A generalization of Siegmund’s normal forms theorem to systems with µ-dichotomies
Abstract: In [2], S. Siegmund made a significant breakthrough by extending Poincaré’s normal forms result. Siegmund’s approach involved eliminating nonresonant Taylor terms up to a certain order, and he introduced the spectrum of the exponential dichotomy as a substitute for the classical notion of eigenvalues, thereby defining the nonresonance con-edition.
In this work [1], we establish a generalized normal forms theorem by exploring the recently introduced concept of µ-dichotomy [3]. We reframe the nonresonance condition in terms of this new spectrum.
To achieve this, we introduce several novel concepts, including the notion of eligibility of perturbations and trumpet neighborhoods. With these developments, we are able to eliminate nonresonant terms in more generalized settings, encompassing cases such as unbounded perturbations and linear systems that do not exhibit exponential dichotomy.
Joint work with: Álvaro Castañeda
References
[1] Castañeda. Á; Jara, N. A generalization of Siegmund’s normal forms theorem to systems with µ-dichotomies. J. Differential Equations 410, 449-480 (2024).
[2] Siegmund, S. Normal forms for nonautonomous differential equations. J. Differential Equations 178, 541–573 (2002).
[3] Silva, C. M. Nonuniform µ-dichotomy spectrum and kinematic similarity. J. Differential Equations, 375, 618-652 (2023).