Smooth Invariant Manifolds for Differential Equations with Infinite Delay
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Smooth Invariant Manifolds for Differential Equations with Infinite Delay Lokesh Singh1
· Dhirendra Bahuguna1
Received: 11 July 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this article, we give the existence of a smooth stable manifold which is invariant under the semiflows of the delay differential equation x = Ax(t) + Lxt + f (t, xt , λ), with the assumption that the corresponding linear differential equation admits a nonuniform exponential dichotomy and the perturbation f (t, xt , λ) is small and smooth enough. We also show that the obtained manifold is Lipschitz in the parameter λ. Keywords Functional differential equation · Nonuniform exponential dichotomy · Invariant manifold theory · Parameter dependence Mathematics Subject Classification (2010) Primary 34D09 · 34K19 · 34K20 · 37D10 · 37D25
1 Introduction The theory of invariant manifolds in the field of dynamical systems can be mapped to the works of Hadamard [8], Perron [15], Lyapunov [13], and Hirsch, Pugh, and Shub [12], also in recent time L. Barreira and C. Valls [4] and others. Due to the significant works in the decade of 1970s, invariant manifold theory blossomed in this period. Physical models described by the dynamical systems were leading to practical results. Therefore, the invariant manifold theory became an essential tool for the applied mathematicians, physicists, engineers, and also biologists as it simplifies the analysis of complex systems by considerably reducing the relevant dimensions of the space. There are two main approaches to obtain the invariant manifolds for dynamical systems. One is developed by Hadamard in 1901 [8] which is geometrical; according to that, the graphs in linearized stable and unstable spaces are considered the stable and unstable
Lokesh Singh
[email protected] Dhirendra Bahuguna [email protected] 1
Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208016, India
Lokesh Singh and Dhirendra Bahuguna
manifolds, respectively. While the other method is developed by Perron [15] and Lyapunov [13], which is functional analytic and the invariant manifolds are obtained as fixed-point functions of integral operators associated with differential equations on some appropriate spaces. Perron ensured the existence of Lipschitz manifolds, which are invariant under the semiflows generated by linear differential equations with the assumption that the solution semigroup satisfies uniform exponential dichotomy (defined in Section 2). Subsequently, Pesin in 1976 [16] established the smoothness of those invariant manifolds for uniformly hyperbolic trajectories. Note that, the uniform exponential dichotomy is too restrictive on nonautonomous systems. For instance, when we assume that the solution operator satisfies uniform exponential dichotomy, we neglect the possibility of the dependence of the norm of the solution on the initial time. But this dependency is quite natural as almost all the solutions of smooth dynamical systems with nonzero Lyapunov
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