Differential Invariants of Feedback Transformations for Quasi-Harmonic Oscillation Equations

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).
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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.
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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.
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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 $J_{22}$.
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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.
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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 $f = f(y,u)$.
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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.