20 research outputs found

    Weak topologies for Carath\'eodory differential equations. Continuous dependence, exponential Dichotomy and attractors

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    We introduce new weak topologies and spaces of Carath\'eodory functions where the solutions of the ordinary differential equations depend continuously on the initial data and vector fields. The induced local skew-product flow is proved to be continuous, and a notion of linearized skew-product flow is provided. Two applications are shown. First, the propagation of the exponential dichotomy over the trajectories of the linearized skew-product flow and the structure of the dichotomy or Sacker-Sell spectrum. Second, how particular bounded absorbing sets for the process defined by a Carath\'eodory vector field ff provide bounded pullback attractors for the processes with vector fields in the alpha-limit set, the omega-limit set or the whole hull of ff. Conditions for the existence of a pullback or a global attractor for the skew-product semiflow, as well as application examples are also given.Comment: 34 page

    Resilience of dynamical systems

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    Stability is among the most important concepts in dynamical systems. Local stability is well-studied, whereas determining how "globally stable" a nonlinear system is very challenging. Over the last few decades, many different ideas have been developed to address this issue, primarily driven by concrete applications. In particular, several disciplines suggested a web of concepts under the headline "resilience". Unfortunately, there are many different variants and explanations of resilience, and often the definitions are left relatively vague, sometimes even deliberately. Yet, to allow for a structural development of a mathematical theory of resilience that can be used across different areas, one has to ensure precise starting definitions and provide a mathematical comparison of different resilience measures. In this work, we provide a systematic review of the most relevant indicators of resilience in the context of continuous dynamical systems, grouped according to their mathematical features. The indicators are also generalized to be applicable to any attractor. These steps are important to ensure a more reliable, quantitatively comparable and reproducible study of resilience in dynamical systems. Furthermore, we also develop a new concept of resilience against certain non-autonomous perturbations to demonstrate, how one can naturally extend our framework. All the indicators are finally compared via the analysis of a classic scalar model from population dynamics to show that direct quantitative application-based comparisons are an immediate consequence of a detailed mathematical analysis.Comment: 54 pages, 18 figure

    Rate-induced tracking for concave or d-concave transitions in a time-dependent environment with application in ecology

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    This paper investigates biological models that represent the transition equation from a system in the past to a system in the future. It is shown that finite-time Lyapunov exponents calculated along a locally pullback attractive solution are efficient indicators (early-warning signals) of the presence of a tipping point. Precise time-dependent transitions with concave or d-concave variation in the state variable giving rise to scenarios of rate-induced tracking are shown. They are classified depending on the internal dynamics of the set of bounded solutions. Based on this classification, some representative features of these models are investigated by means of a careful numerical analysis.Comment: 20 pages, 7 figures; typos corrected, language improved, summary of symbols adde

    Critical transitions for scalar nonautonomous systems with concave nonlinearities: some rigorous estimates

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    The global dynamics of a nonautonomous Carath\'eodory scalar ordinary differential equation x=f(t,x)x'=f(t,x), given by a function ff which is concave in xx, is determined by the existence or absence of an attractor-repeller pair of hyperbolic solutions. This property, here extended to a very general setting, is the key point to classify the dynamics of an equation which is a transition between two nonautonomous asypmtotic limiting equations, both with an attractor-repeller pair. The main focus of the paper is to get rigorous criteria guaranteeing tracking (i.e., connection between the attractors of the past and the future) or tipping (absence of connection) for the particular case of equations x=f(t,xΓ(t))x'=f(t,x-\Gamma(t)), where Γ\Gamma is asymptotically constant. Some computer simulations show the accuracy of the obtained estimates, which provide a powerful way to determine the occurrence of critical transitions without relying on a numerical approximation of the (always existing) locally pullback attractor.Comment: 43 pages, 5 figure

    Tracking nonautonomous attractors in singularly perturbed systems of ODEs with dependence on the fast time

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    Producción CientíficaNew results on the behaviour of the fast motion in slow-fast systems of ODEs with dependence on the fast time are given in terms of tracking of nonautonomous attractors. Under quite general assumptions, including the uniform ultimate boundedness of the solutions of the layer problems, inflated pullback attractors are considered. In general, one cannot disregard the inflated version of the pullback attractor, but it is possible under the continuity of the fiber projection map of the attractor. The problem of the limit of the solutions of the slow-fast system at each fixed positive value of the slow time is also treated and in this formulation the critical set is given by the union of the fibers of the pullback attractors. The results can be seen as extensions of the classical Tikhonov theorem to the nonautonomous setting.Este trabajo forma parte de los proyectos de investigación: MICIIN/FEDER Grant PID2021-125446NB-I00, Universidad de Valladolid Grant PIP-TCESC-2020, y UKRI Grant EP/X027651/1

    Charged-particle distributions at low transverse momentum in s=13\sqrt{s} = 13 TeV pppp interactions measured with the ATLAS detector at the LHC

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    Search for dark matter in association with a Higgs boson decaying to bb-quarks in pppp collisions at s=13\sqrt s=13 TeV with the ATLAS detector