Computing the solutions of the van der Pol equation to arbitrary precision

We describe an extension of the Taylor method for numerical solution ODEs that uses Padé approximants to obtain extremely precise results. The accuracy results is essentially limited only by computer time and memory, provided one works in arbitrary precision. In this stepsize adjusted achieve desired (variable stepsize), while order expansion can be either fixed or changed at each iteration order). As application, we have calculated periodic solutions (limit cycle) van der Pol equation with unprecedented a large set couplings (well beyond values currently found literature) used these validate asymptotic behavior period, amplitude Lyapunov exponent reported literature. also infer formulas fast component period maximum velocity, which never been before.