arXiv subjects

Marcus A. M. de Aguiar

Publications and source records attributed to Marcus A. M. de Aguiar.

At least 37 records · Page 2Linked to original sources

arXiv2024

Kuramoto variables as eigenvalues of unitary matrices

We generalize the Kuramoto model by interpreting the $N$ variables on the unit circle as eigenvalues of a $N$-dimensional unitary matrix $U$, in three versions: general unitary, symmetric unitary and special orthogonal. The time evolution is generated by $N^2$ coupled differential equations for the matrix elements of $U$, and synchronization happens when $U$ evolves into a multiple of the identity. The Ott-Antonsen ansatz is related to the Poisson kernels that are so useful in quantum transport, and we prove it in the case of identical natural frequencies. When the coupling constant is a matrix, we find some surprising new dynamical behaviors.

nlin.PS↗

arXiv2024

Chaos in undamped, forced oscillators via stroboscopic maps

Non-linear dynamics is not a usually covered topic in undergraduate physics courses. However, its importance within classical mechanics and the general theory of dynamical systems is unquestionable. In this work we show that this subject can be included in the schedule of an introductory classical mechanics course without the need to develop a robust theory of chaotic dynamics. To do this, we take as examples conservative non-linear oscillators subject to time-dependent periodic forces. By introducing the concept of stroboscopic maps we show that it is possible to visualize the appearance of chaos in these systems. We also address the example of the forced simple pendulum applying the same treatment. Finally, we briefly comment on the more general theory of chaos in conservative Hamiltonian systems.

physics.class-ph↗

arXiv2024

Dynamics of matrix coupled Kuramoto oscillators on modular networks: excitable behavior and global decoherence

Synchronization is observed in many natural systems, with examples ranging from neuronal activation to walking pedestrians. The models proposed by Winfree and Kuramoto stand as the classic frameworks for investigating these phenomena. The Kuramoto model, in particular, has been extended in different ways since its original formulation to account for more general scenarios. One such extension replaces the coupling parameter with a coupling matrix, describing a form of generalized frustration with broken rotational symmetry. A key feature of this model is the existence of {\it phase tuned states}, characterized by having the phase of the order parameter pointing in the direction of the dominant eigenvector of the coupling matrix. Here we investigate the matrix coupled Kuramoto model on networks with two modules, such that one module is in the phase tuned state and the other in a state where the order parameter rotates. We identified different regimes in which one or the other module dominates the dynamics. We found, in particular, that the phase tuned module can create a bottleneck for the oscillation of the rotating module, leading to a behavior similar to the charge and fire regimes of excitable systems. We also found an extended region in the parameter space where motion is globally disordered, even though one of the modules presented high levels of synchronization when uncoupled.

physics.soc-ph↗

arXiv2024

Third order interactions shift the critical coupling in multidimensional Kuramoto models

The study of higher order interactions in the dynamics of Kuramoto oscillators has been a topic of intense recent research. Arguments based on dimensional reduction using the Ott-Antonsen ansatz show that such interactions usually facilitate synchronization, giving rise to bi-stability and hysteresis. Here we show that three body interactions shift the critical coupling for synchronization towards higher values in all dimensions, except $D=2$, where a cancellation occurs. After the transition, three and four body interactions combine to facilitate synchronization. Similar to the 2-dimensional case, bi-stability and hysteresis develop for large enough higher order interactions. We show simulations in $D=3$ and $4$ to illustrate the dynamics.

nlin.PS↗

arXiv2024

Nuclear compensatory evolution driven by mito-nuclear incompatibilities

Mitochondrial function relies on the coordinated expression of mitochondrial and nuclear genes, exhibiting remarkable resilience regardless the susceptibility of mitochondrial DNA (mtDNA) to accumulate harmful mutations. A suggested mechanism for preserving this mito-nuclear compatibility is the nuclear compensation, where deleterious mitochondrial alleles drive compensatory changes in nuclear genes. However, prevalence and conditioning factors for this phenomenon remain debated, with empirical evidence supporting and refuting its existence. Here, we investigate how mito-nuclear incompatibilities impact nuclear and mitochondrial substitutions in a model for species radiation under selection for mito-nuclear compatibility, similar to the process of mtDNA introgression. Mating eligibility relies on genetic (nuclear DNA) and spatial proximity, with populations evolving from partially compatible mito-nuclear states. Mutations do not confer advantages nor disadvantages, with no optimal nuclear or mitochondrial types, but individual fitness decreases with increasing incompatibilities, driving the demand for mito-nuclear genetic coordination. We find that selection consistently promotes compensation on incompatible nuclear genes, resulting in more substitutions than compatible or non-interacting genes. Surprisingly, low mitochondrial mutation rates favor compensation, as do increased selective pressure or a higher number of mismatches. High mitochondrial mutation rates boost substitutions in initially compatible nuclear genes, relaxing the selection against mito-nuclear incompatibilities and mirroring the compensatory evolution. Moreover, the presence of incompatibilities accelerates species radiation, but richness at equilibrium is not directly correlated with substitutions' response, revealing the complex dynamics triggered by mitochondrial introgression and mito-nuclear co-evolution.

q-bio.PE↗

arXiv2023

A solvable two-dimensional swarmalator model

Swarmalators are oscillators that swarm through space as they synchronize in time. Introduced a few years ago to model many systems which mix synchrony with self-assembly, they remain poorly understood theoretically. Here we obtain the first analytic results on swarmalators moving in two-dimensional (2D) plane by enforcing periodic boundary conditions; this simpler topology allows expressions for order parameters, stabilities, and bifurcations to be derived exactly. We suggest some future directions for swarmalator research and point out some connections to the Kuramoto model and the Vicsek model from active matter; these are intended as a call-to-arms for the sync community and other researchers looking for new problems and puzzles to work on.

nlin.AO↗

arXiv2023

Order, chaos, and dimensionality transition in a system of swarmalators

Similar to sperm, where individuals self-organize in space while also striving for coherence in their tail swinging, several natural and engineered systems exhibit the emergence of swarming and synchronization. The arising and interplay of these phenomena have been captured by collectives of hypothetical particles named swarmalators, each possessing a position and a phase whose dynamics are affected reciprocally and also by the space-phase states of their neighbors. In this work, we introduce a solvable model of swarmalators able to move in two-dimensional spaces. We show that several static and active collective states can emerge and derive necessary conditions for each to show up as the model parameters are varied. These conditions elucidate, in some cases, the displaying of multistability among states. Notably, in the active regime, individuals behave chaotically, maintaining spatial correlation under certain conditions, and breaking it under others on what we interpret as a dimensionality transition.

nlin.AO↗

arXiv2023

Exploring the phase diagrams of multidimensional Kuramoto models

The multidimensional Kuramoto model describes the synchronization dynamics of particles moving on the surface of D-dimensional spheres, generalizing the original model where particles were characterized by a single phase. In this setup, particles are more easily represented by $D$-dimensional unit vectors than by $D-1$ spherical angles, allowing for the coupling constant to be extended to a coupling matrix acting on the vectors. As in the original Kuramoto model, each particle has a set of $D(D-1)/2$ natural frequencies, drawn from a distribution. The system has a large number of independent parameters, given by the average natural frequencies, the characteristic widths of their distributions plus $D^2$ constants of the coupling matrix. General phase diagrams, indicating regions in parameter space where the system exhibits different behaviors, are hard to derive analytically. Here we obtain the complete phase diagram for $D=2$ and Lorentzian distributions of natural frequencies using the Ott-Antonsen ansatz. We also explore the diagrams numerically for different distributions and some specific choices of parameters for $D=2$, $D=3$ and $D=4$. In all cases the system exhibits at most four different phases: disordered, static synchrony, rotation and active synchrony. Existence of specific phases and boundaries between them depend strongly on the dimension $D$, the coupling matrix and the distribution of natural frequencies.

nlin.PS↗

arXiv2023

On the numerical integration of the multidimensional Kuramoto model

The Kuramoto model, describing the synchronization dynamics of coupled oscillators, has been generalized in many ways over the past years. One recent extension of the model replaces the oscillators, originally characterized by a single phase, by particles with D-1 internal phases, represented by a point on the surface of the unit D-sphere. Particles are then more easily represented by D-dimensional unit vectors than by D-1 spherical angles. However, numerical integration of the state equations should ensure that the propagated vectors remain unit and that particles rotate on the sphere as predicted by the dynamical equations. As discussed in [1] integration of the three-dimensional Kuramoto model using Euler's method with time step $Δt$ not only changes the norm of the vectors but produces a small rotation of the particles around the wrong axis. Importantly, the error in the axis' direction does not vanish in the limit $Δt \rightarrow 0$. Therefore, instead of displacing the unit vectors in the direction of the velocity one should performed a sequence of direct small rotations, as dictated by the equations of motion. This keeps the particles on the sphere at all times, ensuring exact norm preservation, and rotates the particles around the proper axis for small $Δt$ [1]. Here I propose an alternative way to do such integration by rotations in 3D that can be generalized to more dimensions using Cayley-Hamilton's theorem. Explicit formulas are provided for 2, 3 and 4 dimensions. I also compare the results with the forth order Runge-Kutta method, which seems to provide accurate results even requiring renormalization of the vectors after each integration step.

nlin.AO↗

arXiv2023

Intermittent migration can induce pulses of speciation in a two-island system

Geographic barriers prevent migration between populations, thereby facilitating speciation through allopatry. However, these barriers can exhibit dynamic behavior in nature, promoting cycles of expansion and contraction of populations. Such oscillations cause temporal variations in migration that do not necessarily prevent speciation; on the contrary, they have been suggested as a driving force for diversification. Here we present a study on a two-island neutral speciation model in scenarios with intermittent migration driven by sea-level fluctuations. Mating is constrained to genetically compatible individuals inhabiting the same island, and offspring inherit nuclear genomes from both parents with recombination. We observe pulses of speciation that would not occur in strict isolation or continuous migration. According to the seabed height, which modulates the duration of the isolation and connection periods, the maximum richness occurs at different times and in an ephemeral fashion. The expansion-contraction dynamics can accelerate diversification, but a long time in isolation can reduce the richness to one species per island, resembling patterns described by the taxon pulse hypothesis of diversification. Together with other studies, our results support the relevance of research on the impact of variable migration on diversification, suggested to be related to regions of high diversity.

q-bio.PE↗

arXiv2023

Sakaguchi Swarmalators

Swarmalators are phase oscillators that cluster in space, like fireflies flashing on a swarm to attract mates. Interactions between particles, which tend to synchronize their phases and align their motion, decrease with the distance and phase difference between them, coupling the spatial and phase dynamics. In this work, we explore the effects of disorder induced by phase frustration on a system of Swarmalators that move on a one-dimensional ring. Our model is inspired by the well-known Kuramoto-Sakaguchi equations. We find, numerically and analytically, the ordered and disordered states that emerge in the system. The active states, not present in the model without disorder, resemble states found previously in numerical studies for the 2D Swarmalators system. One of these states, in particular, shows similarities to turbulence generated in a flattened media. We show that all ordered states can be generated for any values of the coupling constants by tuning the phase frustration parameters only. Moreover, many of these combinations display multi-stability.

nlin.PS↗

arXiv2023

Generalized frustration in the multidimensional Kuramoto model

The Kuramoto model was recently extended to arbitrary dimensions by reinterpreting the oscillators as particles moving on the surface of unit spheres in a D-dimensional space. Each particle is then represented by a D-dimensional unit vector. For $D=2$ the particles move on the unit circle and the vectors can be described by a single phase, recovering the original Kuramoto model. This multidimensional description can be further extended by promoting the coupling constant between the particles to a matrix that acts on the unit vectors, representing a type of generalized frustration. In a recent paper we have analyzed in detail the role of the coupling matrix for $D=2$. Here we extend this analysis to arbitrary dimensions, presenting a study of synchronous states and their stability. We show that when the natural frequencies of the particles are set to zero, the system converges either to a stationary synchronized state with well defined phase, or to an effective two-dimensional dynamics, where the synchronized particles rotate on the sphere. The stability of these states depend on the eigenvalues and eigenvectors of the coupling matrix. When the natural frequencies are not zero, synchronization depends on whether $D$ is even or odd. In even dimensions the transition to synchronization is continuous and rotating states are replaced by active states, where the order parameter rotates while its module oscillates. If $D$ is odd the phase transition is discontinuous and active states are suppressed, occurring only for a restricted class of coupling matrices.

nlin.CD↗

arXiv2022

Matrix coupling and generalized frustration in Kuramoto oscillators

The Kuramoto model describes the synchronization of coupled oscillators that have different natural frequencies. Among the many generalizations of the original model, Kuramoto and Sakaguchi (KS) proposed a {\it frustrated} version that resulted in dynamic behavior of the order parameter, even when the average natural frequency of the oscillators is zero. Here we consider a generalization of the frustrated KS model that exhibits new transitions to synchronization. The model is identical in form to the original Kuramoto model, but written in terms of unit vectors. Replacing the coupling constant by a coupling matrix breaks the rotational symmetry and forces the order parameter to point in the direction of the eigenvector with highest eigenvalue, when the eigenvalues are real. For complex eigenvalues the module of order parameter oscillates while it rotates around the unit circle, creating active states. We derive the complete phase diagram for Lorentzian distribution of frequencies using the Ott-Antonsen ansatz. We also show that changing the average value of the natural frequencies leads to further phase transitions where the module of the order parameter goes from oscillatory to static.

nlin.AO↗

arXiv2022

Mito-nuclear selection induces a trade-off between species ecological dominance and evolutionary lifespan

Mitochondrial and nuclear genomes must be co-adapted to ensure proper cellular respiration and energy production. Mito-nuclear incompatibility reduces individual fitness and induces hybrid infertility, suggesting a possible role in reproductive barriers and speciation. Here we develop a birth-death model for evolution in spatially extended populations under selection for mito-nuclear co-adaptation. Mating is constrained by physical and genetic proximity, and offspring inherit nuclear genomes from both parents, with recombination. The model predicts macroscopic patterns including a community's long-term species diversity, its species abundance distribution, speciation and extinction rates, as well as intra- and inter-specific genetic variation. We explore how these long-term outcomes depend upon the microscopic parameters of reproduction: individual fitness governed by mito-nuclear compatibility, constraints on mating compatibility, and ecological carrying capacity. We find that strong selection for mito-nuclear compatibility reduces the equilibrium number of species after a radiation, increases the species' abundances, while simultaneously increasing both speciation and extinction rates. The negative correlation between species diversity and diversification rates in our model agrees with the broad empirical pattern of lower species diversity and higher speciation/extinction rates in temperate regions, compared to the tropics. We therefore suggest that these empirical patterns may be caused in part by latitudinal variation in metabolic demands, and corresponding variation in selection on mito-nuclear function.

q-bio.PE↗

arXiv2022

Diversity patterns and speciation processes in a two-island system with continuous migration

Geographic isolation is a central mechanism of speciation, but perfect isolation of populations is rare. Although speciation can be hindered if gene flow is large, intermediate levels of migration can enhance speciation by introducing genetic novelty in the semi-isolated populations or founding small communities of migrants. Here we consider a two island neutral model of speciation with continuous migration and study diversity patterns as a function of the migration probability, population size, and number of genes involved in reproductive isolation (dubbed as genome size). For small genomes, low levels of migration induce speciation on the islands that otherwise would not occur. Diversity, however, drops sharply to a single species inhabiting both islands as the migration probability increases. For large genomes, sympatric speciation occurs even when the islands are strictly isolated. Then species richness per island increases with the probability of migration, but the total number of species decreases as they become cosmopolitan. For each genome size, there is an optimal migration intensity for each population size that maximizes the number of species. We discuss the observed modes of speciation induced by migration and how they increase species richness in the insular system while promoting asymmetry between the islands and hindering endemism.

q-bio.PE↗

arXiv2021

Shannon information criterion for low-high diversity transition in Moran and Voter models

Mutation and drift play opposite roles in genetics. While mutation creates diversity, drift can cause gene variants to disappear, especially when they are rare. In the absence of natural selection and migration, the balance between the drift and mutation in a well-mixed population defines its diversity. The Moran model captures the effects of these two evolutionary forces and has a counterpart in social dynamics, known as the Voter model with external opinion influencers. Two extreme outcomes of the Voter model dynamics are consensus and coexistence of opinions, which correspond to low and high diversity in the Moran model. Here we use a Shannon's information-theoretic approach to characterize the smooth transition between the states of consensus and coexistence of opinions in the Voter model. Mapping the Moran into the Voter model we extend the results to the mutation-drift balance and characterize the transition between low and high diversity in finite populations. Describing the population as a network of connected individuals we show that the transition between the two regimes depends on the network topology of the population and on the possible asymmetries in the mutation rates.

q-bio.PE↗

arXiv2021

Ott-Antonsen ansatz for the D-dimensional Kuramoto model: a constructive approach

Kuramoto's original model describes the dynamics and synchronization behavior of a set of interacting oscillators represented by their phases. The system can also be pictured as a set of particles moving on a circle in two dimensions, which allows a direct generalization to particles moving on the surface of higher dimensional spheres. One of the key features of the 2D system is the presence of a continuous phase transition to synchronization as the coupling intensity increases. Ott and Antonsen proposed an ansatz for the distribution of oscillators that allowed them to describe the dynamics of the transition's order parameter with a single differential equation. A similar ansatz was later proposed for the D-dimensional model by using the same functional form of the 2D ansatz and adjusting its parameters. In this paper we develop a constructive method to find the ansatz, similarly to the procedure used in 2D. The method is based on our previous work for the 3D Kuramoto model where the ansatz was constructed using the spherical harmonics decomposition of the distribution function. In the case of motion in a D-dimensional sphere the ansatz is based on the hyperspherical harmonics decomposition. Our result differs from the previously proposed ansatz and provides a simpler and more direct connection between the order parameter and the ansatz.

nlin.CD↗

arXiv2021

Modeling viral mutations in the spread of epidemics

Although traditional models of epidemic spreading focus on the number of infected, susceptible and recovered individuals, a lot of attention has been devoted to integrate epidemic models with population genetics. Here we develop an individual-based model for epidemic spreading on networks in which viruses are explicitly represented by finite chains of nucleotides that can mutate inside the host. Under the hypothesis of neutral evolution we compute analytically the average pairwise genetic distance between all infecting viruses over time. We also derive a mean-field version of this equation that can be added directly to compartmental models such as SIR or SEIR to estimate the genetic evolution. We compare our results with the inferred genetic evolution of SARS-CoV-2 at the beginning of the epidemic in China and found good agreement with the analytical solution of our model. Finally, using genetic distance as a proxy for different strains, we use numerical simulations to show that the lower the connectivity between communities, e.g., cities, the higher the probability of reinfection.

q-bio.PE↗