By using a series of canonical transformations (Birkhoff's series), an approximate integral of a conservative compound pendulum is evaluated. Level lines of this approximate integral are compared with the numerica...By using a series of canonical transformations (Birkhoff's series), an approximate integral of a conservative compound pendulum is evaluated. Level lines of this approximate integral are compared with the numerical simulation results. It is seen clearly that with a raised energy level, the nearly integrable system becomes non-integrable, i.e. the regular motion pattern changes to the chaotic one. Experiments with such a pendulum device display the behavior mentioned above.展开更多
A multiple shooting method combined with a Newton iteration for determining bifurcation curves of two-dimensional maps is presented.This method is so effective that the high period bifurcation curves which could not b...A multiple shooting method combined with a Newton iteration for determining bifurcation curves of two-dimensional maps is presented.This method is so effective that the high period bifurcation curves which could not be estimated with any other methods can be determined.Some P8 bifurcation curves of the Henon map are calculated as an example.展开更多
The calculation of the bifurcation behavior of one-dimensional maps by Feigenbaum enabled him to discover the important universality and scaling properties. Feigenbaum calculated a particular root of a polynomial of o...The calculation of the bifurcation behavior of one-dimensional maps by Feigenbaum enabled him to discover the important universality and scaling properties. Feigenbaum calculated a particular root of a polynomial of order 2<sup>17</sup> with a Newton’s method coupled with the conjectured scaling relation. Later on, based on the sym-展开更多
The discovery of the chaos extends our knowledge about the steady state motion, i. e. except for the unbounded solution and the three bounded solutions (equilibrium, periodic and quasiperiodic), a new bounded solutio...The discovery of the chaos extends our knowledge about the steady state motion, i. e. except for the unbounded solution and the three bounded solutions (equilibrium, periodic and quasiperiodic), a new bounded solution——chaotic solution is added. It is remarkable that two or more steady state attractors may coexist in many cases and have their own attracting basins with non-zero Lebesgue measure in the展开更多
文摘By using a series of canonical transformations (Birkhoff's series), an approximate integral of a conservative compound pendulum is evaluated. Level lines of this approximate integral are compared with the numerical simulation results. It is seen clearly that with a raised energy level, the nearly integrable system becomes non-integrable, i.e. the regular motion pattern changes to the chaotic one. Experiments with such a pendulum device display the behavior mentioned above.
基金Project supported by the National Science Foundation of China。
文摘A multiple shooting method combined with a Newton iteration for determining bifurcation curves of two-dimensional maps is presented.This method is so effective that the high period bifurcation curves which could not be estimated with any other methods can be determined.Some P8 bifurcation curves of the Henon map are calculated as an example.
基金Project supported by the Science Fund of the Chinese Academy of Sciences and the Natural Science Fund of the Ministry of Education of China.
文摘The calculation of the bifurcation behavior of one-dimensional maps by Feigenbaum enabled him to discover the important universality and scaling properties. Feigenbaum calculated a particular root of a polynomial of order 2<sup>17</sup> with a Newton’s method coupled with the conjectured scaling relation. Later on, based on the sym-
基金Project partially supported by the National Natural Science Foundation of China
文摘The discovery of the chaos extends our knowledge about the steady state motion, i. e. except for the unbounded solution and the three bounded solutions (equilibrium, periodic and quasiperiodic), a new bounded solution——chaotic solution is added. It is remarkable that two or more steady state attractors may coexist in many cases and have their own attracting basins with non-zero Lebesgue measure in the
