arXiv subjects

Pedro Soares

Publications and source records attributed to Pedro Soares.

9 recordsLinked to original sources

arXiv2026

Invariant Sphere Theorem and Ring-Coupled Systems

In this work, we show how heteroclinic networks can arise in a simple class of network dynamical systems through the application of the Invariant Sphere Theorem. Ring-coupled systems are ODE networks in $\mathbb{R}^n$ in which each variable $x_i$ interacts only with its predecessor $x_{i-1}$. We derive conditions on the coefficients of a cubic polynomial that guarantee the existence of a globally attracting invariant sphere via the Invariant Sphere Theorem. Moreover, setting one of these coefficients to zero, we identify conditions on the remaining coefficients that guarantee the existence of a heteroclinic network on the invariant sphere. Focusing on the case $n=3$, we investigate perturbations of the vanishing coefficient in a neighbourhood of the heteroclinic network. Under such perturbations, the heteroclinic network is destroyed and periodic orbits emerge. When the original heteroclinic network is asymptotically stable, the resulting periodic orbits shadow the network. In one parameter regime, a unique attracting periodic orbit appears and shadows the entire heteroclinic network. By contrast, when the heteroclinic network is not stable, repelling periodic orbits arise that shadow only part of the heteroclinic structure, namely half of the heteroclinic connections. Symmetry plays a fundamental role throughout the analysis. Exploiting the symmetries of the system, we reduce the heteroclinic network to two homoclinic orbits. Furthermore, the local dynamics near the heteroclinic network can be studied on a quotient space consisting of an annulus attached to a M\"obius band, with each homoclinic orbit lying on one of these components. This reduction provides a geometric framework for understanding the bifurcations and the emergence of the periodic dynamics.

math.DS↗

arXiv2026

Heteroclinic networks in coupled cell systems

A coupled cell system is an ODE system associated with a coupled cell network, where the dimension is determined by the number of cells. A heteroclinic connection is a set of solution trajectories between two equilibria of an ODE system. A realization of a heteroclinic network is an ODE system that exhibits equilibria corresponding to the nodes and heteroclinic connections between them according to the heteroclinic network. This paper investigates the realization of heteroclinic networks within coupled cell systems, focusing on embedding heteroclinic connections in 2D and 3D invariant subspaces. We adapt Field's method of embedding each heteroclinic connection in distinct 2D synchrony subspaces to support multiple connections within the same subspace. Using the concept of book embedding from graph theory, we demonstrate that any heteroclinic network can be realized using a coupled cell system with a number of cells proportional to the network's book-thickness. Additionally, we extend our analysis to 3D synchrony subspaces, allowing for more complex realizations. In this case, the number of cells necessary for a realization is proportional to the number of nodes in the heteroclinic network.

math.DS↗

arXiv2024

Steady-state bifurcations for three-cell networks with asymmetric inputs

We consider homogeneous coupled cell networks with asymmetric inputs. We obtain general results concerning codimension-one steady-state bifurcations for networks with any number of cells and any number of asymmetric inputs. These results rely solely on the network adjacency matrices eigenvalue structure and the existence, or not, of network synchrony subspaces. For networks with three-cells, we describe the possible lattices of synchrony subspaces annotated with the eigenvalues on each synchrony subspace. Applying the previous results, we classify the synchrony-breaking steady-state bifurcations that can occur for three-cell minimal networks with one, two or six asymmetric inputs.

math.DS↗

arXiv2023

The Bifurcation Coalescence Problem for Feedforward Coalescence Networks

Consider two networks and combine them through the coalescence operation to get a larger network. Is it possible to study the steady-state bifurcations of the coalescence network by studying the steady-state bifurcations of the component networks? We conclude that this is not possible for general coalescence networks. We show, however, that in the case of feedforward coalescence networks this is possible and we cover the simplest cases. In particular, we prove how the growth rate of the bifurcation branches in the feedforward coalescence network depends on the connections from one network to the other.

math.DS↗

arXiv2020

Classification of Networks with Asymmetric Inputs

Coupled cell systems associated with a coupled cell network are determined by (smooth) vector fields that are consistent with the network structure. Here, we follow the formalisms of Stewart, Golubitsky and Pivato (Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM J. Appl. Dyn. Syst. 2 (4) (2003) 609--646), Golubistky, Stewart and Torok (Patterns of synchrony in coupled cell networks with multiple arrows, SIAM J. Appl. Dynam. Sys. 4 (1) (2005) 78--100) and Field (Combinatorial dynamics, Dynamical Systems 19 (2004) (3) 217--243). It is known that two non-isomorphic n-cell coupled networks can determine the same sets of vector fields -- these networks are said to be ODE-equivalent. The set of all n-cell coupled networks is so partitioned into classes of ODE-equivalent networks. With no further restrictions, the number of ODE-classes is not finite and each class has an infinite number of networks. Inside each ODE-class we can find a finite subclass of networks that minimize the number of edges in the class, called minimal networks. In this paper, we consider coupled cell networks with asymmetric inputs. That is, if k is the number of distinct edges types, these networks have the property that every cell receives k inputs, one of each type. Fixing the number n of cells, we prove that: the number of ODE-classes is finite; restricting to a maximum of n(n-1) inputs, we can cover all the ODE-classes; all minimal n-cell networks with n(n-1) asymmetric inputs are ODE-equivalent. We also give a simple criterion to test if a network is minimal and we conjecture lower estimates for the number of distinct ODE-classes of n-cell networks with any number k of asymmetric inputs. Moreover, we present a full list of representatives of the ODE-classes of networks with three cells and two asymmetric inputs.

math.DS↗

arXiv2017

Characterization of Fundamental Networks

In the framework of coupled cell systems, a coupled cell network describes graphically the dynamical dependencies between individual dynamical systems, the cells. The fundamental network of a network reveals the hidden symmetries of that network. Subspaces defined by equalities of coordinates which are flow-invariant for any coupled cell system consistent with a network structure are called the network synchrony subspaces. Moreover, for every synchrony subspaces, each network admissible system restricted to that subspace is a dynamical systems consistent with a smaller network. The original network is then said to be a lift of the smaller network. We characterize networks such that: its fundamental network is a lift of the network; the network is a subnetwork of its fundamental network, and the network is a fundamental network. The size of cycles in a network and the distance of a cell to a cycle are two important properties concerning the description of the network architecture. In this paper, we relate these two architectural properties in a network and its fundamental network.

math.CO↗

arXiv2017

Revisiting concurrent separation logic

We present a new soundness proof of Concurrent Separation Logic (CSL) based on a structural operational semantics (SOS). We build on two previous proofs and develop new auxiliary notions to achieve the goal. One uses a denotational semantics (based on traces). The other is based on SOS, but was obtained only for a fragment of the logic - the Disjoint CSL - which disallows modifying shared variables between concurrent threads. In this work, we lift such a restriction, proving the soundness of full CSL with respect to a SOS. Thus contributing to the development of tools able of ensuring the correctness of realistic concurrent programs. Moreover, given that we used SOS, such tools can be well-integrated in programming environments and even incorporated in compilers.

cs.LO↗

arXiv2017

The Lifting Bifurcation Problem on Feed-Forward Networks

We consider feed-forward networks, that is, networks where cells can be divided into layers, such that every edge targeting a layer, excluding the first one, starts in the prior layer. A feed-forward system is a dynamical system that respects the structure of a feed-forward network. The synchrony subspaces for a network, are the subspaces defined by equalities of some cells coordinates, that are flow-invariant by all the network systems. The restriction of each network system to each synchrony subspace is a system associated with a smaller network, which may be, or not, a feed-forward network. The original network is then said to be a lift of the smaller network. We show that a feed-forward lift of a feed-forward network is given by the composition of two types of lifts: lifts that create new layers and lifts inside a layer. Furthermore, we address the lifting bifurcation problem on feed-forward systems. More precisely, the comparison of the possible codimension-one local steady-state bifurcations of a feed-forward system and those of the corresponding lifts is considered. We show that for most of the feed-forward lifts, the increase of the center subspace is a sufficient condition for the existence of additional bifurcating branches of solutions, which are not lifted from the restricted system. However, when the bifurcation condition is associated with the internal dynamics and the lifts occurs inside an intermediate layer, we prove that the existence of bifurcating branches of solutions that are not lifted from the restricted system does depend generically on the particular feed-forward system.

math.DS↗

arXiv2017

Synchrony Branching Lemma for Regular Networks

Coupled cell systems are dynamical systems associated to a network and synchrony subspaces, given by balanced colorings of the network, are invariant subspaces for every coupled cell systems associated to that network. Golubitsky and Lauterbach (SIAM J. Applied Dynamical Systems, 8 (1) 2009, 40-75) prove an analogue of the Equivariant Branching Lemma in the context of regular networks. We generalize this result proving the generic existence of steady-state bifurcation branches for regular networks with maximal synchrony. We also give necessary and sufficient conditions for the existence of steady-state bifurcation branches with some submaximal synchrony. Those conditions only depend on the network structure, but the lattice structure of the balanced colorings is not sufficient to decide which synchrony subspaces support a steady-state bifurcation branch.

math.DS↗