On the boundedness and global asymptotic stability of the Liénard equation with restoring terms (Q2767719)

The author studies the solution properties of the nonautonomous Liénard equation. Here, the author considers a more general form of this two-dimensional ordinary differential equation (ODE). Indeed, if the damping term \( F(x)\) is classical (a nonlinear increasing function of the dependant variable \(x\)), the restoring force is the product of a nonlinear increasing function \(g(x)\) and a time function \(a(t)\). Moreover, the system described by this equation is submitted to an external force \(p(t)\). The text purpose is the establishment of sufficient conditions for which the solutions are on one hand bounded, on the other hand asymptotically and globally stable. NEWLINENEWLINENEWLINEWith respect to the previous results of other authors, the study is made without using the second Lyapunov method. Here, the paper introduces a function \(A(n)\) of the lower bound \(n\) of the Lipschitz term related to \(g(x)\), and a second one \(B(N)\) of the upper bound \(N\) of this term, for having a measure of the behavior difference with respect to the solution to the two-dimensional linear nonautonomous ODE without external force. NEWLINENEWLINENEWLINEThe first theorem gives the conditions for having a bounded solution from an initial condition in a bounded region of the phase plane. The second theorem deals with the uniform asymptotic stability, the uniform stability, and the instability of the ``associated'' two-dimensional linear nonautonomous ODE without external force. Theorems 3 and 4 give a sufficient condition for having the solution to the Liénard equation to be globally asymptotical stable. The fifth and last theorem is related to the convergence of all the solutions toward a unique periodic solution. Each of these results is accompanied by observations showing a generalization of previous theorems.