Abstract
We shape the results on the formal Gevrey normalization. More precisely, we investigate the better expression of \({{\hat{\alpha }}}\), which makes the formal Gevrey-\({{\hat{\alpha }}}\) coordinates substitution turning the Gevrey-\(\alpha \) smooth vector fields X into their normal forms in several cases. Such results show that the ‘loss’ of the Gevrey smoothness is not always necessary even under Siegel type small divisor conditions, which are different from others.
Highlights
The study of normal form theory has a long history, which is original from Poincaré
The celebrated Poincaré-Dulac scheme ensures the existence of formal normal forms
The convergence of formal normal forms plays the central role of the whole research
Summary
The study of normal form theory has a long history, which is original from Poincaré. The theory has extended its domain over various systems such as random dynamical systems, control systems and so on. It does great importance to bifurcations, stability theory and others. The celebrated Poincaré-Dulac scheme ensures the existence of formal normal forms. The convergence of formal normal forms plays the central role of the whole research. In Poincaré domain the system analytically conjugates to its polynomial normal forms. In Siegel domain the system can be analytically linearized under some small divisor conditions. By the dichotomy method or the result of Yoccoz there exists a large gap between formal and analytic normal forms. In the rougher topology Hartman, Sternberg and Chen proved C0, Ck and C∞ conjugacy under the hyper-
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