非线性电路通向混沌的演化过程

给出了四阶非线性电路通向复杂性的两种演化模式,指出这两种模式与三个共存的平衡点有关.在第一种模式中,不稳定的平衡点由Hopf分岔导致了稳定的周期运动,经过倍周期分岔通向混沌,其所有的吸引子都保持对称结构;而在第二种模式中,另两个平衡点由Hopf分岔产生相互对称的极限环,并分别导致了两个混沌吸引子,其分岔过程步调一致,而且所有的吸引子都相互对称.随着参数的变化,这两个混沌吸引子相互作用形成一个扩大的混沌吸引子,导致与第一种分岔模式中定性一致的混沌运动.

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.