American Institute of Mathematical Sciences

We discuss the topological equivalence between evolution families with a generalized exponential dichotomy. These can occur for example when all Lyapunov exponents are infinite or all Lyapunov exponents are zero. In particular, we show that any evolution family admitting a generalized exponential dichotomy is topologically equivalent to a certain normal form, in the which the exponential behavior in the stable and unstable directions are multiples of the identity. Moreover, we show that the topological equivalence between two evolution families admitting generalized exponential dichotomies, possibly with different growth rates, can be completely characterized in terms of a new notion of equivalence between these rates.

Higher-dimensional nonlinear and perturbed systems of implicit ordinary differential equations are studied by means of methods of dynamical systems. Namely, the persistence of solutions are studied under nonautonomous perturbations connecting either impasse points with IK-singularities or two impasse points. Important parts of the paper are applications of the theory to concrete perturbed fully nonlinear RLC circuits.

We give necessary integral conditions and sufficient ones for the existence of a general concept of $μ$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $μ$-dichotomies using these Lyapunov functions.

In this paper, we give a criterion on instability of an equilibrium of a nonlinear Caputo fractional differential system. More precisely, we prove that if the spectrum of the linearization has at least one eigenvalue in the sector

\begin{document}$\left\{ \lambda \in \mathbb{C}\setminus \{0\}:|\arg (\lambda )| < \frac{\alpha \pi }{2} \right\},$ \end{document}

where \begin{document}$α∈ (0,1)$\end{document} is the order of the fractional differential system, then the equilibrium of the nonlinear system is unstable.

We consider an $n$ dimensional dynamical system with discontinuous right-hand side (DRHS), whereby the vector field changes discontinuously across a co-dimension 1 hyperplane \begin{document}$S$\end{document}. We assume that this DRHS system has an asymptotically stable periodic orbit \begin{document}$γ$\end{document}, not fully lying in \begin{document}$S$\end{document}. In this paper, we prove that also a regularization of the given system has a unique, asymptotically stable, periodic orbit, converging to \begin{document}$γ$\end{document} as the regularization parameter goes to $0$.

In this paper, we construct an open and dense set in the space of bounded linear random dynamical systems (both discrete and continuous time) equipped with the essential sup norm such that the Lyapunov exponents depend analytically on the coefficients in this set. As a consequence, analyticity for Lyapunov exponents of bounded linear random dynamical systems is a generic property.

We establish a sufficient condition for existence and uniqueness of periodic solutions to partial functional differential equations of the form \begin{document} $\dot{u}=A(t)u+F(t)(u_t)+g(t,u_t)$ \end{document} on a Banach space \begin{document} $X$ \end{document} where the operator-valued functions \begin{document} $t\mapsto A(t)$ \end{document} and \begin{document} $t\mapsto F(t)$ \end{document} are \begin{document} $1$ \end{document}-periodic, the nonlinear operator \begin{document} $g(t,φ)$ \end{document} is \begin{document} $1$ \end{document}-periodic with respect to \begin{document} $t$ \end{document} for each fixed \begin{document} $φ∈ {\mathcal{C}}:=C([-r,0],X)$ \end{document}, and satisfying \begin{document} $\|g(t,φ_1)-g(t,φ_2)\|≤\varphi(t)\|φ_1-φ_2\|_C$ \end{document} for \begin{document} $φ_1, φ_2∈ {\mathcal{C}}$ \end{document} with \begin{document} $\varphi$ \end{document} being a positive function such that \begin{document} $\sup_{t≥0}∈t_{t}^{t+1}\varphi(τ)dτ < ∞$ \end{document}. We then apply the results to study the existence, uniqueness, and conditional stability of periodic solutions to the above equation in the case that the family \begin{document} $(A(t))_{t≥ 0}$ \end{document} generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.

We establish the existence of a stable foliation in the vicinity of a traveling front solution for systems of reaction diffusion equations in one space dimension that arise in the study of chemical reactions models and solid fuel combustion. In this way we complement the orbital stability results from earlier papers by A. Ghazaryan, S. Schecter and Y. Latushkin. The essential spectrum of the differential operator obtained by linearization at the front touches the imaginary axis. In spaces with exponential weights, one can shift the spectrum to the left. We study the nonlinear equation on the intersection of the unweighted and weighted spaces. Small translations of the front form a center unstable manifold. For each small translation we prove the existence of a stable manifold containing the translated front and show that the stable manifolds foliate a small ball centered at the front.

The properties of stability of a compact set \begin{document} $\mathcal{K}$ \end{document} which is positively invariant for a semiflow \begin{document} $(\Omega× {W^{1,\infty }}([-r,0],\mathbb{R}^n),Π,\mathbb{R}^+)$ \end{document} determined by a family of nonautonomous FDEs with state-dependent delay taking values in \begin{document} $[0,r]$ \end{document} are analyzed. The solutions of the variational equation through the orbits of \begin{document} $\mathcal{K}$ \end{document} induce linear skew-product semiflows on the bundles \begin{document} $\mathcal{K}×{W^{1,\infty }}([-r,0],\mathbb{R}^n)$ \end{document} and \begin{document} $\mathcal{K}× C([-r,0],\mathbb{R}^n)$ \end{document}. The coincidence of the upper-Lyapunov exponents for both semiflows is checked, and it is a fundamental tool to prove that the strictly negative character of this upper-Lyapunov exponent is equivalent to the exponential stability of \begin{document} $\mathcal{K}$ \end{document} in \begin{document} $\Omega×{W^{1,\infty }}([-r,0],\mathbb{R}^n)$ \end{document} and also to the exponential stability of this compact set when the supremum norm is taken in \begin{document} ${W^{1,\infty }}([-r,0],\mathbb{R}^n)$ \end{document}. In particular, the existence of a uniformly exponentially stable solution of a uniformly almost periodic FDE ensures the existence of exponentially stable almost periodic solutions.

The aim of this paper is to present a new and very general method for the study of the uniform exponential trichotomy of nonautonomous dynamical systems defined on the whole axis. We consider a discrete dynamical system and we introduce the property of \begin{document} $(r, p)$ \end{document}-admissibility relative to an associated control system, where \begin{document} $r, p∈ [1, ∞]$ \end{document}. In several constructive steps, we obtain full descriptions of the sufficient conditions and respectively of the necessary criteria for uniform exponential trichotomy based on the \begin{document} $(r, p)$ \end{document}-admissibility of the pair \begin{document} $(\ell^∞(\mathbb{Z}, X), \ell^1(\mathbb{Z}, X))$ \end{document}. In the same time, we provide a complete diagram of the \begin{document} $\ell^p$ \end{document}-spaces which can be considered in the admissible pairs for the study of the uniform exponential trichotomy of discrete dynamical systems. We present illustrative examples in order to motivate the hypotheses and the generality of our method. Finally, we apply the main results to obtain new criteria for uniform exponential trichotomy of dynamical systems modeled by evolution families using the admissibility of various pairs of \begin{document} $\ell^p$ \end{document}-spaces.

In this paper we consider the existence of monotone traveling waves for a class of general integral difference model for populations that allows the dispersal probability to have no continuous density functions but the fecundity functions to generate a monotone dynamical systems. In this setting we deal with the non-compactness of the evolution operator by using the monotone iteration method.