Abstract
The qualitative solutions of dynamical system expressed with nonlinear differential equation can be divided into two categories. One is that the motion of phase point may approach infinite or stable equilibrium point eventually. Neither periodic excited source nor self-excited oscillation exists in such nonlinear dynamic circuits, so its solution cannot be treated as the synthesis of multiharmonic. And the other is that the endless vibration of phase point is limited within certain range, moreover possesses character of sustained oscillation, namely the bounded nonlinear oscillation. It can persistently and repeatedly vibration after dynamic variable entering into steady state; moreover the motion of phase point will not approach infinite at last; system has not stable equilibrium point. The motional trajectory can be described by a bounded space curve. So far, the curve cannot be represented by concretely explicit parametric form in math. It cannot be expressed analytically by human. The chaos is a most universally common form of bounded nonlinear oscillation. A number of chaotic systems, such as Lorenz equation, Chua’s circuit and lossless system in modern times are some examples among thousands of chaotic equations. In this work, basic properties related to the bounded space curve will be comprehensively summarized by analyzing these examples.
Highlights
The motional trajectory of phase point in 3-dimensions phase space can be expressed by a space curve
Describing the motional trajectory of phase point using bounded space curve, chaotic functions which are not introduced in math manual are universal, while some functions introduced in math manual are special
Selecting properly three dynamic variables in nonlinear systems constitute a three-dimension phase space, the continuous chaos can be described by a bounded space curve
Summary
The parametric equation of circular helix is shown in (2) It is a space curve denoted by rectangular coordinates. The motional form of phase point will neither tend to be infinite, nor return to stable equilibrium point at last (or system has not stable equilibrium point), but always wander endlessly within definite range of phase space belong to bounded oscillation solution. This kind of oscillation can be expressed as Fourier series Another one is called nonconstant periodic oscillation or aperiodic oscillation for short. The orbit in phase portrait is always without repetition, the sustained oscillation with infinite or sufficient length is deterministic. Such a phenomenon is called orbital chaos of continuous time system
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