In dynamical systems theory, the Olech theorem establishes sufficient conditions for global asymptotic stability of a two-equation system of non-linear differential equations. The result was established by Czesław Olech in 1963, based on joint work with Philip Hartman.
Theorem The differential equations , , where , for which is an equilibrium point, is uniformly globally asymptotically stable if: (a) the trace of the Jacobian matrix is negative, for all , (b) the Jacobian determinant is positive, for all , and (c) the system is coupled everywhere with either
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