1 Introduction

Consider the system of differential equations

$$\begin{aligned} \dot{x}=f(x,y,\varepsilon ), \end{aligned}$$
(1)
$$\begin{aligned} \varepsilon \dot{y}=g(x,y,\varepsilon ), \end{aligned}$$
(2)

where \(x \in \mathbb {R}^{n}\), \(y \in \mathbb {R}^{n}\), \(\varepsilon \) is a small positive parameter, \(0<\varepsilon \ll 1\), functions f and g are continuous with respect to (x, y) for all \(x\in \mathbb {R}^{n}\), \(y \in D \subset \mathbb {R}^{m}\) (\(D \subset \mathbb {R}^{m}\)). 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]):

  1. (i)

    the equation \(g(x,y,0)=0\) has an isolated solution \(y =\psi _0(x)\) for \(x\in \mathbb {R}^{n}\);

  2. (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 (\(k\ge 0\)) in some region \(\Omega _0 =\{(x, y,\varepsilon ): x\in \mathbb {R}^n,\, \Vert y-\psi _0(x)\Vert <\rho ,\, 0\le \varepsilon \le \varepsilon _0\}\);

  3. (iii)

    the eigenvalues of the matrix

    $$ B(x)=\frac{\partial g}{\partial y}(x,\psi _0( x),0) $$

    satisfy the inequality \(Re\lambda _i (x)\le -2\gamma <0\).

Note that some interesting aspects of the theory of slow integral manifolds and the behavior of solutions in their neighborhood were presented in [2, 3].

The degenerated system regarding to (1), (2) has a form

$$\begin{aligned} \begin{array}{l} \dot{x}=f(x,y,0), \\ 0=g(x,y,0). \end{array} \end{aligned}$$
(3)

It should be noted that the equations of 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 \(y=\psi _0(x)\). In these cases, for the system-order reduction, it is possible to use a parametric form for the representation of the slow invariant manifolds [4, 5]. Below we consider the three major cases, where 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.

2 The Case \(\varvec{n=m}\)

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 \(x=\varphi _0(y)\). In this case, the fast vector-variable y can play a role of a parameter for the representation of the slow invariant manifolds in the parametric form

$$\begin{aligned} x=\varphi (y,\varepsilon )=\varphi _0(y)+\varepsilon \varphi _1(y)+\cdots +\varepsilon ^k \varphi _k(y)+\cdots ~. \end{aligned}$$
(4)

The corresponding invariance equation is obtained by substituting (4) in (1):

$$\begin{aligned} \frac{\partial \varphi }{\partial y}g(\varphi , y,\varepsilon )=\varepsilon f(\varphi , y,\varepsilon ). \end{aligned}$$
(5)

For all functions in (5), we write the formal asymptotic expansions in the powers of the small parameter \(\varepsilon \):

$$ f\Big (\sum _{k\ge 0}\varepsilon ^{k}\varphi _{k},y,\varepsilon \Big ) =\sum _{k\ge 0}\varepsilon ^{k}f^{(k)}(\varphi _{0},\ldots ,\varphi _{k},y), $$
$$ g\Big ( \sum _{k\ge 0}\varepsilon ^{k}\varphi _{k},y, \varepsilon \Big ) = g^{(0)}(\varphi _0,y) +B(y)\sum _{k\ge 1}\varepsilon ^{k}\varphi _{k}+\sum _{k\ge 1}\varepsilon ^{k}g^{(k)} (\varphi _{0},\ldots ,\varphi _{k-1},y), $$

where \(g^{(0)}(\varphi _0,y)=g(\varphi _{0},y,0)\) and the nondegenerate matrix \(B(y)=g_{x}(\varphi _{0},y,0)\); see [1, 4]. Taking these expansions into account, the invariance equation (5) takes the form:

$$ \sum _{k\ge 0}\varepsilon ^k\frac{\partial \varphi _k}{\partial y}\Big ( g^{(0)}+B\sum _{k\ge 1}\varepsilon ^k \varphi _k+ \sum _{k\ge 1}\varepsilon ^k g^{(k)}\Big )= \varepsilon \sum _{k\ge 0}\varepsilon ^k f^{(k)}. $$

Equating the coefficients at the same of like powers of \(\varepsilon \) in the last equation, we get the expressions, which uniquely define the coefficients in (4) when \(\det \left( \partial \varphi _0/\partial y \right) \ne 0\).

Indeed, for \(\varepsilon ^0\) we have \(g(\varphi _0,y,0)=0\), which give the function \(\varphi _0(y)\). For \(\varepsilon ^1\), we get

$$ \varphi _1=\Big ( \frac{\partial \varphi _0}{\partial y}B\Big )^{-1} \Big ( f^{(0)}- \frac{\partial \varphi _0}{\partial y}g^{(1)}\Big ). $$

Likewise, for \( \varepsilon ^k\) we obtain

$$ \varphi _k=\Big ( \frac{\partial \varphi _0}{\partial y}B \Big )^{-1} \Big [ f^{(k-1)}- \frac{\partial \varphi _0}{\partial y}g^{(k)}- \sum \limits _{i= 1}^{k-1}\frac{\partial \varphi _i}{\partial y}\left( B\varphi _i+g^{(k-i)}\right) \Big ]. $$

Thus, the parametric representation of the slow invariant manifold of (1), (2) is found in the form (4).

3 The Case \(\varvec{n<m}\)

Consider the case where the number of fast variables in the system (1), (2) exceeds the number of slow variables. Then, the system (3) contains m equations for n unknowns and \(n<m\). We take all components of vector x (\(\dim (x)=n\)) complemented by \(m-n\) components of vector y, as the 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

$$ x=\varphi _0(y_2), \quad y_1=\psi _0(y_2), $$

with a parameter \(y_2\), where \(y=(y_1,y_2)^T\), \(\dim y_1=m-n\), \(\dim y_2=n\). The system (1), (2) start the sentence with this can be rewritten in a more convenient form:

$$\begin{aligned} \dot{x}=f(x,y_1,y_2,\varepsilon ), \end{aligned}$$
(6)
$$\begin{aligned} \varepsilon \dot{y}_1=g_1(x,y_1,y_2,\varepsilon ), \end{aligned}$$
(7)
$$\begin{aligned} \varepsilon \dot{y}_2=g_2(x,y_1,y_2,\varepsilon ). \end{aligned}$$
(8)

We will find the slow integral manifold in the form

$$\begin{aligned} x=\varphi (y_2,\varepsilon ), \end{aligned}$$
(9)
$$\begin{aligned} y_1=\psi (y_2,\varepsilon ). \end{aligned}$$
(10)

Substituting (9), (10) into (6) and (7), and taking into account (8), we obtain the invariance equations

$$ \frac{\partial \varphi }{\partial y_2}g_2(\varphi , \psi , y_2,\varepsilon )=\varepsilon f(\varphi ,\psi , y_2,\varepsilon ), $$
$$ \frac{\partial \psi }{\partial y_2}g_2(\varphi , \psi , y_2,\varepsilon )=g_1(\varphi ,\psi , y_2,\varepsilon ). $$

For the functions \(\varphi (y_2,\varepsilon )\), and \(\psi (y_2,\varepsilon )\) we write the formal asymptotic expansions:

$$\begin{aligned} \varphi (y_2,\varepsilon )=\varphi _0(y_2)+\varepsilon \varphi _1(y_2)+\cdots +\varepsilon ^k \varphi _k(y_2)+\cdots , \end{aligned}$$
(11)
$$\begin{aligned} \psi (y_2,\varepsilon )=\psi _0(y_2)+\varepsilon \psi _1(y_2)+\cdots +\varepsilon ^k \psi _k(y_2)+\cdots \end{aligned}$$
(12)

Equating the coefficients of the same powers of \(\varepsilon \) in the invariance equations, we get the expressions, which uniquely define the coefficients in (11) and (12) when \(\det \partial \varphi _0/\partial y_2 \ne 0\) and \(\det \partial \psi _0/\partial y_2 \ne 0\).

4 The Case \(\varvec{n>m}\)

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 for 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 \(x_1=\varphi _0(x_2,y)\), where \(x=(x_1,x_2)^T\), \(\dim x_1=m\), \(\dim x_2=n-m\).

The system (1), (2) in this case can be rewritten in more convenient form as

$$ \dot{x}_1=f_1(x_1,x_2,y,\varepsilon ), $$
$$\begin{aligned} \dot{x}_2=f_2(x_1,x_2,y,\varepsilon ), \end{aligned}$$
(13)
$$ \varepsilon \dot{y}=g_2(x_1,x_2,y,\varepsilon ). $$

We will find the slow integral manifold in the form

$$\begin{aligned} x_1=\varphi (x_2,y,\varepsilon )=\varphi _0(x_2,y)+\varepsilon \varphi _1(x_2,y)+\cdots + \varepsilon ^k \varphi _k(x_2,y)+\cdots . \end{aligned}$$
(14)

The invariance equation

$$\begin{aligned} \varepsilon \frac{\partial \varphi }{\partial x_2}f_2(\varphi ,x_2, y,\varepsilon )+ \frac{\partial \varphi }{\partial y}g(\varphi ,x_2, y,\varepsilon )=\varepsilon f_1(\varphi ,x_2, y,\varepsilon ) \end{aligned}$$
(15)

is yielded from (13) and (14). Equating the coefficients at the same powers of \(\varepsilon \) in the last equation, we get the expressions, which uniquely define the coefficients in (14) for the case when \(\det \Big ( \partial \varphi _0/\partial y\,\, G\Big )\ne 0\).

Thus, formula (14) defines the slow integral manifold of the system in the parametric form.