Differential systems under small perturbations

Abstract

The stability, changes and mobility bounds of generalized Lyapunov exponents of linear homogeneous systems of differential equations with continuous and tending to zero coefficients under small perturbations was studied with respect to the generalized central and generalized singular exponents. An example of the instability of generalized Lyapunov exponents, under small perturbations tends to zero was given. Using the generalized upper central exponents the exact upper limit of change of generalized Lyapunov exponents of linear systems under small perturbations was determined in a certain class of nonlinear systems of differential equations. Using the generalized lower central exponents the exact lower bounds of change of generalized Lyapunov exponents of linear systems under small perturbations was determined in a certain class of nonlinear systems of differential equations. Using a special upper generalized exponent the upper bounds of change of generalized Lyapunov exponents of linear systems under small perturbations was determined in a certain class of nonlinear systems of differential equations. Using the generalized lower special exponent the lower bounds of changes of generalized Lyapunov exponents of linear systems under small perturbations was determined in a certain class of nonlinear systems of differential equations. Mutual relations between generalized upper and lower central exponents, upper and lower generalized singular exponents was established. A brief description of the generalized Lyapunov exponents of the generalized upper and lower central exponents, generalized upper and lower singular exponents as Baire functions in a certain metric space was given.