Relative asymptotic equivalence of dynamic equations on time scales

This paper aims to study the relative equivalence of the solutions of the following dynamic equations $y^{\Delta }(t)=A(t)y(t)$ y Δ ( t ) = A ( t ) y ( t ) and $x^{\Delta }(t)=A(t)x(t)+f(t,x(t))$ x Δ ( t ) = A ( t ) x ( t ) + f ( t , x ( t ) ) in the sense that if $y(t)$ y ( t ) is a given solution of the unperturbed system, we provide sufficient conditions to prove that there exists a family of solutions $x(t)$ x ( t ) for the perturbed system such that $\Vert y(t)-x(t) \Vert =o( \Vert y(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ y ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ , and conversely, given a solution $x(t)$ x ( t ) of the perturbed system, we give sufficient conditions for the existence of a family of solutions $y(t)$ y ( t ) for the unperturbed system, and such that $\Vert y(t)-x(t) \Vert =o( \Vert x(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ x ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ ; and in doing so, we have to extend Rodrigues inequality, the Lyapunov exponents, and the polynomial exponential trichotomy on time scales.

Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach

<p style='text-indent:20px;'>We consider linear stability of steady states of 1<inline-formula><tex-math id="M1">\begin{document}$ \frac{1}{2} $\end{document}</tex-math></inline-formula> and 3DVlasov-Maxwell systems for collisionless plasmas. The linearized systems canbe written as separable Hamiltonian systems with constraints. By using ageneral theory for separable Hamiltonian systems, we recover the sharp linearstability criteria obtained previously by different approaches. Moreover, weobtain the exponential trichotomy estimates for the linearized Vlasov-Maxwellsystems in both relativistic and nonrelativistic cases.</p>

Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs

Consider a general linear Hamiltonian system ∂ t u = J L u \partial _{t}u=JLu in a Hilbert space X X . We assume that L : X → X ∗ \ L:X\rightarrow X^{\ast } induces a bounded and symmetric bi-linear form ⟨ L ⋅ , ⋅ ⟩ \left \langle L\cdot ,\cdot \right \rangle on X X , which has only finitely many negative dimensions n − ( L ) n^{-}(L) . There is no restriction on the anti-self-dual operator J : X ∗ ⊃ D ( J ) → X J:X^{\ast }\supset D(J)\rightarrow X . We first obtain a structural decomposition of X X into the direct sum of several closed subspaces so that L L is blockwise diagonalized and J L JL is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of e t J L e^{tJL} . In particular, e t J L e^{tJL} has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relate n − ( L ) n^{-}\left ( L\right ) and the dimensions of generalized eigenspaces of eigenvalues of J L \ JL , some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostly J J was assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schrödinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states.

Shadowing for infinite dimensional dynamics and exponential trichotomies

Let $(A_m)_{m \in {\mathop Z}}$ be a sequence of bounded linear maps acting on an arbitrary Banach space X and admitting an exponential trichotomy and let $f_m:X \to X$ be a Lispchitz map for every $m\in {\mathop Z} $ . We prove that whenever the Lipschitz constants of $f_m$ , $m \in {\mathop Z} $ , are uniformly small, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$ , $m\in {\mathop Z} $ , has various types of shadowing. Moreover, if X is finite dimensional and each $A_m$ is invertible we prove that a converse result is also true. Furthermore, we get similar results for one-sided and continuous time dynamics. As applications of our results, we study the Hyers–Ulam stability for certain difference equations and we obtain a very general version of the Grobman–Hartman's theorem for nonautonomous dynamics.

Characterizations of Generalized Exponential Trichotomies for Linear Discrete-time Systems

The concept of generalized exponential trichotomy for linear time-varying systems is investigated in relationship with the classical notion of uniform exponential trichotomy. Some key properties of generalized exponential trichotomy are explored through supplementary projections. These results are also extended to the case of projection sequences, while certain applications for adjoint systems are suggested.

On uniform exponential trisplitting for cocycles of linear operators in Banach spaces

The aim of this paper is to study the concept of uniform exponential trisplitting for skew-product semiflow in Banach spaces. This concept is a generalisation of the well-known concept of uniform exponential trichotomy. We obtain necessary and sufficient conditions for this concept of Datko’s type. a character-isation in terms of Lyapunov functions is provided. The results are obtained from the point of view of the projector families, i.e. invariant and strongly invariant.

The Equivalence of Datko and Lyapunov Properties for (h,k)-Trichotomic Linear Discrete-Time Systems

The aim of this paper is to characterize a general property of(h,k)-trichotomy through some Lyapunov functions for linear discrete-time systems in infinite dimensional spaces. Also, we apply the results to illustrate necessary and sufficient conditions for nonuniform exponential trichotomy and nonuniform polynomial trichotomy.