Stability of Motion of a Nonholonomic System

Perturbation differential equations of motion of a general nonholonomic system subjected to the ideal nonholonomic constraints of Chetaev＇s type are established, and the equation of variation of energy is deduced by using the perturbation equations of the system. A criterion of the stability is obtained and an example is given to illustrate the application of the result.     

Poincaré Map Based on Splitting Methods

Firstly, by using the Liouville formula, we prove that the Jacobian matrix determinants of splitting methods are equal to that of the exact flow. However, for the explicit Runge-Kutta methods, there is an error term of order p ＋ I for the Jacobian matrix determinants. Then, the volume evolution law of a given region in phase space is discussed for different algorithms. It is proved that splitting methods can exactly preserve the sum of Lyapunov exponents invariable. Finally, a Poincaré map and its energy distribution of the Duffing equation are computed by using the second-order splitting method and the Heun method （a second-order Runge-Kutta method）. Computation illustrates that the results by splitting methods can properly represent systems＇ chaotic phenomena.     

Structure-Preserving Algorithms for the Lorenz System

Based on a splitting method and a composition method, we construct some structure-preserving algorithms with first-order, second-order and fourth-order accuracy for a Lorenz system. By using the Liouville＇s formula, it is proven that the structure-preserving algorithms exactly preserve the volume of infinitesimal cube for the Lorenz system. Numerical experimental results illustrate that for the conservative Lorenz system, the qualitative behaviour of the trajectories described by the classical explicit fourth-order Runge-Kutta （RK4） method and the fifth-order Runge-Kutta-Fehlberg （RKF45） method is wrong, while the qualitative behaviour derived from the structure-preserving algorithms with different orders of accuracy is correct. Moreover, for the small dissipative Lorenz system, the norm errors of the structure-preserving algorithms in phase space axe less than those of the Runge-Kutta methods.