By Yanzhu Liu
Attitude dynamics is the theoretical foundation of perspective keep watch over of spacecrafts in aerospace engineering. With the improvement of nonlinear dynamics, chaos in spacecraft angle dynamics has drawn nice cognizance because the 1990's. the matter of the predictability and controllability of the chaotic perspective movement of a spacecraft has a pragmatic value in astronautic technological know-how. This e-book goals to summarize simple thoughts, major methods, and up to date development during this quarter. It specializes in the examine paintings of the writer and different chinese language scientists during this box, supplying new equipment and viewpoints within the research of spacecraft angle movement, in addition to new mathematical versions, with yes engineering backgrounds, for additional analysis.
Professor Yanzhu Liu used to be the Director of the Institute of Engineering Mechanics, Shanghai Jiao Tong collage, China. Dr. Liqun Chen is a Professor on the division of Mechanics, Shanghai collage, China.
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Extra info for Chaos in Attitude Dynamics of Spacecraft
If meanwhile the unstable manifold W u(ps1) of ps1 coincides with the stable manifold W s(ps2) of ps2 so that there is another heteroclinic orbit, the two heteroclinic orbits form a heteroclinic cycle. A heteroclinic cycle may consist of more than two heteroclinic orbits connecting a few saddle points. Fig. 1 shows examples of homoclinic orbits, heteroclinic orbits, and heteroclinic cycles on a plane. 1 Homoclinic orbits, heteroclinic orbits, and heteroclinic cycles (a) a homoclinic orbit (b) two homoclinic orbits (c) heteroclinic cycle consisting of two heteroclinic orbits (d) heteroclinic cycle consisting of three heteroclinic orbits If the stable manifold and the unstable manifold do not coincide with each other, they may intersect each other.
A chaotic motion is represented by a trajectory that never closes and repeats itself because of the aperiodicity of the motion, and the trajectory is located in a bounded region due to the recurrence of the motion. Therefore, the trajectory of chaos in the phase plane usually occupies a part of the phase space. However, the trajectory of a quasiperiodic motion does not close on itself either, although it looks much more regular than a chaotic trajectory. In addition, it is difficult in practice to differentiate a trajectory of chaos from that of a periodic motion with a sufficient large period.
Some numerical characteristics associated with the motion of a system can be used to identify chaos. These characteristics include Lyapunov exponents, fractal dimensions, power spectra, and entropies. If one or more of these characteristics satisfy certain conditions, the motion may be chaotic. As explained in the previous section, chaos can be described in different aspects. Quantifying these descriptions leads to corresponding numerical characteristics. To specify the sensitivity of chaos to initial states, Lyapunov exponents are introduced.