Abstract:
The classification problem for a control-parameter-dependent second-order differential equations is considered. The algebra of the differential invariants with respect to Lie pseudo-group of feedback transformations is calculated. The equivalence problem for a control-parameter-dependent quasi-harmonic oscillation equation is solved. Some canonical forms of this equation are constructed.
Consider the problems of equivalence and classification for the differential equation:
$$
\tag{1}
\frac{d^{2}y}{d x^{2}}+f(y,u)=0,
$$
with respect to the feedback transformations [1]:
$$
\tag{2}
\varphi\colon (x,y,u)\longmapsto (X(x,y),Y(x,y),U(u)),
$$
where the function$f(y,u)$ is smooth. Here $u$ is a scalar control
parameter. We will call an equation of form (1)
{control-parameter-dependent quasi-harmonic oscillator equation}
(QHO).
$ $ Definition. Operator
$$
\tag{3}
\nabla = M\frac{d}{dy} + N\frac{d}{du}
$$
is called $G$-invariant differentiation if it commutes with every element of any prolongation of Lie algebra $\mathcal{G}$, where $M$ and $N$ are the functions on the jet space.
$ $ Theorem.Differential operators \begin{align}
\tag{4}
\nabla_1 = \frac{z}{z_{y}} \frac{d}{dy},
\\
\tag{5}
\nabla_2 = \frac{z}{z_{u}} \frac{d}{du}
\end{align} are$G$-invariant differentiations.
$ $ Theorem.Functions $$
J_{21} =\frac{z_{yy}z}{z_y^2},\quad J_{22} =\frac{z_{yu}z}{z_{y}z_u}
$$ form a complete set of the basic second-order differential invariants, i.e.any other second-order differential invariants are the functions of$J_{21}$and$J_{22}$.
$ $ Theorem.Quasi-harmonic oscillation equation differential invariants algebra is generated by second-order differential invariants$J_{21}$, $J_{22}$and invariant differentiations$\nabla_1$and$\nabla_2$. This algebra separates regular orbits. $ $ Let us call an equation $\mathcal{E}_f$regular, if
$$
dJ_{21}(f)\wedge dJ_{22}(f)\neq 0.
$$
Here $J(f)$ is the value of the differential invariant $J$ on the function $f = f(y,u)$.
$ $ Theorem.Suppose that the functions$f$and$g$are real-analytical. Two regular equations$\mathcal{E}_f$and$\mathcal{E}_g$are locally$G$-equivalent if and only if the functions$\Phi_{if}$and$\Phi_{ig}$identically equal ($i=1,2,3$) and 3-jets of the functions$f$and$g$belong to the same connection component.