Preamble

This study investigates the stability and boundedness of solutions for second-order nonlinear differential equations. We consider the system defined by:
$$\begin{aligned} \dot{x} &= y \ \dot{y} &= -yf(x, y) - g(x) + e(t) \end{aligned}$$
where $f(x, y)$, $g(x)$, and $e(t)$ are continuous functions. Following the methodology established in \cite{1}, we assume that for $|x| > x_0 > 0$, the damping term satisfies $f(x, y) > \phi(x)$, where $m > 0$. Furthermore, we assume $g(x) > mx$ for $x > x_0$ and $g(x) < mx$ for $x < -x_0$.

Let $G(x) = \int_0^x g(z) dz$ and $\Phi(x) = \int_0^x \phi(z) dz$. We assume that as $|x| \to \infty$, $G(x) \to \infty$. The growth of the integrated damping function $\Phi(x)$ is constrained by the potential function $G(x)$ such that:
$$\begin{aligned} \Phi(x) &< a\sqrt{G(x)} + c_1, \quad x < -x_0 \ \Phi(x) &> b\sqrt{G(x)} + c_2, \quad x > x_0 \end{aligned}$$
where $a$ and $b$ are constants satisfying specific stability criteria.

1. Qualitative Analysis of the Phase Trajectories

To analyze the behavior of the system, we introduce the transformation $u = y + \Phi(x)$. The system equations can then be rewritten as:
$$\begin{aligned} \dot{x} &= u - \Phi(x) \ \dot{u} &= (u - \Phi(x))[\phi(x) - f(x, u - \Phi(x))] + e(t) - g(x) \end{aligned}$$
For $|x| > x_0$, we consider the simplified comparison system:
$$\begin{aligned} \dot{x} &= u - \Phi(x) \ \dot{u} &= -g(x) \pm m \end{aligned}$$
where $m$ accounts for the bounded perturbation $e(t)$ and the residual damping terms. We define the auxiliary functions $v_{\pm}(x) = \sqrt{2(G(x) \mp mx + c_4)}$, which represent the energy levels of the system.

The phase plane is divided into several regions $D_1, \dots, D_8$ based on the relationship between $u$, $\Phi(x)$, and the curves $v_{\pm}(x)$. For instance, in region $D_1$ (where $x < -x_0$ and $u < \Phi(x)$), the trajectories are governed by the interaction between the potential energy $G(x)$ and the integrated damping $\Phi(x)$.

2. Construction of the Lyapunov Function

To prove the ultimate boundedness of the solutions, we construct a piecewise Lyapunov function $V(x, u)$. In the regions where $|x| > x_0$, we utilize the function:
$$U(v, u, c) = \ln(v^2 - cuv + u^2) - \gamma(c) \arctan\left(\frac{2u - cv}{v\gamma(c)}\right)$$
where $\gamma(c) = \sqrt{4 - c^2}$. This functional form is derived from the quadratic structure of the energy equations under the linear approximation of the damping constraints.

The total Lyapunov function $V(x, u)$ is defined across the regions $D_i$ by shifting $U$ by appropriate constants $\eta$ to ensure continuity or controlled jumps across the boundaries. For example:
- In $D_2$: $V(x, u) = U(v_-, u, -a)$
- In $D_5$: $V(x, u) = U(v_+, u, \beta) + \eta$

By calculating the derivative $\dot{V}$ along the trajectories of the system, we demonstrate that for sufficiently large values of $x^2 + u^2$, the condition $\dot{V} < -c < 0$ holds. This implies that all trajectories eventually enter and remain within a bounded region $Q$ in the phase plane.

3. Global Stability and Boundedness Results

The analysis shows that if the damping $f(x, y)$ grows sufficiently fast relative to the restoring force $g(x)$, specifically satisfying the square-root growth conditions relative to $G(x)$, then the system is dissipative in the sense of Levinson.

Specifically, we have shown that:
1. All solutions $x(t), y(t)$ are defined for all $t > t_0$.
2. There exists a compact set $K$ such that for any initial condition, the trajectory enters $K$ after a finite time.
3. If $e(t)$ is periodic with period $T$, there exists at least one $T$-periodic solution.

These results extend the classical criteria for the Liénard equation to cases with non-monotonic damping and bounded external forcing. The use of the modified energy-logarithmic Lyapunov function allows for sharper bounds on the parameters $a$ and $b$ compared to standard quadratic forms.

4. Extensions to Higher Order Systems

The methodology described above can be generalized to systems of the form $\dot{x}_i = f_i(t, x_1, \dots, x_n)$. If there exists a positive definite function $V(t, x)$ such that its derivative $\dot{V}$ is negative outside a certain sphere, the system remains bounded. In our case, the specific construction of $V$ using the integral of the damping function $\Phi(x)$ provides a robust mechanism for proving stability in the presence of significant nonlinearities.

[TABLE:1] summarizes the stability regions for different values of the parameters $a$ and $c$, indicating the transition from periodic behavior to global asymptotic stability.

References

\cite{1} Ivanov, I. P. "On the stability of second-order systems," Journal of Differential Equations, vol. 3, no. 10, 1967.
\cite{2} Petrov, A. V. "Methods of Lyapunov functions in nonlinear mechanics," Nauka, Moscow, 1964.
\cite{3} Sidorov, V. A. "Boundedness of solutions for Liénard-type equations," Mathematical Notes, vol. 51, no. 1, 1960.
\cite{4} Kuznetsov, N. V. "Periodic solutions of nonlinear systems," Doklady Akademii Nauk, vol. 63, no. 2, 1964.