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PublisherThe University of Arizona.
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AbstractThe Lyapunov function method is used in proving stability, asymptotic or globally asymptotic stability of discrete dynamic systems. We show that the slightly relaxed versions of the well known sufficient conditions are also necessary. The stability of the equilibria of time-invariant nonlinear dynamical systems with discrete time scale is investigated. We present an elementary proof showing that in the case of a stable equilibrium and continuously differentiable state transition function, all eigenvalues of the Jacobian computed at the equilibrium must be inside or on the unit circle. We also demonstrate via numerical examples that if some eigenvalues are on the unit circle and all other eigenvalues are inside the unit circle, then the equilibrium maybe unstable, or stable, or even asymptotically stable, which show that the necessary condition cannot be further restricted in general. In addition, the necessary condition is given in terms of spectral radius and matrix norms. The asymptotic stability of equilibria in a number of discrete dynamic oligopolies is analyzed. First the equivalence of the equilibrium problem of a large class of nonlinear games and the equilibrium problem of a class of discrete dynamic systems is verified. Stability conditions are then derived for a certain class of dynamic models, and these results are finally applied to single-product oligopolies, multiproduct oligopolies, and labor-managed oligopolies. The economic interpretation of the stability conditions are also presented. The stability properties of a special class of homogeneous dynamic economic systems are examined. The nonlinearity of the models and the presence of eigenvalues with zero real parts in a normally hyperbolic invariant set make the application of the classical theory impossible. Some principles of the modern theory of dynamical systems and invariant manifolds are applied. The local and global strong attractivity of the set of equilibria is verified under mild conditions. As an application, special labor-managed oligopolies are investigated.
Degree ProgramGraduate College