Center manifolds for impulsive equations under nonuniform hyperbolicity. (English) Zbl 1215.34049
The authors establish the existence of smooth center manifolds under sufficiently small perturbations of an impulsive linear equation. They prove the center manifold theorem by using the concept of “nonuniform exponential trichotomy” defined in the paper.
Reviewer: Gani T. Stamov (Sliven)
MSC:
| 34C45 | Invariant manifolds for ordinary differential equations |
| 34D09 | Dichotomy, trichotomy of solutions to ordinary differential equations |
| 34D10 | Perturbations of ordinary differential equations |
| 34A37 | Ordinary differential equations with impulses |
References:
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| [2] | Samoilenko, A.; Perestyuk, N., (Impulsive Differential Equations. Impulsive Differential Equations, Nonlinear Science Series A: Monographs and Treatises, vol. 14 (1995), World Scientific) · Zbl 0837.34003 |
| [3] | Barreira, L.; Valls, C., (Stability of Nonautonomous Differential Equations. Stability of Nonautonomous Differential Equations, Lect. Notes. in Math., vol. 1926 (2008), Springer) · Zbl 1152.34003 |
| [4] | Elbialy, M., On sequences of \(C_b^{k, \delta}\) maps which converge in the uniform \(C^0\)-norm, Proc. Amer. Math. Soc., 128, 3285-3290 (2000) · Zbl 0947.37016 |
| [5] | Henry, D., (Geometric Theory of Semilinear Parabolic Equations. Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math., vol. 840 (1981), Springer) · Zbl 0456.35001 |
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