to chaotic attractors in a nonlinear electrical circuit

For a fourth-order autonomous nonlinear electric circuit, we present two evolution patterns to complexity associated with the three coexisting equilibrium points. In the first pattern, stable periodic movement with symmetric structure can be observed by Hopf bifurcation from the unstable equilibrium point, which may lead to chaos via cascading of period-doubling bifurcations. All the attractors, including the chaos, keep the symmetric property. While in the second evolution pattern, two limit cycles symmetric to each other may occur via Hopf bifurcations from the other two stable equilibrium points, which may also lead to two chaotic attractors, respectively. Comparing with the two evolution procedures associated with the two stable equilibrium points, not only the bifurcations keep the same pace, but also the attractors including the two final chaotic attractors are still symmetric to each other. With further variation of the parameters, the two chaotic attractors may interact with each other to form another enlarged chaotic attractor, which is qualitatively equivalent to the chaos in the first evolution pattern.