Special Issue DCDS (Series B)



Together with Ken Palmer we coedited a special issue of Discrete and Continuous Dynamical Systems (Series B) devoted to

Nonautonomous Hyperbolicity and Related Concepts

The main goal of the special issue is be to publish original contributions developing different aspects, from the theoretical, applied and computational points of view, of “nonautonomous hyperbolicity” together with discussion of related areas of nonautonomous dynamics (bifurcations, Lyapunov exponents, spectra, etc.), with particular emphasis on systems of ordinary differential equations and difference equations.

The notion of hyperbolicity is a central and well-established concept in the area of dynamical systems, both in bifurcation and chaos theory. However, recent decades saw an increasing interest in models subject to temporally fluctuating perturbations and thus the field of nonautonomous dynamical systems. Here, various notions of hyperbolicity (or exponential dichotomy) are in use covering aspects such as different spectra (Sacker-Sell, Morse, Lyapunov, exponential and integral separation, etc.), finite time dynamics, and diverse approaches to nonautonomous bifurcation. Even in autonomous problems such time-dependent problems also arise  when linearizing along nonconstant solutions.

In this special issue we collect 11 original contributions by 29 authors on this topic:

  1. Observing that solutions to time-dependent equations need not grow exponentially, Barriera, Popescu and Valls study generalized exponential dichotomies and characterize topological equivalence. This leads to a certain normal form whose stable and unstable components are positive or negative multiples of the identity.

  1. The contribution by Battelli and Fečkan deals with a fully nonlinear perturbed RLC circuit, which is described by a quaslinear implicit differential equation. It is shown that solutions connecting so-called IK singularities persist. The exponential dichotomy properties of a certain linearized equation play an important role in the proof.

  1. Also Bento, Lupa, Megan and Silva study nonexponential growth rates, which include nonuniform exponential rates and polynomial growth rates. They provide necessary, as well as sufficient conditions for a related dichotomy concept. These conditions rely on a particular integrability property of the Green's function, or on Lyapunov functions.

  1. The paper by Cong, Doan, Siegmund and Tuan provides an instability result for Caputo fractional differential equations based on linearization: If an eigenvalue of the linearization is contained in a sector given by the order of the fractional derivative, then the equilibrium is unstable.

  1. Dieci, Elia and Pi investigate systems of ordinary differential equations, whose right-hand side is discontinous across a codimension 1 hyperplane. It is shown that an asymptotically stable periodic orbit ɣ of the discontinuous system persists under regularization and converges to ɣ as the regularization parameter becomes arbitrarily small.

  1. Doan constructs an open and dense subset of the space of linear random dynamical systems on which Lyapunov exponents depend analytically on coefficients. This establishes genericity of analytic Lyapunov exponents.

  1. Huy and Dang provide sufficient conditions for the existence and uniqueness of periodic solutions to partial functional differential equations with periodic right-hand sides. Also a local stable manifold near this periodic solution is constructed. A key hypothesis is that the linear part of the equation has an exponential dichotomy.

  1. The existence of a stable foliation near a travelling front for systems of reaction diffusion equations in one spatial dimension is shown by Latushkin, Schnaubelt and Yang. The Lyapunov-Perron method, often associated with exponential dichotomy, plays an important role here.

  1. Nonautonomous functional differential equations with state-dependent delay are addressed in the paper by Maroto, Núñez and Obaya. In particular, based on upper Lyapunov exponents, it is proved that the existence of an exponentially stable solution implies the existence of such almost periodic solutions (provided the equation is uniformly almost periodic).

  1. Based on a new admissibility concept in sequence spaces, Sasu and Sasu establish necessary and sufficient conditions for nonuniform exponential trichotomy (on the full line) of nonautonomous difference equations in Banach spaces.

  1. Finally, Le and Minh apply a monotone iteration method to prove the existence of monotone travelling waves for an integral difference equation with an application in population dynamics.

Christian Pötzsche (Dec 2017)