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Abstract.A new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with computer-assisted computations. As an application of these methods it is proven that for an explicit parameter value the Lorenz equations exhibit chaotic dynamics. IntroductionThe purpose of this note is to briefly describe a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems. The technique combines abstract existence results based on topological invariants (the Conley index) with finite, computer-assisted computations necessary to verify the assumptions of the theorems in a concrete example. There are at least three new aspects of this technique. It applies to concrete differential equations; it can provide a relatively strong description of global dynamics (in terms of semiconjugacies); and the necessary computer-assisted computations are small enough to be performed on currently available computers.To focus the presentation of the ideas, an outline of a proof that the Lorenz equations,

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