An autonomous circuit without dissipative elements is a conservative system, whose phase portrait of oscillation is a Hamiltonian cycle rather than a limit cycle. The size of cycle is closed related to the initial energy stored in the network. The total energy stored in the network keeps always constant after the starting instant. It is a periodic oscillation but not an attractor. It is a lossless system that the non-autonomous circuit has excited source but without dissipative elements. The oscillation solution not only depends on the excited source, but also on the initial conditions sensitively. The change process of stored energy of every reactance elements is distinct in every excited period. The boundness and aperiodicity of phase portrai show the basic characteristics of the chaotic oscillation. This paper introduces an experimental circuit of chaotic signal generator structured by lossless system. It is demonstrated that the chaos produced by the lossless systems is a kind of aperiodic oscillation. It has not attractiveness for neighboring trajectory so that it is not an attractor either.

The solutions of dynamical system expressed with nonlinear differential equation usually is shown by using time function () u t. But this is not unique mode, when particularly () u t cannot be solved. In the modern theory of circuit and system, we can select three dynamical variables in the nonlinear system to constitute 3-dimension phase space. The mutual nonlinear relation among three dynamical variables can be described by a bounded space curve. This is 3-dimension phase portrait. The nonlinear dynamical systems of regarding 3 N > variation may constitute N-dimension Euclidean space. The bounded space curve cannot be represented by concretely explicit parametric form in math. It cannot be solved analytically by human. However, the graphic solution can be plotted by numerical simulation. If the bounded space curve is non-periodic in simulation interval, this is orbital chaos of continuous time system. This paper researches the produce and property of chaos by means of the analysis method of frequency domain and theorem of power balance. We prove that the second order differential circuit which is constituted by mixing of multi-excited source with different frequency also can produce chaos.

Chaos is the most universally common form of bounded nonlinear function. It commonly exists in the various subject areas of nature. Chaos is the universal term of variously bounded nonlinear aperiodic oscillation. This paper proves that the first order differential circuit which is constituted by mixing of three harmonic sources with different frequency also can produce chaos. It sufficiently explains that the extensiveness of chaotic functions exists in nature. The main harmonic components in the differential equations can be solved by using the harmonic balance principle and power balance theorem. Their correctness of solving results can be verified by phase portrait plotted by simulation. In last century, the era when chaos theory was first published, chaos was considered as a singular attractor in a lot of literatures. The recognition is obviously unilateral and wrong. In fact, people can also make a completely opposite conclusion, the motional trajectory of phase point will neither diverge to be infinite nor converge to stable limit cycle. The phase point freely and arbitrarily wandering in phase space is ordinary and universal motional form, but it is non-random. Chaotic phase portraits on which trajectories are not repeated are pervasive phenomenon. The constant periodic oscillation which continuously repeats original orbit is an individual and special motion form.