Abstract
A system of equations
\begin{equation}
dx=\sum_{i=1}^np_i(x)\,dt^i, \tag{1} \label{1}
\end{equation}
is considered, where $x$ and $p_i(x)$ are $(n+1)$-dimensional vectors for which the conditions of complete integrability are satisfied. It is assumed that the system \eqref{1} possesses a closed trajectory $\gamma$. It is proved that at least one of the vectors $p_i(x)$ is non-zero at all points of this closed trajectory.
In the neighborhood of $\gamma$, a system of local coordinates $(z,s)$ is introduced, where $z$ is an $n$-dimensional vector and $s$ is a scalar. It is shown that the conditions of complete integrability are also satisfied for the system corresponding to system \eqref{1} in these local coordinates. The simplest case is examined, where the system \eqref{1} in local coordinates corresponds to a linear system
\begin{equation}
dz=\sum_{i=1}^nB_i(s)z\,dt^i+A(s)z\,ds, \tag{2} \label{2}
\end{equation}
where $B_i(s)$ and $A(s)$ are $n \times n$ matrices with period $1$. For the case where the Jordan normal form for the matrices $B_i(0)$ ($i=1,2,\dots,n-1$) is diagonal, conditions are provided under which two systems of the form \eqref{1} possessing closed trajectories are topologically equivalent in the neighborhoods of these trajectories.
Bibliography: 8 items.
Full Text
Preamble
This work investigates the properties of systems of differential equations of the form $\frac{dx}{dt_i} = p_i(x)$ for $i=1, \dots, n$, following the foundational approaches established in \cite{1, 2, 3}. We consider the $(n+1)$-dimensional system:
$$ \frac{dx}{dt_i} = p_i(x), \quad i = 1, 2, \dots, n $$
where the functions $p_i(x)$ satisfy the commutativity condition:
$$ \frac{\partial p_i(x)}{\partial x} p_j(x) = \frac{\partial p_j(x)}{\partial x} p_i(x) $$
for all $i, j = 1, \dots, n$. Here, $x$ belongs to a domain $D$ in $(n+1)$-dimensional space, and $p_i(x)$ are assumed to be of class $C^r$ ($r \ge 1$).
Section 1. Structural Properties and Commutativity
Let $Q$ be a subdomain where $p_i(x) \neq 0$. For any $x_0 \in Q$, there exist functions $k_j(x)$ such that $p_j(x) = k_j(x) p_i(x)$. From the commutativity conditions, it follows that:
$$ \frac{dk_j(x)}{dx} p_i(x) = 0 $$
This implies that the coefficients $k_j(x)$ are constant along the trajectories of the system. Consequently, in the domain $Q$, the vectors $p_j(x)$ are proportional to $p_i(x)$, which simplifies the integration of the system. If we consider a sequence of subdomains $Q_s$ ($s = 1, 2, \dots, m$), the relationship between the vector fields can be expressed as $p_{is}(x) = M(s) p_{is}(x)$, where $M(s)$ is a transition matrix.
Section 2. Transformation and Integration
We consider the transformation of the system using the variables $z$ and $s$. The derivatives with respect to these variables are governed by the following relations:
$$ \begin{aligned} \frac{\partial q_i(z, s)}{\partial z} q_j(z, s) &= \theta^*(s) \left( \frac{\partial p_i(x)}{\partial x} - \frac{\Pi_i(x)}{\Pi_n(x)} \frac{\partial p_n(x)}{\partial x} - p_n(x) \frac{d}{dx} \left( \frac{\Pi_i(x)}{\Pi_n(x)} \right) \right) \ &\times \left( p_j(x) - \frac{\Pi_j(x)}{\Pi_n(x)} p_n(x) \right) \end{aligned} $$
By applying the conditions from (50) and (51), we establish the equivalence of the mixed partial derivatives. The system can then be reduced to the form:
$$ dz = \sum_{i=1}^{n-1} B_i(s) z \, dt_i + A(s) z \, ds $$
where $B_i(s)$ are $n \times n$ matrices. The consistency of this system requires that the matrices satisfy the commutation relation $C B_i(0) = B_i(0) C$, where $C = Z(1)$ is the fundamental matrix solution at $s=1$.
Section 3. Global Solutions and Mapping
The general solution of the transformed system (52) can be expressed using the exponential mapping. Specifically, if $Z(s)$ is the solution to the initial value problem $dz/ds = A(s)z$ with $Z(0) = E$, then the state at any point can be related to the initial state $z_0$ via:
$$ z(s, t) = Z(s) \exp\left( \sum B_i(0) t_i \right) z_0 $$
This formulation allows us to map the solutions across different regions of the domain. We define a mapping $\mu_x: z \to z'$ such that the components transform as $z'_j = \text{sign}(z_j) |z_j|^\chi$.
Section 4. Convergence and Stability Conditions
We analyze the behavior of the system as $x \to \infty$ or $s \to 0$. The existence of a stable solution depends on the signs of the coefficients $\alpha_j$ and the magnitudes of the eigenvalues of the matrices $B_i$. Specifically, if $\prod |c_j|^{\alpha_j} = 1$, the system exhibits specific asymptotic properties.
For the case where $k = n-1$, the conditions for the existence of a unique solution simplify significantly. The mapping $v(z, s)$ defined by:
$$ v(z, s) = Z(s) \mu_x Z^{-1}(s) z $$
provides a continuous link between the states at $s=0$ and $s=1$. This ensures that the structural properties of the differential equations are preserved under the group of transformations defined by (52) and (57).
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