arXiv subjects

Per Sebastian Skardal

Publications and source records attributed to Per Sebastian Skardal.

At least 55 records · Page 3Linked to original sources

arXiv2017

Collective phenomena emerging from the interactions between dynamical processes in multiplex networks

We introduce a framework to intertwine dynamical processes of different nature, each with its own distinct network topology, using a multilayer network approach. As an example of collective phenomena emerging from the interactions of multiple dynamical processes, we study a model where neural dynamics and nutrient transport are bidirectionally coupled in such a way that the allocation of the transport process at one layer depends on the degree of synchronization at the other layer, and vice versa. We show numerically, and we prove analytically, that the multilayer coupling induces a spontaneous explosive synchronization and a heterogeneous distribution of allocations, otherwise not present in the two systems considered separately. Our framework can find application to other cases where two or more dynamical processes such as synchronization, opinion formation, information diffusion, or disease spreading, are interacting with each other.

nlin.AO↗

arXiv2016

Evolving network structure of academic institutions

Today's colleges and universities consist of highly complex structures that dictate interactions between the administration, faculty, and student body. These structures can play a role in dictating the efficiency of policy enacted by the administration and determine the effect that curriculum changes in one department have on other departments. Despite the fact that the features of these complex structures have a strong impact on the institutions, they remain by-and-large unknown in many cases. In this paper we study the academic structure of our home institution of Trinity College in Hartford, CT using the major and minor patterns between graduating students to build a temporal multiplex network describing the interactions between different departments. Using recent network science techniques developed for such temporal networks we identify the evolving community structures that organize departments' interactions, as well as quantify the interdisciplinary centrality of each department. We implement this framework for Trinity College, finding practical insights and applications, but also present it as a general framework for colleges and universities to better understand their own structural makeup in order to better inform academic and administrative policy.

physics.soc-ph↗

arXiv2016

Optimal phase synchronization in networks of phase-coherent chaotic oscillators

We investigate the existence of an optimal interplay between the natural frequencies of a group chaotic oscillators and the topological properties of the network they are embedded in. We identify the conditions for achieving phase synchronization in the most effective way, i.e., with the lowest possible coupling strength. Specifically, we show by means of numerical and experimental results that it is possible to define a synchrony alignment function linking the natural frequencies of a set of non-identical phase-coherent chaotic oscillators with the topology of the Laplacian matrix $L$, the latter accounting for the specific organization of the network of interactions between oscillators. We use the classical Rössler system to show that the synchrony alignment function obtained for phase oscillators can be extended to phase-coherent chaotic systems. Finally, we carry out a series of experiments with nonlinear electronic circuits to show the robustness of the theoretical predictions despite the intrinsic noise and parameter mismatch of the electronic components.

nlin.AO↗

arXiv2016

Spectral properties of the hierarchical product of graphs

The hierarchical product of two graphs represents a natural way to build a larger graph out of two smaller graphs with less regular and therefore more heterogeneous structure than the Cartesian product. Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. Introducing a coupling parameter describing the relative contribution of each of the two smaller graphs, we perform an asymptotic analysis for the full spectrum of eigenvalues of the adjacency matrix of the hierarchical product. Specifically, we derive the exact limit points for each eigenvalue in the limits of small and large coupling, as well as the leading-order relaxation to these values in terms of the eigenvalues and eigenvectors of the two smaller graphs. Given its central roll in the structural and dynamical properties of networks, we study in detail the Perron-Frobenius, or largest, eigenvalue. Finally, as an example application we use our theory to predict the epidemic threshold of the Susceptible-Infected-Susceptible model on a hierarchical product of two graphs.

nlin.AO↗

arXiv2016

Synchronization of heterogeneous oscillators under network modifications: Perturbation and optimization of the synchrony alignment function

Synchronization is central to many complex systems in engineering physics (e.g., the power-grid, Josephson junction circuits, and electro-chemical oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms). Despite these widespread applications---for which proper functionality depends sensitively on the extent of synchronization---there remains a lack of understanding for how systems evolve and adapt to enhance or inhibit synchronization. We study how network modifications affect the synchronization properties of network-coupled dynamical systems that have heterogeneous node dynamics (e.g., phase oscillators with non-identical frequencies), which is often the case for real-world systems. Our approach relies on a synchrony alignment function (SAF) that quantifies the interplay between heterogeneity of the network and of the oscillators and provides an objective measure for a system's ability to synchronize. We conduct a spectral perturbation analysis of the SAF for structural network modifications including the addition and removal of edges, which subsequently ranks the edges according to their importance to synchronization. Based on this analysis, we develop gradient-descent algorithms to efficiently solve optimization problems that aim to maximize phase synchronization via network modifications. We support these and other results with numerical experiments.

nlin.AO↗

arXiv2016

Optimal synchronization of directed complex networks

We study optimal synchronization of networks of coupled phase oscillators. We extend previous theory for optimizing the synchronization properties of undirected networks to the important case of directed networks. We derive a generalized synchrony alignment function that encodes the interplay between network structure and the oscillators' natural frequencies and serves as an objective measure for the network's degree of synchronization. Using the generalized synchrony alignment function, we show that a network's synchronization properties can be systematically optimized. This framework also allows us to study the properties of synchrony-optimized networks, and in particular, investigate the role of directed network properties such as nodal in- and out-degrees. For instance, we find that in optimally rewired networks the heterogeneity of the in-degree distribution roughly matches the heterogeneity of the natural frequency distribution, but no such relationship emerges for out-degrees. We also observe that a network's synchronization properties are promoted by a strong correlation between the nodal in-degrees and the natural frequencies of oscillators, whereas the relationship between the nodal out-degrees and the natural frequencies has comparatively little effect. This result is supported by our theory, which indicates that synchronization is promoted by a strong alignment of the natural frequencies with the left singular vectors corresponding to the largest singular values of the Laplacian matrix.

nlin.AO↗

arXiv2016

On controlling networks of limit-cycle oscillators

The control of network-coupled nonlinear dynamical systems is an active area of research in the nonlinear science community. Coupled oscillator networks represent a particularly important family of nonlinear systems, with applications ranging from the power grid to cardiac excitation. Here we study the control of network-coupled limit cycle oscillators, extending previous work that focused on phase oscillators. Based on stabilizing a target fixed point, our method aims to attain complete frequency synchronization, i.e., consensus, by applying control to as few oscillators as possible. We develop two types of control. The first type directs oscillators towards to larger amplitudes, while the second does not. We present numerical examples of both control types and comment on the potential failures of the method.

nlin.AO↗

arXiv2016

Collective frequency variation in network synchronization and reverse PageRank

A wide range of natural and engineered phenomena rely on large networks of interacting units to reach a dynamical consensus state where the system collectively operates. Here we study the dynamics of self-organizing systems and show that for generic directed networks the collective frequency of the ensemble is {\it not} the same as the mean of the individuals' natural frequencies. Specifically, we show that the collective frequency equals a weighted average of the natural frequencies, where the weights are given by an out-flow centrality measure that is equivalent to a reverse PageRank centrality. Our findings uncover an intricate dependence of the collective frequency on both the structural directedness and dynamical heterogeneity of the network, and also reveal an unexplored connection between synchronization and PageRank, which opens the possibility of applying PageRank optimization to synchronization. Finally, we demonstrate the presence of collective frequency variation in real-world networks by considering the UK and Scandinavian power grids.

nlin.AO↗

arXiv2015

Nonlinear Transport in the Stochastic Standard Map

We study a stochastically-driven standard map. The addition of a noise term destroys the invariant manifolds that organize the phase space which allows for more widespread transport than in the noiseless case. Using appropriately defined hitting times to quantify the dynamics, we identify two qualitatively different classes of transport: linear and nonlinear. Linear transport is primarily driven by the stochasticity in the system, while nonlinear transport results from a combination of the nonlinear dynamics and stochasticity and provides a significant speed-up in transport.

nlin.CD↗

arXiv2015

Control of coupled oscillator networks with application to microgrid technologies

The control of complex systems and network-coupled dynamical systems is a topic of vital theoretical importance in mathematics and physics with a wide range of applications in engineering and various other sciences. Motivated by recent research into smart grid technologies we study here control of synchronization and consider the important case of networks of coupled phase oscillators with nonlinear interactions--a paradigmatic example that has guided our understanding of self-organization for decades. We develop a method for control based on identifying and stabilizing problematic oscillators, resulting in a stable spectrum of eigenvalues, and in turn a linearly stable synchronized state. Interestingly, the amount of control, i.e., number of oscillators, required to stabilize the network is primarily dictated by the coupling strength, dynamical heterogeneity, and mean degree of the network, and depends little on the structural heterogeneity of the network itself.

nlin.AO↗

arXiv2015

Frequency assortativity can induce chaos in oscillator networks

We investigate the effect of preferentially connecting oscillators with similar frequency to each other in networks of coupled phase oscillators (i.e., frequency assortativity). Using the network Kuramoto model as an example, we find that frequency assortativity can induce chaos in the macroscopic dynamics. By applying a mean-field approximation in combination with the dimension reduction method of Ott and Antonsen, we show that the dynamics can be described by a low dimensional system of equations. We use the reduced system to characterize the macroscopic chaos using Lyapunov exponents, bifurcation diagrams, and time-delay embeddings. Finally, we show that the emergence of chaos stems from the formation of multiple groups of synchronized oscillators, i.e., meta-oscillators.

nlin.CD↗

arXiv2015

Erosion of synchronization: Coupling heterogeneity and network structure

We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.

nlin.AO↗

arXiv2015

Erosion of synchronization in networks of coupled oscillators

We report erosion of synchronization in networks of coupled phase oscillators, a phenomenon where perfect phase synchronization is unattainable in steady-state, even in the limit of infinite coupling. An analysis reveals that the total erosion is separable into the product of terms characterizing coupling frustration and structural heterogeneity, both of which amplify erosion. The latter, however, can differ significantly from degree heterogeneity. Finally, we show that erosion is marked by the reorganization of oscillators according to their node degrees rather than their natural frequencies.

nlin.AO↗

arXiv2014

Coexisting chaotic and multi-periodic dynamics in a model of cardiac alternans

The spatiotemporal dynamics of cardiac tissue is an active area of research for biologists, physicists, and mathematicians. Of particular interest is the study of period-doubling bifurcations and chaos due to their link with cardiac arrhythmogenesis. In this paper we study the spatiotemporal dynamics of a recently developed model for calcium-driven alternans in a one dimensional cable of tissue. In particular, we observe in the cable coexistence of regions with chaotic and multi-periodic dynamics over wide ranges of parameters. We study these dynamics using global and local Lyapunov exponents and spatial trajectory correlations. Interestingly, near nodes -- or phase reversals -- low-periodic dynamics prevail, while away from the nodes the dynamics tend to be higher-periodic and eventually chaotic. Finally, we show that similar coexisting multi-periodic and chaotic dynamics can also be observed in a detailed ionic model.

nlin.CD↗

arXiv2014

Optimal synchronization of complex networks

We study optimal synchronization in networks of heterogeneous phase oscillators. Our main result is the derivation of a synchrony alignment function that encodes the interplay between network structure and oscillators' frequencies and can be readily optimized. We highlight its utility in two general problems: constrained frequency allocation and network design. In general, we find that synchronization is promoted by strong alignments between frequencies and the dominant Laplacian eigenvectors, as well as a matching between the heterogeneity of frequencies and network structure.

nlin.AO↗

arXiv2014

Disorder induces explosive synchronization

We study explosive synchronization, a phenomenon characterized by first-order phase transitions between incoherent and synchronized states in networks of coupled oscillators. While explosive synchronization has been the subject of many recent studies, in each case strong conditions on either the heterogeneity of the network, its link weights, or its initial construction are imposed to engineer a first-order phase transition. This raises the question of how robust explosive synchronization is in view of more realistic structural and dynamical properties. Here we show that explosive synchronization can be induced in mildly heterogeneous networks by the addition of quenched disorder to the oscillators' frequencies, demonstrating that it is not only robust to, but moreover promoted by, this natural mechanism. We support these findings with numerical and analytical results, presenting simulations of a real neural network as well as a self-consistency theory used to study synthetic networks.

nlin.AO↗

arXiv2014

Spatiotemporal Dynamics of Calcium-Driven Cardiac Alternans

We investigate the dynamics of spatially discordant alternans (SDA) driven by an instability of intracellular calcium cycling using both amplitude equations [P. S. Skardal, A. Karma, and J. G. Restrepo, Phys. Rev. Lett. 108, 108103 (2012)] and ionic model simulations. We find that, close to the alternans bifurcation, SDA is manifested as a smooth wavy modulation of the amplitudes of both repolarization and calcium transient (CaT) alternans, similarly to the well-studied case of voltage-driven alternans. In contrast, further away from the bifurcation, the amplitude of CaT alternans jumps discontinuously at the nodes separating out-of-phase regions, while the amplitude of repolarization alternans remains smooth. We show that node motion of discontinuous SDA patterns is strongly hysteretic even in homogeneous tissue due to the novel phenomenon of "unidirectional pinning": node movement can only be induced towards, but not away from, the pacing site in response to a change of pacing rate or physiological parameter. In addition, we show that the wavelength of discontinuous SDA patterns scales linearly with the conduction velocity restitution length scale, in contrast to the wavelength of smooth patterns that scales sub-linearly with this length scale. Those results are also shown to be robust against cell-to-cell fluctuations owing to the property that unidirectional node motion collapses multiple jumps accumulating in nodal regions into a single jump. Amplitude equation predictions are in good overall agreement with ionic model simulations. Finally, we briefly discuss physiological implications of our findings. In particular, we suggest that due to the tendency of conduction blocks to form near nodes, the presence of unidirectional pinning makes calcium-driven alternans potentially more arrhythmogenic than voltage-driven alternans.

q-bio.TO↗

arXiv2012

Complex macroscopic behavior in systems of phase oscillators with adaptive coupling

Using recent dimensionality reduction techniques in large systems of coupled phase oscillators exhibiting bistability, we analyze complex macroscopic behavior arising when the coupling between oscillators is allowed to evolve slowly as a function of either macroscopic or local system properties. For example, we observe macroscopic excitability and intermittent synchrony in a system of time-delayed Kuramoto oscillators with Hebbian and anti-Hebbian learning. We demonstrate the robustness of our findings by considering systems with increasing complexity, including time-delayed oscillators with adaptive network structure and community interaction, as well as a system with bimodally distributed frequencies.

nlin.AO↗