On systems of differential equations containing parameters.
(Über Systeme von Differentialgleichungen, die Parameter enthalten.)

*(Russian)*Zbl 0041.42316Translation of the author’s introduction: Consider a system of differential equations

\[ \frac{dy_i}{dt} = f_i(t,y,z)\quad (i=1,2,\ldots,n), \]

\[ \mu_j\,\frac{dz_j}{dt} = F_j(t,y,z)\quad (j=1,2,\ldots,m), \]

and the solution to this system, determined by the conditions

\[ y_i(t^{(0)}) = y_i^{(0)},\quad z_j(t^{(0)}) = z_j^{(0)}, \]

This solution depends on the parameters \(\mu_j>0\).

The purpose of this article is to study functions \(y_i(t,\mu_j)\) and \(z_j(t,\mu_j)\) when all \(\mu_j\rightarrow 0\). Moreover, we will assume that \(\mu_j\) tends to zero can be characterized by some parameter \(\mu\) (continuous or discrete), so that all \(\mu_j\) are the functions \(\mu\): \(\mu_j = \mu_j(\mu)\) and \(\mu_j(\mu)\rightarrow 0\) for \(\mu\rightarrow 0\).

In addition, we will assume that \(\mu_{j+1}(\mu)\le \mu_j(\mu)\) and there is a limit to the relation \(\frac{\mu_{j+1}(\mu)}{\mu_j(\mu)} = \nu_j\) for \(\mu\rightarrow 0\). Without restriction of generality (by changing the right-hand sides), we can assume that either \(\nu_j = 1\) and generally \(\mu_{j+1}\mu) = \mu_j(\mu)\), or \(v_j = 0\).

Our goal is to establish the conditions under which the limits

\[ \overline y_i(t) = \lim_{\mu\to 0} y(t,\mu_j)\text{ and }\overline z_j(t) = \lim_{\mu\to 0} z(t,\mu_j) \]

defined as a solution to a degenerate system

\[ \frac{d\overline y_i}{dt} = f_i(t,\overline y,\overline z)\quad (i=1,2,\ldots,n), \]

\[ F_j(t,\overline y,\overline z) = 0\quad (j=1,2,\ldots,m), \]

for \(\overline y_i(t^{(0)}) = y_i^{(0)}\).

This problem was considered for the case of one parameter by the author [Mat. Sb., N. Ser. 22(64), 193–204 (1948; Zbl 0037.34401)] and A. B. Vasil’eva [Dokl. Akad. Nauk SSSR, N. Ser. 61, 597–599 (1948; Zbl 0041.42005)]. Exploring multiparameter systems very similar to the study of systems with one parameter was noted by us in the cited paper. However, as articulated there the theorem is false (my attention to this inaccuracy of the formulation of the aforementioned theorem was drawn by I. S. Gradshteĭn), then we give here a more detailed exposition in the case of several parameters.

\[ \frac{dy_i}{dt} = f_i(t,y,z)\quad (i=1,2,\ldots,n), \]

\[ \mu_j\,\frac{dz_j}{dt} = F_j(t,y,z)\quad (j=1,2,\ldots,m), \]

and the solution to this system, determined by the conditions

\[ y_i(t^{(0)}) = y_i^{(0)},\quad z_j(t^{(0)}) = z_j^{(0)}, \]

This solution depends on the parameters \(\mu_j>0\).

The purpose of this article is to study functions \(y_i(t,\mu_j)\) and \(z_j(t,\mu_j)\) when all \(\mu_j\rightarrow 0\). Moreover, we will assume that \(\mu_j\) tends to zero can be characterized by some parameter \(\mu\) (continuous or discrete), so that all \(\mu_j\) are the functions \(\mu\): \(\mu_j = \mu_j(\mu)\) and \(\mu_j(\mu)\rightarrow 0\) for \(\mu\rightarrow 0\).

In addition, we will assume that \(\mu_{j+1}(\mu)\le \mu_j(\mu)\) and there is a limit to the relation \(\frac{\mu_{j+1}(\mu)}{\mu_j(\mu)} = \nu_j\) for \(\mu\rightarrow 0\). Without restriction of generality (by changing the right-hand sides), we can assume that either \(\nu_j = 1\) and generally \(\mu_{j+1}\mu) = \mu_j(\mu)\), or \(v_j = 0\).

Our goal is to establish the conditions under which the limits

\[ \overline y_i(t) = \lim_{\mu\to 0} y(t,\mu_j)\text{ and }\overline z_j(t) = \lim_{\mu\to 0} z(t,\mu_j) \]

defined as a solution to a degenerate system

\[ \frac{d\overline y_i}{dt} = f_i(t,\overline y,\overline z)\quad (i=1,2,\ldots,n), \]

\[ F_j(t,\overline y,\overline z) = 0\quad (j=1,2,\ldots,m), \]

for \(\overline y_i(t^{(0)}) = y_i^{(0)}\).

This problem was considered for the case of one parameter by the author [Mat. Sb., N. Ser. 22(64), 193–204 (1948; Zbl 0037.34401)] and A. B. Vasil’eva [Dokl. Akad. Nauk SSSR, N. Ser. 61, 597–599 (1948; Zbl 0041.42005)]. Exploring multiparameter systems very similar to the study of systems with one parameter was noted by us in the cited paper. However, as articulated there the theorem is false (my attention to this inaccuracy of the formulation of the aforementioned theorem was drawn by I. S. Gradshteĭn), then we give here a more detailed exposition in the case of several parameters.

##### MSC:

34-XX | Ordinary differential equations |