Asymptotic stability for quasi-linear systems whose linear approximation is not assumed to be uniformly attractive

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Sufficient conditions are obtained for uniform stability and asymptotic stability of the zero solution of two-dimensional quasi-linear systems under the assumption that the zero solution of linear approximation is not always uniformly attractive. A class of quasi-linear systems considered in this paper includes a planar system equivalent to the damped pendulum x′′ + h(t)x′ + sin x = 0, where h(t) is permitted to change sign. Some suitable examples are included to illustrate the main results.