Topologies of continuity for Carathéodory differential equations with applications in non-autonomous dynamics

Longo, Iacopo P.

doi:10.35376/10324/44162

Cited by 3 publications

(2 citation statements)

References 38 publications

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“…. As a matter of fact, under the given assumptions, the problem also induces a continuous skew-product flow [56][57][58]. The fundamental idea behind the concept of intensity is to conceive each bounded set of perturbing functions g ∈ L ∞ as a class of controls, and to investigate which parts of the phase space can be reached through them.…”

Section: Intensity Of Attractionmentioning

confidence: 99%

Resilience of dynamical systems

2023

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Stability is among the most important concepts in dynamical systems. Local stability is well-studied, whereas determining the ‘global stability’ of 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 generalised 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 nonautonomous 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.

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Section: Intensity Of Attractionmentioning

confidence: 99%

Resilience of dynamical systems

2023

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“…Last but not least, we note that topology has been used extensively in the construction of semantic models of programming systems, notably of the 𝜆-calculus [16]. This has led to powerful approaches to the computability of functions and type theory [17]. In general, computable topology has focussed on the computability of 'topological objects', rather than on computational processes as we do here.…”

Section: Cross Connectionsmentioning

confidence: 99%

Knowledge, Representation and the Dynamics of Computation

Leeuwen

Wiedermann

2017

Representation and Reality in Humans, Other Living Organisms and Intelligent Machines

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Cognitive processes are often modelled in computational terms. Can this still be done if only minimal assumptions are made about any sort of representation of reality? Is there a purely knowledge-based theory of computation that explains the key phenomena which are deemed to be computational in both living and artificial systems as understood today? We argue that this can be done by means of techniques inspired by the modelling of dynamical systems. In this setting, computations are defined as curves in suitable metaspaces and knowledge is generated by virtue of the operation of the underlying mechanism, whatever it is. Desirable properties such as compositionality will be shown to fit naturally. The framework also enables one to formally characterize the computational behaviour of both knowledge generation and knowledge recognition. The approach may be used in identifying when processes or systems can be viewed as being computational in general. Several further questions pertaining to the philosophy of computing are considered.

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

Longo,

Núñez,

Obaya

2024

Nonlinearity

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The global dynamics of a nonautonomous Carathéodory scalar ordinary differential equation x ′ = f ( t , x ) , given by a function f which is concave in x, 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 asymptotic 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 , x − Γ ( t ) ) , where Γ 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.

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Topologies of continuity for Carathéodory differential equations with applications in non-autonomous dynamics

Cited by 3 publications

References 38 publications

Resilience of dynamical systems

Resilience of dynamical systems

Knowledge, Representation and the Dynamics of Computation

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

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