Abstract: A quantitative analysis of limit cycles and homoclinic orbits, and the bifurcation curve for the Bogdanov-Takens system are discussed. The parameter incremental method for approximate analytical-expressions of these problems is given. These analytical-expressions of the limit cycle and homoclinic orbit are shown as the generalized harmonic functions by employing a time transformation. Curves of the parameters and the stability characteristic exponent of the limit cycle versus amplitude are drawn. Some of the limit cycles and homoclinic orbits phase portraits are plotted. The relationship curves of parameters μand αwith amplitude a and the bifurcation diagrams about the parameter are also given. The numerical accuracy of the calculation results is good.

Key words: parameter incremental method, analytical- expressions, Bogdanov-Takens system, limit cycle, homoclinic orbit, bifurcation diagrams