Smooth Normal Form And Strenberg Theorem Of Almost Periodic Difference Systems

This thesis mainly studies the Sternberg theorem of almost periodic difference systems,based on the classical normal form theory.It is mainly divided into two parts: the first part studies the smooth conjugate theorem of Taylor expansion and Sternberg theorem of almost periodic difference systems.The second part discusses the smooth conjugate theorem of Taylor expansion and Sternberg theorem of classical almost periodic difference systems.In the first part,we study the smooth conjugate theorem of Taylor expansion and Sternberg theorem of almost periodic difference systems.Based on the hyperbolic conditions of averaged system corresponding to the studied almost periodic difference system,then using invariant manifold theory and the homotopy method,we can find the formal solution satisfying the homotopy equation.Combined with Gronwall inequality,we can obtain the smooth conjugate theorem of Taylor expansion of almost periodic difference system.Furthermore,based on the non-real resonance condition satisfied by the real part of the eigenvalue of the averaged system,the Sternberg theorem of almost periodic difference systems is obtained by using the homotopy method on the basis of the formal normal form.The second part,we discuss the smooth conjugate theorem of Taylor expansion and Sternberg theorem of classical almost periodic difference systems.We prove the theorem through four steps: directness,decomposition,globalization and the application of homotopy method.Combined with the classical normal form theory,we obtain the smooth conjugate theorem and Sternberg theorem of the classical almost periodic difference system.Finally,taking the two dimensional classical almost periodic difference system as an example,the corresponding smooth normal form is studied.