arXiv subjects

Carmen Núñez

Publications and source records attributed to Carmen Núñez.

At least 19 recordsLinked to original sources

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.

math.DS↗

arXiv2026

Averaging and tracking of local attractors in slowly varying systems with two time scales

The paper analyzes to what extent the dynamics of a nonautonomous $n$-dimensional dynamical system with two time scales, formulated in the slow time as $dx/dt=f(t/\varepsilon, t, x)$, can be approximated for small values of $\varepsilon$ by the dynamics of the averaged system $dz/dt=\hat f(t,z)$. Assuming that the skewproduct flow associated with the averaged system admits a local attractor $\mathcal{A}$, we prove that the solutions of the original system whose initial data lie in the basin of attraction of $\mathcal{A}$ track the fibers of the inflated attractor for all positive times. If the fiber map of $\mathcal{A}$ is continuous, inflation is no longer required. Alternative tracking results with a more classical formulation are also presented, under assumptions involving uniformly asymptotically stable solutions or uniform local attractors for the nonautonomous process, rather than for the skewproduct flow. Several examples illustrate the scope and applicability of the results. The twofold extension of the classical averaging results (to the doubly nonautonomous setting and to the whole positive halfline) is expected to be relevant to a broad range of application.

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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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arXiv2025

Global bifurcation diagrams for coercive third-degree polynomial ordinary differential equations with recurrent nonautonomous coefficients

Nonautonomous bifurcation theory is a growing branch of mathematics, for the insight it provides into radical changes in the global dynamics of realistic models for many real-world phenomena, i.e., into the occurrence of critical transitions. This paper describes several global bifurcation diagrams for nonautonomous first order scalar ordinary differential equations generated by coercive third degree polynomials in the state variable. The conclusions are applied to a population dynamics model subject to an Allee effect that is weak in the absence of migration and becomes strong under a migratory phenomenon whose sense and intensity depend on a threshold in the number of individuals in the population.

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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

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

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 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 $x'=f(t,x-Γ(t))$, 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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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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arXiv2021

Critical transitions in piecewise uniformly continuous concave quadratic ordinary differential equations

A critical transition for a system modelled by a concave quadratic scalar ordinary differential equation occurs when a small variation of the coefficients changes dramatically the dynamics, from the existence of an attractor-repeller pair of hyperbolic solutions to the lack of bounded solutions. In this paper, a tool to analyze this phenomenon for asymptotically nonautonomous ODEs with bounded uniformly continuous or bounded piecewise uniformly continuous coefficients is described, and used to determine the occurrence of critical transitions for certain parametric equations. Some numerical experiments contribute to clarify the applicability of this tool.

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arXiv2021

Rate-induced tipping and saddle-node bifurcation for a class of quadratic differential equations with nonautonomous asymptotic dynamics

An in-depth analysis of nonautonomous bifurcations of saddle-node type for scalar differential equations $x'=-x^2+q(t)\,x+p(t)$, where $q\colon\mathbb{R}\to\mathbb{R}$ and $p\colon\mathbb{R}\to\mathbb{R}$ are bounded and uniformly continuous, is fundamental to explain the absence or occurrence of rate-induced tipping for the differential equation $y' =(y-(2/π)\arctan(ct))^2+p(t)$ as the rate $c$ varies on $[0,\infty)$. A classical attractor-repeller pair, whose existence for $c=0$ is assumed, may persist for any $c>0$, or disappear for a certain critical rate $c=c_0$, giving rise to rate-induced tipping. A suitable example demonstrates that one can have more than one critical rate, and the existence of the classical attractor-repeller pair may return when $c$ increases.

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arXiv2019

On the solvability of the Yakubovich linear-quadratic infinite horizon minimization problem

The Yakubovich Frequency Theorem, in its periodic version and in its general nonautonomous extension, establishes conditions which are equivalent to the global solvability of a minimization problem of infinite horizon type, given by the integral in the positive half-line of a quadratic functional subject to a control system. It also provides the unique minimizing pair "solution, control" and the value of the minimum. In this paper we establish less restrictive conditions under which the problem is partially solvable, characterize the set of initial data for which the minimum exists, and obtain its value as well a minimizing pair. The occurrence of exponential dichotomy and the null character of the rotation number for a nonautonomous linear Hamiltonian system defined from the minimization problem are fundamental in the analysis.

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arXiv2019

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.

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arXiv2019

Li-Yorke chaos in nonautonomous Hopf bifurcation patterns - I

We analyze the characteristics of the global attractor of a type of dissipative nonautonomous dynamical systems in terms of the Sacker and Sell spectrum of its linear part. The model gives rise to a pattern of nonautonomous Hopf bifurcation which can be understood as a generalization of the classical autonomous one. We pay special attention to the dynamics at the bifurcation point, showing the possibility of occurrence of Li-Yorke chaos in the corresponding attractor and hence of a high degree of unpredictability.

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arXiv2017

Exponential stability for nonautonomous functional differential equations with state-dependent delay

The properties of stability of compact set $\mathcal{K}$ which is positively invariant for a semiflow $(Ω\times W^{1,\infty}([-r,0],\mathbb{R}^n),Π,\mathbb{R}^+)$ determined by a family of nonautonomous FDEs with state-dependent delay taking values in $[0,r]$ are analyzed. The solutions of the variational equation through the orbits of $\mathcal{K}$ induce linear skew-product semiflows on the bundles $\mathcal{K}\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and $\mathcal{K}\times C([-r,0],\mathbb{R}^n)$. The coincidence of the upper-Lyapunov exponents for both semiflows is checked, and it is a fundamental tool to prove that the strictly negative character of this upper-Lyapunov exponent is equivalent to the exponential stability of $\mathcal{K}$ in $Ω\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and also to the exponential stability of this minimal set when the supremum norm is taken in $W^{1,\infty}([-r,0],\mathbb{R}^n)$. In particular, the existence of a uniformly exponentially stable solution of a uniformly almost periodic FDE ensures the existence of exponentially stable almost periodic solutions.

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arXiv2017

The bosonic string on string-size tori from double field theory

We construct the effective action for toroidal compactifications of bosonic string theory from generalized Scherk-Schwarz reductions of double field theory. The enhanced gauge symmetry arising at special points in moduli space is incorporated into this framework by promoting the $O(k,k)$ duality group of $k$-tori compactifications to $O(n,n)$, $n$ being the dimension of the enhanced gauge group, which allows to account for the full massless sector of the theory. We show that the effective action reproduces the right masses of scalar and vector fields when moving sligthly away from the points of maximal symmetry enhancement. The neighborhood of the enhancement points in moduli space can be neatly explored by spontaneous symmetry breaking. We generically discuss toroidal compactifications of arbitrary dimensions and maximally enhanced gauge groups, and then inspect more closely the example of $T^2$ at the $SU(3)_L \times SU(3)_R$ point, which is the simplest setup containing all the non trivialities of the generic case. We show that the entire moduli space can be described in a unified way by considering compactifications on higher dimensional tori.

hep-th↗

arXiv2003

Type IIB orientifolds on Gepner points

We study various aspects of orientifold projections of Type IIB closed string theory on Gepner points in different dimensions. The open string sector is introduced, in the usual constructive way, in order to cancel RR charges carried by orientifold planes. Moddings by cyclic permutations of the internal N=2 superconformal blocks as well as by discrete phase symmetries are implemented. Reduction in the number of generations, breaking or enhancements of gauge symmetries and topology changes are shown to be induced by such moddings. Antibranes sector is also considered; in particular we show how non supersymmetric models with antibranes and free of closed and open tachyons do appear in this context. A systematic study of consistent models in D=8 dimensions and some illustrative examples in D=6 and D=4 dimensions are presented.

hep-th↗

arXiv1999

Interacting strings on AdS_3

We consider string theory on AdS_3 in terms of the Wakimoto free field representation. The scattering amplitudes for N unitary tachyons are analysed in the factorization limit and the poles corresponding to the mass-shell conditions for physical states are extracted. The vertex operators for excited levels are obtained from the residues and their properties are examined. Negative norm states are found at the second mass level.

hep-th↗