Abstract: With definition of DHT(distinguished hyperbolic trajectory) and existing measure function in phase space,a measure function in extended phase space is presented.Existing algorithms with constant accuracy parameters is laborious as high precision is required.In order to overcome this shortage,a variable-step convergence algorithm is proposed.The main idea is to estimate initial region with the help of ISP(instantaneous stagnation points) and adopt variable-step grids to increase efficiency.With theoretical analysis and numerical calculation,an optimal range of key parameter is given.Two-dimensional and three-dimensional Duffing systems are used to test the performance.It shows that convergence route gained by the developed measure function is smooth and stable.And the variable-step convergence algorithm is more efficient.

Key words: non-autonomous system, time dependent vector fields, distinguished trajectory