The stability-preserving trajectory-reduction methodology for analyzing nonlinear dynamics

Stability-preserving trajectory-reduction (SPTR) is proposed for analyzing nonlinear dynamics. The key is to get the time response curves at first, and divide the curves into complementary pairs respectively, and then map them into a series of spaces with 1 or 2 DOF by using suitable linear transformation. Thus, all dynamic information contained in the original system is transformed to these mapping systems of lower-dimension. Based on SPTR, the complementary-cluster energy-barrier criterion (CCEBC) was set up for dealing with bounded stability in higher-space. It defines a transformation to transform multirigid trajectory into the extended phase plane corresponding to the relative motion of the complementary clusters. The stability margin can be defined rigorously for the image trajectory. The coordinate-plane projection method is put forward for studying bifurcation and chaos in higher-dimensional dynamics systems. The n-dimensional X is divided into a pair of complementary subsets 2D and (n-2)D. The specified trajectory is investigated by presenting X₁(t) on phase plane and X₂(t) on parameter space