arXiv subjects

Alejandro López-Nieto

Publications and source records attributed to Alejandro López-Nieto.

4 recordsLinked to original sources

arXiv2025

Transient rebellions in the Kuramoto oscillator: Morse-Smale structural stability and connection graphs of finite 2-shift type

The celebrated 1975 Kuramoto model of $N$ identical oscillators with phase angle vector $\boldsymbol{\vartheta}=(\vartheta_1,\ldots,\vartheta_N)$ and all-to-all coupling reads \begin{equation} \label{*} \dot\vartheta_j\,=\tfrac{1}{N}\sum_{k=1}^N \sin(\vartheta_k-\vartheta_j). \tag{*} \end{equation} Here we have passed to co-rotating coordinates in normalized time scale. The model is highly accessible to rigorous mathematical analysis, and has been studied as a paradigm for effects like total and partial synchronization. Most initial conditions $\boldsymbol{\vartheta}$ lead to total synchronization. The plethora of $2^N-1$ (circles of) partially synchronized states, however, is unstable. The precise behavior of transitions to synchrony seems to have eluded description. In the present paper, we address this gap. By the gradient structure of (*), the global dynamics decompose into equilibria and heteroclinic orbits between them. Except for the extremes of total instability and total synchrony, all equilibria are 2-cluster solutions: their phase angles $\vartheta_j$ attain only two values, with a phase difference of $π$ between them. Any heteroclinic orbit between 2-cluster equilibria, or towards synchrony, can be realized as a 3-cluster rebellion. Cluster rebellions split the smaller, "slim", minority cluster of the source equilibrium and send the rebellious part to join the bigger, "fat", majority cluster. Heteroclinic transversality identifies the Kuramoto model as a structurally stable Morse-Smale system. In particular, heteroclinic orbits can be concatenated in finite time. The options involved in successive cluster rebellions amount to finite symbol sequences of 2-shift type. The paper is dedicated to Professor Yoshiki Kuramoto, with admiration.

math.DS↗

arXiv2025

Classification of global attractors for $\mathbb{S}^1$-equivariant parabolic equations: a survey

We survey the global dynamics of semiflows generated by scalar semilinear parabolic equations which are $\mathbb{SO}(2)$ equivariant under spatial shifts of $x\in \mathbb{S}^1=\mathbb{R}/2π\mathbb{Z}$, i.e. $$ u_t = u_{xx} + f(u,u_x),\qquad x\in \mathbb{S}^1.$$ For dissipative $C^2$ nonlinearities $f$, the semiflow possesses a compact global attractor $\mathcal{A}=\mathcal{A}^\mathcal{P}$ which we call Sturm attractor. The Sturm attractor $\mathcal{A}^\mathcal{P}$ decomposes as $$ \mathcal{A}^\mathcal{P}=\mathcal{E}\cup\mathcal{F}^\mathcal{P}\cup\mathcal{R}^\mathcal{P}\cup\mathcal{H}^\mathcal{P},$$ where $\mathcal{H}^\mathcal{P}$ denotes heteroclinic orbits between distinct elements of spatially homogeneous equilibria $\mathcal{E}$, rigidly rotating waves $\mathcal{R}^\mathcal{P}$ and, as their non-rotating counterparts, frozen waves $\mathcal{F}^\mathcal{P}$. We therefore represent $\mathcal{A}^\mathcal{P}$ by its connection graph $\mathcal{C}^\mathcal{P}$, with vertices in $\mathcal{E},\mathcal{F}^\mathcal{P},\mathcal{R}^\mathcal{P}$ and edges $\mathcal{H}^\mathcal{P}$. Under mild hyperbolicity assumptions, the directed graphs $\mathcal{C}^\mathcal{P}$ are finite and transitive. For illustration, we enumerate all 21 connection graphs $\mathcal{C}^\mathcal{P}$ with up to seven vertices. The result uses a lap signature of period maps associated to integrable versions of the steady state ODE of our PDE. As an example, we freeze and reconstruct the connection graph of the Vas tulip attractor, known from delay differential equations, in our PDE setting.

math.DS↗

arXiv2025

Periodicity in delayed self-regulation is a predator-prey process

We unify two different periodicity mechanisms: delayed self-regulation and planar predator-prey feedback. We consider scalar delay differential equations $\dot x(t) = rf(x(t), x(t - 1))$ where $f$ is monotone in the delayed component. Due to a Poincaré--Bendixson theorem for monotone delayed feedback systems, the typical global dynamics present periodic orbits as the delay parameter $r$ increases. In this article, we show that, as we vary the delay, each connected component of periodic orbits is an annulus with global coordinates given by the time and the amplitude of the corresponding periodic solutions. On each annulus, the variables $x(t)$ and $x(t - 1)$ solve an integrable ordinary differential equation that satisfies a predator-prey feedback relation. Moreover, we completely characterize the set of periodic solutions of the delay differential equation in terms of two time maps generated by the underlying predator-prey system.

math.DS↗

arXiv2024

Hybrid Bifurcations: Periodicity from Eliminating a Line of Equilibria

We describe a new mechanism that triggers periodic orbits in smooth dynamical systems. To this end, we introduce the concept of hybrid bifurcations: Such bifurcations occur when a line of equilibria with an exchange point of normal stability vanishes. Our main result is the existence and stability criteria of periodic orbits that bifurcate from breaking a line of equilibria. As an application, we obtain stable periodic coexistent solutions in an ecosystem for two competing predators with Holling's type II functional response.

math.DS↗