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 \lq\lq solution, control\rq\rq~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.Ministerio de Economía y Competitividad / FEDER, MTM2015-66330-PMinisterio de Ciencia, Innovación y Universidades, RTI2018-096523-B-I00European Commission, H2020-MSCA-ITN-2014INDAM -- GNAMPA Project 201

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.MINECO/FEDER, MTM2015-66330-PEuropean Commission, H2020-MSCA-ITN-201

Non-Atkinson perturbations of nonautonomous linear Hamiltonian systems: exponential dichotomy and nonoscillation

Producción CientíficaWe analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions $M^\pm$ for one-parametric families of finite-dimensional nonautonomous linear Hamiltonian systems defined along the orbits of a compact metric space, which are perturbed from an initial one in a direction which does not satisfy the classical Atkinson condition: either they do not have ED for any value of the parameter; or they have it for at least all the nonreal values, in which case the Weyl functions exist and are Herglotz. When the parameter varies in the real line, and if the unperturbed family satisfies the properties of exponential dichotomy and global existence of $M^+$, then these two properties persist in a neighborhood of 0 which agrees either with the whole real line or with an open negative half-line; and in this last case, the ED fails at the right end value. The properties of ED and of global existence of $M^+$ are fundamental to guarantee the solvability of classical minimization problems given by linear-quadratic control processes.MINECO/FEDER, MTM2015-66330-PEuropean Commission, H2020-MSCA-ITN-201

Rate-induced tipping and saddle-node bifurcation for 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 $q\colon\R\to\R$ and $p\colon\R\to\R$ are bounded and uniformly continuous, is fundamental to explain the absence or occurrence of rate-induced tipping for the differential equation $y' =(y-(2/\pi)\arctan(ct))^2+p(t)$ 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 this tipping phenomenon may be reversible.Marie Skłodowska-Curie grant agreement No 643073Ministerio de Ciencia, Innovación y Universidades, RTI2018-096523-B-I00Marie Skłodowska-Curie grant agreement No 75446

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.Ministerio de Economía y Competitividad / FEDER, MTM2015-66330-PMinisterio de Ciencia, Innovación y Universidades, RTI2018-096523-B-I00European Commission, H2020-MSCA-ITN-201

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

Producción CientíficaThe 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.MINECO-Feder (RTI2018-098850-B-I00)Junta de Andalucía (PY18-RT-2422 y B-FQM-580-UGR20)Ministerio de Ciencia e Innovación (PID2021-125446NB-100)Universidad de Valladolid (PIP-TCESC-2020

Bifurcation theory of attractors and minimal sets in d-concave nonautonomous scalar ordinary differential equations

Producción CientíficaTwo one-parametric bifurcation problems for scalar nonautonomous ordinary differential equations are analyzed assuming the coercivity of the time-dependent function determining the equation and the concavity of its derivative with respect to the state variable. The skewproduct formalism leads to the analysis of the number and properties of the minimal sets and of the shape of the global attractor, whose abrupt variations determine the occurrence of local saddle-node, local transcritical and global pitchfork bifurcation points of minimal sets and of discontinuity points of the global attractor.Ministerio de Ciencia, Innovación - Ministerio de Universidades (RTI2018- 096523-B-I00)Universidad de Valladolid (PIP-TCESC-2020)Ministerio de Universidades (FPU20/01627

Critical transitions in piecewise uniformly continuous concave quadratic ordinary differential equations

Producción CientíficaA 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.Ministerio de Ciencia, Innovación y Universidades (under project RTI2018-096523-B-I00)Universidad de Valladolid (under proyect PIP-TCESC-2020)European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 754462Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

Dynamical properties of nonautonomous functional differential equations with state-dependent delay

The properties of stability of a compact set $K$ which is positively invariant for a semiflow (\W\times W^\infty([-r,0],\mathbb{R}^n),\Pi,\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 $K$ induce linear skew-product semiflows on the bundles $K\times W^\infty([-r,0],\R^n)$ and $K\times C([-r,0],\R^n)$. 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 \mK in \W\times W^\infty([-r,0],\R^n) and also to the exponential stability of this compact set when the supremum norm is taken in $W^\infty([-r,0],\R^n)$. 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.Ministerio de Economía, Industria y Competitividad (MTM2015-66330-P

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

The properties of stability of a compact set $K$ which is positively invariant for a semiflow (\W\times W^{1,\infty}([-r,0],\mathbb{R}^n),\Pi,\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 $K$ induce linear skew-product semiflows on the bundles $K\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and \mK\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 $K$ in \W\times W^{1,\infty}([-r,0],\mathbb{R}^n) and also to the exponential stability of this compact 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.Ministerio de Economía, Industria y Competitividad (MTM2015-66330-P