Comparative analysis of two asymptotic approaches based on integral manifolds

摘要

A comparative analysis of the two powerful asymptotic methods, ILDM and MIM (intrinsic low-dimensional manifolds; method of invariant manifold), is presented in the paper. The two methods are based on the general theory of integral manifolds. The ILDM method is able to handle large systems of ODEs, whereas the MIM method treats systems with a limited number of unknown variables. The MIM method allows one to conduct analytical exploration of the original system and to obtain final expressions in compact form, whereas the ILDM method is a numerical approach that yields the numerical form of the desired surface. The ILDM method works well in a region where a rough splitting of the initial system exists. Regions of the phase space where splitting does not exist are problematic for the ILDM method. In these regions the MIM method provides additional information regarding the dynamical behaviour of the system. A number of simple examples are considered and analysed. It is shown that for the Semenov model (singularly perturbed system of ODEs) the ILDM method gives a surface which appears close to the first order (with respect to the corresponding small parameter) approximation of the stable (attracting) invariant manifolds. The complementary properties of the two asymptotic approaches suggests a feasible combination of the two methods, which is the subject of a future work.