Lyapunov functions for linear nonautonomous dynamical equations on time scales

The existence of a Lyapunov function is established following a method of Yoshizawa for the uniform exponential asymptotic stability of the zero solution of a nonautonomous linear dynamical equation on a time scale with uniformly bounded graininess

Rate-induced transitions for parameter shift systems

Rate-induced transitions have recently emerged as an identifiable type of instability of attractors in nonautonomous dynamical systems. In most studies so far, these attractors can be associated with equilibria of an autonomous limiting system, but this is not necessarily the case. For a specific class of systems with a parameter shift between two autonomous systems, we consider how the breakdown of the quasistatic approximation for attractors can lead to rate-induced transitions, where nonautonomous instability can be characterised in terms of a critical rate of the parameter shift. We find a number of new phenomena for non-equilibrium attractors: weak tracking where the pullback attractor of the system limits to a proper subset of the attractor of the future limit system, partial tipping where certain phases of the pullback attractor tip and others track the quasistatic attractor, em invisible tipping where the critical rate of partial tipping is isolated and separates two parameter regions where the system exhibits end-point tracking. For a model parameter shift system with periodic attractors, we characterise thresholds of rate-induced tipping to partial and total tipping. We show these thresholds can be found in terms of certain periodic-to-periodic and periodic-to-equilibrium connections that we determine using Lin's method for an augmented system. Considering weak tracking for a nonautonomous Rossler system, we show that there are infinitely many critical rates at which a pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor

A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions

In this survey paper we review several aspects related to Navier-Stokes models when some hereditary characteristics (constant, distributed or variable delay, memory, etc) appear in the formulation. First some results concerning existence and/or uniqueness of solutions are established. Next the local stability analysis of steady-state solutions is studied by using the theory of Lyapunov functions, the Razumikhin-Lyapunov technique and also by constructing appropriate Lyapunov functionals. A Gronwall-like lemma for delay equations is also exploited to provide some stability results. In the end we also include some comments concerning the global asymptotic analysis of the model, as well as some open questions and future lines for research

Boundedness criteria for a class of second order nonlinear differential equations with delay

summary:We consider certain class of second order nonlinear nonautonomous delay differential equations of the form $$a(t)x^{\prime \prime } + b(t)g(x,x^\prime ) + c(t)h(x(t-r))m(x^\prime ) = p(t,x,x^\prime )$$ and $$(a(t)x^\prime )^\prime + b(t)g(x,x^\prime ) + c(t)h(x(t-r))m(x^\prime ) = p(t,x,x^\prime ),$$ where $a$, $b$, $c$, $g$, $h$, $m$ and $p$ are real valued functions which depend at most on the arguments displayed explicitly and $r$ is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovski\v ı functional to establish our results. This work extends and improve on some results in the literature

Robustness of time-dependent attractors in H1-norm for nonlocal problems

In this paper, the existence of regular pullback attractors as well as their upper semicontinuous behaviour in H1-norm are analysed for a parameterized family of non-autonomous nonlocal reaction-diffusion equations without uniqueness, improving previous results [Nonlinear Dyn. 84 (2016), 35–50].Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de Andalucí

Existence of global attractor for a nonautonomous state-dependent delay differential equation of neuronal type

The analysis of the long-term behavior of the mathematical model of a neural network constitutes a suitable framework to develop new tools for the dynamical description of nonautonomous state-dependent delay equations (SDDEs). The concept of global attractor is given, and some results which establish properties ensuring its existence and providing a description of its shape, are proved. Conditions for the exponential stability of the global attractor are also studied. Some properties of comparison of solutions constitute a key in the proof of the main results, introducing methods of monotonicity in the dynamical analysis of nonautonomous SDDEs. Numerical simulations of some illustrative models show the applicability of the theory.Ministerio de Economía y Competitividad / FEDER, MTM2015-66330-PMinisterio de Ciencia, Innovación y Universidades, RTI2018-096523-B-I00European Commission, H2020-MSCA-ITN-201