Symbolic manipulation for some non-algebraic objects and its application in computing the lce of van der pol equation

The applications of symbolic manipulation methods based on algebraic equation solving theory are extended to a class of non-algebraic objects, called analytic equalities, for finding algorithmic solutions to some mathematical problems related to nonlinear dynamics. A symbolic manipulation method for analytic equalities is established, based on which an algorithm for constructing necessary conditions for analytic equality sets is developed to mechanically derive some unknown relations. As an example, the method is used to compute the Lyapunov Characteristic Exponent (LCE) of the chaotic attractor of van der Pol equation. Instead of numerically computing the LCE, we try to find its dependence on the system parameters by symbolic manipulation. Combining our result on the LCE with related qualitative results on the system, we get some conclusions on the system behavior.