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Effective order reduction method based on parametrization of slow invariant manifolds

The method of integral manifolds is used to study the multidimensional systems of differential equations. This approach allows to solve an important problem of order reduction of differential systems. If a slow invariant manifold cannot be described explicitly then its parametrization is used for the system order reduction. In this case, either a part of the fast variables, or all fast variables, supplemented by a certain number of slow variables, can play a role of the parameters.


Introduction
Consider the system of differential equationṡ 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 the situation when the system (1), (2) has an integral manifold, that is, when the following conditions are fulfilled [1]- [5]: (1) the equation g(x, y, 0) = 0 has an isolated solution y = ψ 0 (x) for x ∈ R n ; (2) the functions f and g are uniformly continuous and bounded together with partial derivatives with respect to all variables up to (k + 2)-th order inclusive (k ≥ 0) in some region Ω 0 = {(x, y, ε) : x ∈ R n , y − ψ 0 (x) < ρ, 0 ≤ ε ≤ ε 0 }; (3) the eigenvalues of the matrix Recall, that the origins of the method of integral manifolds are found in the works of J. Hadamard [6], A. Lyapunov [7], H. Poincare [8] and O. Perron [9]. The essence of the method of integral manifolds was realized with amazing depth by A. Lyapunov [7] who developed a constructive procedure for problems of reduction on the stability of multidimensional systems 2 to systems of lower dimension. The possibility of the system order reduction is the essential aspect of the method of integral manifolds. The foundations of the theory were laid by N. Bogolyubov [10] and significant impact on the development of the method was provided by N. Bogolyubov and Yu Mitropolskii [11,12] and J. Hale [13]. Various aspects of the theory of slow integral manifolds and the behavior of solutions in their neighborhood are presented in [14]- [25], see also references therein.
The degenerated system regarding to (1), (2) has a forṁ It should be noted that the equations in the system (3) can often be either transcendental or polynomials of a high degree with respect to y. In these cases a solution of the system cannot be found in explicit form as y = ψ 0 (x). For the system order reduction in these cases it is possible to use a parametric form for the representation of the slow invariant manifolds [26,27,28]. Three main cases, in which either fast variables, or only a fraction of the fast variables, or fast variables supplemented by a certain number of slow variables, play a role of the parameters are considered below.

The case n = m
Consider the case of the dimensions equality of the fast and slow variables. Suppose that the system (3) can be solved with respect to x in the form x = ϕ 0 (y). In this case the fast vectorvariable y can play a role of a parameter for the representation of the slow invariant manifolds in the parametric form The corresponding invariance equation is obtained by substituting (4) in (1): For all functions occurring in (5), we write the formal asymptotic expansions in powers of the small parameter ε : where g (0) (ϕ 0 , y) = g(ϕ 0 , y, 0) and the non-degenerate matrix B(y) = g x (ϕ 0 , y, 0) [4,29,30].
Taking these expansions into account, the invariance equation (5) takes the form: Equating the coefficients of like powers of ε in the last equation, we get the expressions, which uniquely define the coefficients in (4) if det ∂ϕ 0 ∂y = 0. 3 Indeed, for ε 0 we have g(ϕ 0 , y, 0) = 0, which give the function ϕ 0 (y). For ε 1 we get (1) .
By similar way, for ε k we obtain: Thus, the parametric representation of the slow invariant manifold of (1), (2) is found in the form (4).

The case n < m
Consider the case when the number of fast variables in the system (1), (2) exceeds the number of slow variables. Then the system (3) contains m equations and n unknowns, where n < m. We take all components of vector x (dim x = n) complemented by (m − n) components of vector y, as an unknowns. Thereby the number of equations and unknowns in the system (3) will coincide.
Suppose that the solution of (3) can be written in the form with a parameter y 2 , where y = (y 1 , y 2 ) T , dim y 1 = m − n, dim y 2 = n.

The case n > m
Consider the case when the dimension of slow variables is greater than the dimension of fast variables. We call attention to the degenerate subsystem (3). It contains m equations and n unknowns, where n > m. To find the parametric representation of the slow invariant manifold of (1), (2), we take all components of y complemented by (n − m) components of the vector x, as the parameters. Then a solution of the system (3) can be written in the parametric form The system (1), (2) in this case can be rewritten in more convenient form aṡ x 1 = f 1 (x 1 , x 2 , y, ε), εẏ = g 2 (x 1 , x 2 , y, ε).
We will find the slow integral manifold in the form