In a joint work with Palmer we have formulated sufficient conditions under which there exist continuous and invertible transformations of the form $H_n(x,y)$ taking solutions of a coupled system \begin{equation*} x_{n+1} =A_nx_n+f_n(x_n, y_n), \quad y_{n+1}=g_n( y_n), \end{equation*} onto the solutions of the associated partially linearized uncoupled system \begin{equation*} x_{n+1} =A_nx_n, \quad y_{n+1}=g_n( y_n). \end{equation*} In the present work we go one step further and provide conditions under which $H_n$ and $H_n^{-1}$ are smooth in one of the variables $x$ and $y$.
We emphasise that our conditions are of a general form and do not involve any kind of dichotomy, nonresonance or spectral gap assumptions for the linear part which are present on most of the related works.
We prove that smoothness of nonautonomous linearization is of class $C^2.$ Our approach admits the existence of stable and unstable manifolds determined by a family of nonautonomous hyperbolicities. Moreover, our goal is reached without spectral conditions.
The main purpose of this paper is to formulate new conditions for smooth linearization of nonautonomous systems with discrete and continuous time. Our results assume that the linear part admits a nonuniform polynomial dichotomy and that the associated polynomial dichotomy spectrum exhibits appropriate spectral gap and spectral band conditions. This is in sharp contrast to most of the previous results in the literature which assumed that the linear part admits an exponential dichotomy. Our techniques exploit the relationship between polynomial and exponential dichotomies via a suitable reparame…
In this paper, we prove the partial linearization for n-dimensional nonautonomous differential equations. The conditions are formulated in terms of the dichotomy spectrum.
In this paper we give a smooth linearization theorem for nonautonomous difference equations with a nonuniform strong exponential dichotomy. The linear part of such a nonautonomous difference equation is defined by a sequence of invertible linear operators on $\mathbb{R}^d$. Reducing the linear part to a bounded linear operator on a Banach space, we discuss the spectrum and its spectral gaps. Then we obtain a gap condition for $C^1$ linearization of such a nonautonomous difference equation. We finally extend the result to the infinite dimensional case. Our theorems improve known results even in…
In this paper we give a smooth linearization theorem for nonautonomous differential equations with a nonuniform strong exponential dichotomy. In terms of discretized evolution operator with hyperbolic fixed point 0, we formulate its spectrum and then give a spectral bound condition for the linearization of such equations to be simultaneously differentiable at 0 and Hölder continuous near 0. Restricted in the autonomous case, our result is the first one that gives a rigorous proof for simultaneously differentiable and Hölder linearization of hyperbolic systems without any non-resonant condition…
We consider a nonautonomous system \[ \dot x=A(t)x+f(t,x,y),\quad \dot y = g(t,y)\] and give conditions under which there is a transformation of the form $H(t,x,y)=(x+h(t,x,y),y)$ taking its solutions onto the solutions of the partially linearized system \[ \dot x=A(t)x,\quad \dot y = g(t,y).\] Shi and Xiong \cite{SX} proved a special case where $g(t,y)$ was a linear function of $y$ and $\dot x=A(t)x$ had an exponential dichotomy. Our assumptions on $A$ and $f$ are of the general form considered by Reinfelds and Steinberga \cite{RS}, which include many of the generalizations of Palmer's theore…
The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg-Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector fields.
We establish a linearization criterion for skew products of contractions in any dimension. We prove their smooth or holomorphic parameter dependence. In the smooth setting, we use the language of tame Fréchet spaces. We apply our result to the linearization of totally projectively expanding Cantor sets.
We establish a general version of the Siegel-Sternberg linearization theorem for ultradiffentiable maps which includes the analytic case, the smooth case and the Gevrey case. It may regarded as a small divisior theorem without small divisor conditions. Along the way we give an exact characterization of those classes of ultradifferentiable maps which are closed under composition, and reprove regularity results for solutions of ode's and pde's. This will open up new directions in \textsc{kam}-theory and other applications of ultradifferentiable functions.
We study how the "full hierarchy" of invariant manifolds for nonautonomous integrodifference equations on the Banach spaces of $p$-integrable functions behaves under spatial discretization of Galerkin type. These manifolds include the classical stable, center-stable, center, center-unstable and unstable ones, and can be represented as graphs of $C^m$-functions. For kernels with a smoothing property, our main result establishes closeness of these graphs in the $C^{m-1}$-topology under numerical discretizations preserving the convergence order of the method. It is formulated in a quantitative fa…
The averaging method combined with the Lyapunov-Schmidt reduction provides sufficient conditions for the existence of periodic solutions of the following class of perturbative $T$-periodic nonautonomous differential equations $x'=F_0(t,x)+\varepsilon F(t,x,\varepsilon)$. Such periodic solutions bifurcate from a manifold $\mathcal{Z}$ of periodic solutions of the unperturbed system $x'=F_0(t,x)$. Determining the stability of this kind of periodic solutions involves the computation of eigenvalues of matrix-valued functions $M(\varepsilon)$, which can done using the theory of $k$-hyperbolic matri…
In this paper we contribute to qualitative and geometric analysis of planar piecewise smooth vector fields, which consist of two smooth vector fields separated by the straight line $y=0$ and sharing the origin as a non-degenerate equilibrium. In the sense of $Σ$-equivalence, we provide a sufficient condition for linearization and give phase portraits and normal forms for these linearizable vector fields. This condition is hard to be weakened because there exist vector fields which are not linearizable when this condition is not satisfied. Regarding perturbations, a necessary and sufficient con…
In this paper we adapt the method of [P. H. Baptistelli, M. Manoel and I. O. Zeli. Normal form theory for reversible equivariant vector fields. Bull. Braz. Math. Soc., New Series 47 (2016), no. 3, 935-954] to obtain normal forms of a class of smooth bireversible vector fields. These are vector fields reversible under the action of two linear involution and whose linearization has a nilpotent part and a semisimple part with purely imaginary eigenvalues. We show that these can be put formally in normal form preserving the reversing symmetries and their linearization. The approach we use is based…
In the present article, we discuss some aspects of the local stability analysis for a class of abstract functional differential equations. This is done under smoothness assumptions which are often satisfied in the presence of a state-dependent delay. Apart from recapitulating the two classical principles of linearized stability and instability, we deduce the analogon of the Pliss reduction principle for the class of differential equations under consideration. This reduction principle enables to determine the local stability properties of a solution in the situation where the linearization does…
In connection with the optimization problem $$\inf_{x\in argmin Ψ}\{Φ(x)+Θ(x)\},$$ where $Φ$ is a proper, convex and lower semicontinuous function and $Θ$ and $Ψ$ are convex and smooth functions defined on a real Hilbert space, we investigate the asymptotic behavior of the trajectories of the nonautonomous Levenberg-Marquardt dynamical system \begin{equation*}\left\{ \begin{array}{ll} v(t)\in\partialΦ(x(t))\\ λ(t)\dot x(t) + \dot v(t) + v(t) + \nabla Θ(x(t))+β(t)\nabla Ψ(x(t))=0, \end{array}\right.\end{equation*} where $λ$ and $β$ are functions of time controlling the velocity and the penalty …
