Preamble

This work, published in 1967 (Vol. III, No. 10), addresses the construction of Lyapunov functions for autonomous systems of differential equations. We consider a system of the form:
$$\frac{dx}{dt} = P(x, y), \quad \frac{dy}{dt} = Q(x, y)$$
where $P(x, y)$ and $Q(x, y)$ are defined in a domain $G$. We investigate the existence of a function $M(x, y)$ such that the total derivative with respect to time satisfies specific stability criteria. Following the methodologies established in \cite{1, 2, 4}, we define a set of auxiliary functions $\omega_i(x, y) = 0$ for $i = 1, \dots, s$, which characterize the geometric structure of the phase space and the boundaries of stability regions.

The construction of the Lyapunov function is based on the product of these auxiliary functions:
$$\Phi = \omega_0 \omega_1 \dots \omega_p \omega_{p+1} \dots \omega_s \omega_{s+1}$$
where $\omega_0 = 1$ and $\omega_{s+1} = 1$. For indices $i = 1, \dots, r$, the functions $\omega_i = 0$ represent algebraic curves or surfaces. For $i = p+1, \dots, q$, we introduce linear forms $u_i = a_{i1}x + b_{i1}y + c_{i1}$ and $v_i = a_{i2}x + b_{i2}y + c_{i2}$, where the determinant $\Delta_i = a_{i1}b_{i2} - a_{i2}b_{i1} \neq 0$. These forms allow for the local linearization of the system near critical points or specific trajectories.

For the regions $A_j$ defined by the intersection of these surfaces, we analyze the behavior of the partial derivatives of $\omega_i$. The coefficients $\alpha_j, \beta_j, \gamma_j, \delta_j$ are introduced to satisfy the following differential relations:
$$\frac{\partial \omega_j}{\partial x} P + \frac{\partial \omega_j}{\partial y} Q = \omega_j \left( \alpha_j \frac{\partial P}{\partial x} + \beta_j \frac{\partial P}{\partial y} + \gamma_j \frac{\partial Q}{\partial x} + \delta_j \frac{\partial Q}{\partial y} \right)$$
In the case where $\alpha_j = \delta_j = 0$ and $\beta_j = -\gamma_j$, the system exhibits specific rotational or Hamiltonian symmetries. By substituting these into the governing equations, we derive conditions for the existence of a positive definite function $F(x, y)$ that serves as a robust Lyapunov candidate.

The stability analysis further requires examining the quadratic forms associated with these derivatives. Let $u_i$ and $v_i$ be transformed coordinates such that:
$$S_i = a_{i1}^2 + (a_{i1}b_{i2} + a_{i2}b_{i1})(\beta_i + \gamma_i) + 2b_{i1}^2 = 0$$
If the discriminant of this form is negative, the equilibrium point is isolated and potentially stable. We define the function $g_i(x, y)$ to account for higher-order perturbations:
$$g_i(x, y) = [M(x, y) + Q_i] \omega_0 \dots \omega_{i-1} \omega_{i+1} \dots \omega_s$$
where $M(x, y)$ is a scaling factor. In the neighborhood $S(\Gamma_i, \epsilon)$ of a trajectory $\Gamma_i$, we require that the function $f_i(x, y)$ satisfies $|f_i(x, y)| > \eta_i$ for some $\eta_i > 0$.

For the system to be stable in the sense of Lyapunov, the derivative of the constructed function must satisfy $\frac{d\Phi}{dt} < 0$ within the domain of interest. By analyzing the interaction between the components $\omega_j$ and $\omega_m$ in the sub-regions $A_{jm}$, we establish that:
$$\Phi_{jm} [(\alpha_j a_j + \beta_j b_j) \omega_m + (\alpha_m a_m + \beta_m b_m) \omega_j] = M_{jm} (b_j d_m + b_m d_j)$$
This leads to a system of algebraic inequalities for the coefficients $\chi_{jm}$, ensuring that the energy-like function decreases along the trajectories of the system. These results generalize previous findings in \cite{3, 4} regarding the construction of integral invariants and limit cycles for second-order autonomous systems.

References

1. E. [Journal of Applied Mathematics and Mechanics], Vol. XVII, No. 6, pp. 659–670, 1952.
2. M. [Reports of the Academy of Sciences], Vol. I, No. 1, pp. 52–55, 1963.
3. A. [Differential Equations], No. 4, pp. 3–6, 1963.
4. A. [Mathematical Collection], pp. 12–16, 1965.
5. [Bulletin of Moscow University], No. 1, 1957.