Slow Invariant Manifolds in the Problem of Order Reduction of Singularly Perturbed Systems Elena Tropkina and Vladimir Sobolev Abstract The method of integral manifolds is used to study singularly perturbed systems of differential equations. The algorithms for the construction of the slow invariant manifolds in the case with different dimensions of the fast and slow variables was derived. 1 Introduction Consider the system of differential equations ˙ x = f (x , y , ε), (1) ε ˙ y = g(x , y , ε), (2) where x ∈ R n , y ∈ R n , ε is a small positive parameter, 0 <ε ≪ 1, functions f and g are continuous with respect to (x , y ) for all x ∈ R n , y ∈ D ⊂ R m ( D ⊂ R m ). We will consider a situation where the system (1), (2) has an integral manifold, that is, when the following conditions are fulfilled (see [1, 4]): (i) the equation g(x , y , 0) = 0 has an isolated solution y = ψ 0 (x ) for x ∈ R n ; (ii) the functions f and g are uniformly continuous and bounded together with their partial derivatives with respect to all variables up to (k + 2)th order inclusively This work was funded by RFBR and Samara Region (project 16-41-630529-p) and the Ministry of Education and Science of the Russian Federation under the Competitiveness Enhancement Program of Samara University (2013–2020). E. Tropkina (B ) · V. Sobolev Department of Differential Equations and Control Theory, Samara National Research University, Moskovskoye Shosse, 34, Samara 443086, Russian Federation e-mail: elena_a.85@mail.ru V. Sobolev e-mail: v.sobolev@ssau.ru © Springer Nature Switzerland AG 2019 A. Korobeinikov et al. (eds.), Extended Abstracts Spring 2018, Trends in Mathematics 11, https://doi.org/10.1007/978-3-030-25261-8_19 125