arXiv subjects

Rafael Obaya

Publications and source records attributed to Rafael Obaya.

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 $x'=f(t,x)+\lambda$, 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.

math.DS↗

arXiv2026

A new class of generalized ordinary differential equations with applications

The space of parametric b-measures endowed with appropriate topologies is introduced to define a new class of generalized ODEs given by parametric b-measures. This framework offers a new approach for dealing with precompact families of Carath\'eodory ODEs using nonautonomous dynamical systems techniques. An application to the study of the dynamics of the fast variables of a slow-fast system of ODEs, where the fast motion is determined by a Carath\'eodory vector field with equicontinuous $m$-bounds and bounded $l$-bounds, is given.

math.DS↗

arXiv2025

Nonautonomous modelling in Energy Balance Models of climate. Limitations of averaging and climate sensitivity

Starting from a classical Budyko-Sellers-Ghil energy balance model for the average surface temperature of the Earth, a nonautonomous version is designed by allowing the solar irradiance and the cloud cover coefficients to vary with time in a fast timescale, and to exhibit chaos in a precise sense. The dynamics of this model is described in terms of three existing nonautonomous equilibria, the upper one being attracting and representing the present temperature profile. The theory of averaging is used to compare the nonautonomous model and its time-averaged version. We analyse the influence of the qualitative properties of the time-dependent coefficients and develop physically significant error estimates close to the upper attracting solution. Furthermore, previous concepts of two-point response and sensitivity functions are adapted to the nonautonomous context and used to value the increase in temperature when a forcing caused by CO2 and other emissions intervenes.

physics.ao-ph↗

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.

math.DS↗

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.

math.DS↗

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.

math.DS↗

arXiv2024

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

New 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. %In particular this happens when the attractors of the layer problems are copies of the base, which is the counterpart of an asymptotically stable equilibrium point in the autonomous case. 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.

math.DS↗

arXiv2024

Exponential Ordering for Neutral Functional Differential Equations With Non-Autonomous Linear D-Operator

We study neutral functional differential equations with stable linear non-autonomous $D$-operator. The operator of convolution $\hat{D}$ transforms $BU$ into $BU$. We show that, if $D$ is stable, then $\hat{D}$ is invertible and, besides, $\hat{D}$ and $\hat{D}^{-1}$ are uniformly continuous for the compact-open topology on bounded sets. We introduce a new transformed exponential order and, under convenient assumptions, we deduce the 1-covering property of minimal sets. These conclusions are applied to describe the amount of material in a class of compartmental systems extensively studied in the literature.

math.DS↗

arXiv2024

Neutral Functional Differential Equations with Applications to Compartmental Systems

We study the monotone skew-product semiflow generated by a family of neutral functional differential equations with infinite delay and stable D-operator. The stability properties of D allow us to introduce a new order and to take the neutral family to a family of functional differential equations with infinite delay. Next, we establish the 1-covering property of omega-limit sets under the componentwise separating property and uniform stability. Finally, the obtained results are applied to the study of the long-term behavior of the amount of material within the compartments of a neutral compartmental system with infinite delay.

math.DS↗

arXiv2024

Exponential Ordering for Nonautonomous Neutral Functional Differential Equations

We study monotone skew-product semiflows generated by families of nonautonomous neutral functional differential equations with infinite delay and stable D-operator, when the exponential ordering is considered. Under adequate hypotheses of stability for the order on bounded sets, we show that the omega-limit sets are copies of the base to explain the long-term behavior of the trajectories. The application to the study of the amount of material within the compartments of a neutral compartmental system with infinite delay, shows the improvement with respect to the standard ordering.

math.DS↗

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.

math.DS↗

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.

math.DS↗

arXiv2023

Two dynamical approaches to the notion of exponential separation for random systems of delay differential equations

This paper deals with the exponential separation of type II, an important concept for random systems of differential equations with delay, introduced in \JM\ et al.~\cite{MiNoOb1}. Two different approaches to its existence are presented. The state space $X$ will be a separable ordered Banach space with $\dim X\geq 2$, dual space $X^{*}$ and positive cone $X^+$ normal and reproducing. In both cases, appropriate cooperativity and irreducibility conditions are assumed to provide a family of generalized Floquet subspaces. If in addition $X^*$ is also separable, one obtains a exponential separation of type II. When this is not the case, but there is an Oseledets decomposition for the continuous semiflow, the same result holds. Detailed examples are given for all the situations, including also a case where the cone is not normal.

math.DS↗

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.

math.DS↗

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.

math.DS↗

arXiv2022

Uniform stability and chaotic dynamics in nonhomogeneous linear dissipative scalar ordinary differential equations

The paper analyzes the structure and the inner long-term dynamics of the invariant compact sets for the skewproduct flow induced by a family of time-dependent ordinary differential equations of nonhomogeneous linear dissipative type. The main assumptions are made on the dissipative term and on the homogeneous linear term of the equations. The rich casuistic includes the uniform stability of the invariant compact sets, as well as the presence of Li-Yorke chaos and Auslander-Yorke chaos inside the attractor.

math.DS↗

arXiv2022

The exponential ordering for non-autonomous delay systems with applications to compartmental Nycholson systems

The exponential ordering is exploited in the context of non-auto\-no\-mous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which asymptotically attracts every other positive solution.

math.DS↗