arXiv subjects

Jesús Dueñas

Publications and source records attributed to Jesús Dueñas.

8 recordsLinked to original sources

arXiv2026

Asymptotically Autonomous Maps: Heteroclinic connections, Splitting and Phase-induced tipping

We consider discrete-time dynamical systems generated by the iteration of asymptotically autonomous maps, whose asymptotic behavior may be either conservative or dissipative. These systems are of particular interest in the modeling of complex phenomena with time-dependent parameter variation. With an appropriate time compactification, the past and future infinity become normally hyperbolic invariant hyperplanes, and thus the plausible asymptotic states of the system can be determined by considering the appropriate fibers of the stable and unstable foliations or suitable sections of the asymptotic manifolds to their compact invariant subsets. In this context, we explore how tipping induced by a change of phase--namely, how a phase shift affects the evolution of the system--is encoded in the unstable manifolds of compact invariant subsets of the past equation. We further connect the tipping phenomenon to the splitting of invariant manifolds and identify certain system reversibilities able to prevent tipping. The theoretical framework we use clarifies the relationships among the various notions of tipping induced by phase-change introduced in the literature.

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arXiv2026

Loss-of-hyperbolicity and jump bifurcations in scalar nonautonomous d-concave ODEs

The paper studies bifurcation for additively parametrized scalar nonautonomous ODEs of the form $x'=f(t,x)+λ$, where $f$ is d-concave and coercive in the state variable. By formulating the problem through the hull of $f$ and the associated skewproduct flow, the autonomous notion of bifurcation can be extended in terms of loss of hyperbolicity of copies of the base. A fairly complete classification of the possible bifurcation diagrams is provided. The notion of a jump bifurcation, linked to abrupt changes in the global attractor, is introduced and related to the occurrence of critical transitions. Several nontrivial, genuinely nonautonomous examples are presented, illustrating the different bifurcation diagrams that may arise from the lack of unique ergodicity on the hull and showing how critical transitions can result from an underlying jump bifurcation point. The framework has potential applications in climate dynamics, ecology, circuits, optics, biology, and other models involving abrupt transitions.

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arXiv2025

Nonautonomous scalar concave-convex differential equations: conditions for uniform stability or bistability in a model of optical fluorescence

The long-term dynamics of a Bonifacio-Lugiato model of optical superfluorescence is investigated. The scalar ordinary differential equation modelling the phenomenon is given by a concave-convex autonomous function of the state variable that is excited by a time-dependent input, $I(t)$. The system's response is described in terms of the dynamical characteristics of the input function, with particular focus on uniform stability or bistability cases. Building on previous published results, the open interval defined by the constant input values for which the equation exhibits uniform stability or bistability is considered, and it is proved that bistability occurs when $I(t)$ lies within this interval. This condition is sufficient but not necessary. Applying nonautonomous bifurcation methods and imposing more restrictive conditions on the variation of $I(t)$ makes it possible to determine the necessary and sufficient conditions for bistability and to prove that the general response is uniform stability when these conditions are not satisfied. Finally, the case of a periodic input that varies on a slow timescale is analyzed using fast-slow system methods to rigorously establish either a uniformly stable or a bistable response.

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arXiv2024

Concave-convex nonautonomous scalar ordinary differential equations: from bifurcation theory to critical transitions

A mathematical modeling process for phenomena with a single state variable that attempts to be realistic must be given by a scalar nonautonomous differential equation $x'=f(t,x)$ that is concave with respect to the state variable $x$ in some regions of its domain and convex in the complementary zones. This article takes the first step towards developing a theory to describe the corresponding dynamics: the case in which $f$ is concave on the region $x\ge b(t)$ and convex on $x\le b(t)$, where $b$ is a $C^1$ map, is considered. The different long-term dynamics that may appear are analyzed while describing the bifurcation diagram for $x'=f(t,x)+λ$. The results are used to establish conditions on a concave-convex map $h$ and a nonnegative map $k$ ensuring the existence of a value $ρ_0$ giving rise to the unique critical transition for the parametric family of equations $x'=h(t,x)-ρ\,k(t,x)$, which is assumed to approach $x'=h(t,x)$ as time decreases, but for which no conditions are assumed on the future dynamics. The developed theory is justified by showing that concave-convex models fit correctly some laboratory experimental data, and applied to describe a population dynamics model for which a large enough increase on the peak of a temporary higher predation causes extinction.

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arXiv2024

Saddle-node bifurcations for concave in measure and d-concave in measure skewproduct flows with applications to population dynamics and circuits

Concave in measure and d-concave in measure nonautonomous scalar ordinary differential equations given by coercive and time-compactible maps have similar properties to equations satisfying considerably more restrictive hypotheses. This paper describes the generalized simple or double saddle-node bifurcation diagrams for one-parametric families of equations of these types, from which the dynamical possibilities for each of the equations follow. This new framework allows the analysis of ``almost stochastic" equations, whose coefficients vary in very large chaotic sets. The results also apply to the analysis of the occurrence of critical transitions for a range of models much larger than in previous approaches.

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arXiv2023

Critical Transitions for Asymptotically Concave or D-Concave Nonautonomous Differential Equations with Applications in Ecology

The occurrence of tracking or tipping situations for a transition equation $x'=f(t,x,Γ(t,x))$ is analyzed under the assumptions on concavity in $x$ either of the maps giving rise to the asymptotic equations $x'=f(t,x,Γ_\pm(t,x))$ or of their derivatives with respect to the state variable (d-concavity), but without assuming these conditions on the transition equation itself. The approaching condition is just $\lim_{t\to\pm\infty}(Γ(t,x)-Γ_\pm(t,x))=0$ uniformly on compact real sets, and so there is no restriction to the dependence on time of the limit equations. The analysis provides a powerful tool to analyze the occurrence of critical transitions for one-parametric families $x'=f(t,x,Γ^c_\pm(t,x))$. The new approach significatively widens the field of application of the results, since the evolution law of the transition equation can be essentially different from those of the limit equations. Among these applications, some scalar population dynamics models subject to non trivial predation and migration patterns are analyzed, both theoretically and numerically. Some key points in the proofs are: to understand the transition equation as part of an orbit in its hull which approaches the $α$-limit and $ω$-limit sets; to observe that these sets concentrate all the ergodic measures; and to prove that in order to describe the dynamical possibilities of the equation it suffices that the concavity or d-concavity conditions hold for a complete measure subset of the equations of the hull.

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arXiv2023

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

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.

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arXiv2022

Critical Transitions in D-Concave Nonautonomous Scalar Ordinary Differential Equations Appearing in Population Dynamics

A function with finite asymptotic limits gives rise to a transition equation between a "past system" and a "future system". This question is analyzed in the case of nonautonomous coercive nonlinear scalar ordinary differential equations with concave derivative with respect to the state variable. The fundamental hypothesis is the existence of three hyperbolic solutions for the limit systems, in which case the upper and lower ones are attractive. All the global dynamical possibilities are described in terms of the internal dynamics of the pullback attractor: cases of tracking of the two hyperbolic attractive solutions or lack of it (tipping) may arise. This analysis, made in the language of processes and also in terms of the skewproduct formulation of the problem, includes cases of rate-induced critical transitions, as well as cases of phase-induced and size-induced tipping. The conclusions are applied in models of mathematical biology and population dynamics. Rate-induced tracking phenomena causing extinction of a native species or invasion of a non-native one are described, as well as population models affected by a Holling type III functional response to predation where tipping due to the changes in the size of the transition may occur. In all these cases, the appearance of a critical transition can be understood as a consequence of the strength of Allee effect.

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