Smooth normal forms and eliminate nonresonant terms by mean dichotomy spectrum

As the essential instrument for identifying bifurcations and determining qualitative behaviors, the smooth normalizations are provided for nonautonomous systems with linear parts admitting mean hyperbolicity. To be precise, for nonlinear system ẋ=A(t)x+f(t,x), we investigate the theory of smooth normal forms of any finite order through mean dichotomy spectrum. Considering the error function ɛ(t, s), mean hyperbolic systems allow more complex dynamical behaviors, such as the coexistence of hyperbolic and non-hyperbolic phenomena. To overcome the difficulties from weak hyperbolicity, we provide the new criteria to eliminate as many as possible non-resonant terms, and guarantee the bounded and convergent coefficients of resonant polynomials, respectively. Our results present the efficient way of reducing nonlinear systems with mean hyperbolicity, and allow recognition of more intricate dynamics arising in applied sciences.