arXiv subjects

Bob Rink

Publications and source records attributed to Bob Rink.

At least 19 recordsLinked to original sources

arXiv2026

Reconstructing resonant phase oscillator interactions from noisy time series

We present a method for reconstructing resonant interactions in weakly coupled phase oscillator systems from noisy time series. Instead of attempting to recover the full phase equations, which may be non-identifiable in the presence of bounded observational uncertainty, the method reconstructs the resonant normal form terms that determine the leading-order drift dynamics. We develop first-order and second-order reconstruction procedures based on finite libraries of resonant Fourier modes and least-squares estimation. We prove error bounds for the reconstructed coefficients under natural assumptions on the observation noise and the distribution of initial conditions. The second-order method detects effective resonant interactions generated by the interplay of nonresonant first-order couplings. Numerical examples illustrate the reconstruction of resonant subnetworks and emergent higher-order interactions.

nlin.CD↗

arXiv2025

Emergence of higher-order interactions in systems of coupled Kuramoto oscillators with time delay

We show that higher-order interactions naturally emerge from time-delayed pairwise coupling in Kuramoto oscillators. By expanding the delayed pairwise coupling to the second order, we derive a delay-free Kuramoto model possessing both pairwise and three-body interactions. Numerical simulations and stability analysis demonstrate that the three-body Kuramoto model and the time-delayed pairwise Kuramoto model exhibit qualitatively consistent synchronization transitions under appropriate conditions. In particular, the bistability arising in the time-delayed Kuramoto model is accounted for by the three-body interactions. Our findings reveal that time delays can be recast effectively as higher-order interactions, providing an insight into how coupling delays shape collective dynamics.

nlin.AO↗

arXiv2025

Homogeneous coupled cell systems with high-dimensional internal dynamics

The analysis of network dynamics is oftentimes restricted to networks with one-dimensional internal dynamics. Here, we show how symmetry explains the relation between behavior of systems with one-dimensional internal dynamics and with higher dimensional internal dynamics, when the network topology is the same. Fundamental networks of homogeneous coupled cell systems (B. Rink, J. Sanders. Coupled Cell Networks and Their Hidden Symmetries. SIAM J. Math. Anal. 46.2 (2014)) can be expressed in terms of monoid representations, which uniquely decompose into indecomposable subrepresentations. In the high-dimensional internal dynamics case, these subrepresentations are isomorphic to multiple copies of those one computes in the one-dimensional case. We describe the implications of this observation on steady state and Hopf bifurcations in $l$-parameter families of network vector fields. The main results are that (1) generic one-parameter steady state bifurcations are qualitatively independent of the dimension of the internal dynamics and that, (2) in order to observe all generic $l$-parameter bifurcations that may occur for internal dynamics of any dimension, the internal dynamics has to be at least $l$-dimensional for steady state bifurcations and $2l$-dimensional for Hopf bifurcations. Furthermore, we illustrate how additional structure in the network can be exploited to obtain understanding beyond qualitative statements about the collective dynamics. One-parameter steady state bifurcations in feedforward networks exhibit an unusual amplification in the asymptotic growth rates of individual cells, when these are one-dimensional (S. von der Gracht, E. Nijholt, B. Rink. Amplified steady state bifurcations in feedforward networks. Nonlinearity 35.4 (2022)). We prove that (3) the same cells exhibit this amplifying effect with the same growth rates when the internal dynamics is high-dimensional.

math.DS↗

arXiv2025

A coordinate-independent Pontryagin-Rodygin theorem for slow-fast averaging

The slow drift along a manifold of periodic orbits is a key mathematical structure underlying bursting dynamics in many scientific applications. While classical averaging theory, as formalised by the Pontryagin-Rodygin theorem, provides a leading-order approximation for this slow drift, the connection to the underlying geometry described by Geometric Singular Perturbation Theory (GSPT)--also known as Fenichel theory--is often not explicit, particularly at higher orders. This paper makes that connection rigorous and constructive using the parametrisation method. We provide a detailed, self-contained exposition of this functional analytic technique, showing how it synthesizes the geometric insight of invariant manifold theory with a systematic, perturbative algorithm. By treating the manifold's embedding and the reduced flow as coupled unknowns, the method provides a constructive proof of an averaged system that is guaranteed to be geometrically consistent with the persistence of the normally hyperbolic manifold to any order. We translate the abstract theory into a concrete computational procedure using Floquet theory, spectral analysis, and the Fredholm alternative, yielding a practical guide for computing high-accuracy, higher-order averaged models, and we demonstrate its implementation, both analytically and numerically, through specific examples.

math.DS↗

arXiv2025

Non-Birkhoff periodic orbits in symmetric billiards

We study non-Birkhoff periodic orbits in symmetric convex planar billiards. Our main result provides a quantitative criterion for the existence of such orbits with prescribed minimal period, rotation number, and spatiotemporal symmetry. We exploit this criterion to find sufficient conditions for a symmetric billiard to possess infinitely many non-Birkhoff periodic orbits. It follows that arbitrarily small analytical perturbations of the circular billiard have non-Birkhoff periodic orbits of any rational rotation number and with arbitrarily long periods. We also generalize a known result for elliptical billiards to other $\mathbb{D}_2$-symmetric billiards. Lastly, we provide Matlab codes which can be used to numerically compute and visualize the non-Birkhoff periodic orbits whose existence we prove analytically.

math.DS↗

arXiv2025

Quasicrystals in pattern formation. Part II: Spatially almost-periodic profiles and global existence

This paper continues our study of quasicrystals initiated in Part I. We propose a general mechanism for constructing quasicrystals, existing globally in time, in spatially-extended systems (partial differential equations with Euclidean symmetry) and demonstrate it on model examples of the Swift-Hohenberg and Brusselator equations. In contrast to Part I, our approach here emphasises the theory of almost-periodic functions as well as the global solvability of the corresponding equations in classes of spatially non-decaying functions. We note that the existence of such time-evolving quasicrystals with rotational symmetry of all orders, icosahedral symmetry, etc., does not require technical issues such as Diophantine properties and hard implicit function theorems, which look unavoidable in the case of steady-state quasicrystals. This paper can be largely read independently of Part I. Background material and definitions are repeated for convenience, but some elementary calculations from Part I are omitted.

math.DS↗

arXiv2024

Quasicrystals in pattern formation, Part I: Local existence and basic properties

In this paper, we propose a general mechanism for the existence of quasicrystals in spatially extended systems (partial differential equations with Euclidean symmetry). We argue that the existence of quasicrystals with higher order rotational symmetry, icosahedral symmetry, etc, is a natural and universal consequence of spontaneous symmetry breaking, bypassing technical issues such as Diophantine properties and hard implicit function theorems. The diffraction diagrams associated with these quasicrystal solutions are not Delone sets, so strictly speaking they do not conform to the definition of a ``mathematical quasicrystal''. But they do appear to capture very well the features of the diffraction diagrams of quasicrystals observed in nature. For the Swift-Hohenberg equation, we obtain more detailed information, including that the $\ell^2$ norm of the diffraction diagram grows like the square root of the bifurcation parameter.

nlin.PS↗

arXiv2024

Dynamics on invariant tori emerging through forced symmetry breaking in phase oscillator networks

We consider synchrony patterns in coupled phase oscillator networks that correspond to invariant tori. For specific nongeneric coupling, these tori are equilibria relative to a continuous symmetry action. We analyze how the invariant tori deform under forced symmetry breaking as more general network interaction terms are introduced. We first show in general that perturbed tori that are relative equilibria can be computed using a parametrization method; this yields an asymptotic expansion of an embedding of the perturbed torus, as well as the local dynamics on the torus. We then apply this result to a coupled oscillator network, and we numerically study the dynamics on the persisting tori in the network by looking for bifurcations of their periodic orbits in a boundary-value-problem setup. This way we find new bifurcating stable synchrony patterns that can be the building blocks of larger global structures such as heteroclinic cycles.

math.DS↗

arXiv2023

Higher order interactions lead to "reluctant" synchrony breaking

To model dynamical systems on networks with higher order (non-pairwise) interactions, we recently introduced a new class of ODEs on hypernetworks. Here we consider one-parameter synchrony breaking bifurcations in such ODEs. We call a synchrony breaking steady state branch ``reluctant'' if it is tangent to a synchrony space, but does not lie inside it. We prove that reluctant synchrony breaking is ubiquitous in hypernetwork systems, by constructing a large class of examples that support it. We also give an explicit formula for the order of tangency to the synchrony space of a reluctant steady state branch.

math.DS↗

arXiv2023

A parametrisation method for high-order phase reduction in coupled oscillator networks

We present a novel method for high-order phase reduction in networks of weakly coupled oscillators and, more generally, perturbations of reducible normally hyperbolic (quasi-)periodic tori. Our method works by computing an asymptotic expansion for an embedding of the perturbed invariant torus, as well as for the reduced phase dynamics in local coordinates. Both can be determined to arbitrary degrees of accuracy, and we show that the phase dynamics may directly be obtained in normal form. We apply the method to predict remote synchronisation in a chain of coupled Stuart-Landau oscillators.

math.DS↗

arXiv2023

Hypernetworks: cluster synchronisation is a higher-order effect

Many networked systems are governed by non-pairwise interactions between nodes. The resulting higher-order interaction structure can then be encoded by means of a hypernetwork. In this paper we consider dynamical systems on hypernetworks by defining a class of admissible maps for every such hypernetwork. We explain how to classify robust cluster synchronisation patterns on hypernetworks by finding balanced partitions, and we generalize the concept of a graph fibration to the hypernetwork context. We also show that robust synchronisation patterns are only fully determined by polynomial admissible maps of high order. This means that, unlike in dyadic networks, cluster synchronisation on hypernetworks is a higher-order, i.e., nonlinear effect. We give a formula, in terms of the order of the hypernetwork, for the degree of the polynomial admissible maps that determine robust synchronisation patterns. We also demonstrate that this degree is optimal by investigating a class of examples. We conclude by displaying how this effect may cause remarkable synchrony breaking bifurcations that occur at high polynomial degree.

math.DS↗

arXiv2021

Amplified steady state bifurcations in feedforward networks

We investigate bifurcations in feedforward coupled cell networks. Feedforward structure (the absence of feedback) can be defined by a partial order on the cells. We use this property to study generic one-parameter steady state bifurcations for such networks. Branching solutions and their asymptotics are described in terms of Taylor coefficients of the internal dynamics. They can be determined via an algorithm that only exploits the network structure. Similar to previous results on feedforward chains, we observe amplifications of the growth rates of steady state branches induced by the feedforward structure. However, contrary to these earlier results, as the interaction scenarios can be more complicated in general feedforward networks, different branching patterns and different amplifications can occur for different regions in the space of Taylor coefficients.

math.DS↗

arXiv2020

Korteweg-de Vries and Fermi-Pasta-Ulam-Tsingou: asymptotic integrability of quasi unidirectional waves

In this paper we construct a higher order expansion of the manifold of quasi unidirectional waves in the Fermi-Pasta-Ulam (FPU) chain. We also approximate the dynamics on this manifold. As perturbation parameter we use $h^2=1/n^2$, where $n$ is the number of particles of the chain. It is well known that the dynamics of quasi unidirectional waves is described to first order by the Korteweg-de Vries (KdV) equation. Here we show that the dynamics to second order is governed by a combination of the first two nontrivial equations in the KdV hierarchy -- for any choice of parameters in the FPU potential. On the other hand, we find that only if the parameters of the FPU potential satisfy a condition, then a combination of the first three nontrivial equations in the KdV hierarchy determines the dynamics of quasi unidirectional waves to third order. The required condition is satisfied by the Toda chain. Our results suggest why the close-to-integrable behavior of the FPU chain (the FPU paradox) persists on a time scale longer than explained by the KdV approximation, and also how a breakdown of integrability (detachment from the KdV hierarchy) may responsible for the eventual thermalization of the system.

math-ph↗

arXiv2020

Multiple timescales and the parametrisation method in geometric singular perturbation theory

We present a novel method for computing slow manifolds and their fast fibre bundles in geometric singular perturbation problems. This coordinate-independent method is inspired by the parametrisation method introduced by Cabr\'e, Fontich and de la Llave. By iteratively solving a so-called conjugacy equation, our method simultaneously computes parametrisations of slow manifolds and fast fibre bundles, as well as the dynamics on these objects, to arbitrarily high degrees of accuracy. We show the power of this top-down method for the study of systems with multiple (i.e., three or more) timescales. In particular, we highlight the emergence of hidden timescales and show how our method can uncover these surprising multiple timescale structures. We also apply our parametrisation method to several reaction network problems.

math.DS↗

arXiv2020

Quiver representations and dimension reduction in dynamical systems

Dynamical systems often admit geometric properties that must be taken into account when studying their behaviour. We show that many such properties can be encoded by means of quiver representations. These properties include classical symmetry, hidden symmetry and feedforward structure, as well as subnetwork and quotient relations in network dynamical systems. A quiver equivariant dynamical system consists of a collection of dynamical systems with maps between them that send solutions to solutions. We prove that such quiver structures are preserved under Lyapunov-Schmidt reduction, center manifold reduction, and normal form reduction.

math.DS↗

arXiv2020

More on the parameterization method for center manifolds

In a previous paper we generalized the parameterization method of Cabré, Fontich and De la Llave to center manifolds of discrete dynamical systems. In this paper, we extend this result to several different settings. The natural setting in which center manifolds occur is at bifurcations in dynamical systems with parameters. Our first results will show that we can find parameter-dependent center manifolds near bifurcation points. Furthermore, we will generalize the parameterization method to center manifolds of fixed points of ODEs. Finally, we will apply our method to a reaction diffusion equation. In our application, we will show that the freedom to obtain the conjugate dynamics in normal form makes it possible to obtain detailed qualitative information about the center dynamics.

math.DS↗

arXiv2019

A new algorithm for computing idempotents of R-trivial monoids

The authors of [Primitive orthogonal idempotents for R-trivial monoids, Journal of Algebra] provide an algorithm for finding a complete system of primitive orthogonal idempotents for CM, where M is any finite R-trivial monoid. Their method relies on a technical result stating that R-trivial monoids are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an R-trivial monoid may be realised by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where L-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We also show that our algorithm only has to be performed for ZM, after which a complete system of primitive orthogonal idempotents may be obtained for RM, where R is any ring with a known complete system of primitive orthogonal idempotents. In particular, the algorithm works if R is any field.

math.RT↗

arXiv2019

The parameterization method for center manifolds

In this paper, we present a generalization of the parameterization method, introduced by Cabré, Fontich and De la Llave, to center manifolds associated to non-hyperbolic fixed points of discrete dynamical systems. As a byproduct, we find a new proof for the existence and regularity of center manifolds. However, in contrast to the classical center manifold theorem, our parameterization method will simultaneously obtain the center manifold and its conjugate center dynamical system. Furthermore, we will provide bounds on the error between approximations of the center manifold and the actual center manifold, as well as bounds for the error in the conjugate dynamical system.

math.DS↗