Qualitative study of autonomous systems of difference equations

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This dissertation explores several classes of nonlinear difference equations and nonlinear fuzzy difference equations, with a primary emphasis on the qualitative properties of their solutions. The work begins with a comprehensive study of nonlinear higher-order fuzzy difference equations. Under suitable initial conditions, we prove the existence and uniqueness of positive fuzzy solutions. In addition, we establish conditions ensuring their boundedness, persistence, and convergence to a single equilibrium point. The convergence rate is analyzed, and numerical simulations are provided to support and validate the theoretical results. The concluding part addresses a three-dimensional nonlinear system of difference equations involving arbitrary constants and non-identical parameters. By performing a linearization around its equilibrium, we investigate the local dynamics and derive sufficient criteria for asymptotic stability. Moreover, it is shown that the solutions may exhibit oscillatory, bounded, or persistent behaviors depending on the parameter choices and initial conditions.

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