Computation of two-dimensional invariant manifolds with radial growth factor

In order to balance growth rate of manifold in all directions and construct global manifold structure of a dynamical system, a radial control factor is adopted to normalize the original dynamical system. Taking radius component of the tangent vector as a standard, this method controls manifold expanding at same speed in all directions. Theoretical analysis and example calculation demonstrate that manifolds before and after normalization have same orbit with the original one, which means their global manifold structures are consistent. Lorenz and Duffing systems are taken for examples to demonstrate effectiveness of the proposed approach. It indicates that the method not only get same effect as geodesic process but also present manifold in discrete flow way, which avoids many complicated boundary value problems.