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Zachary G. Nicolaou

Publications and source records attributed to Zachary G. Nicolaou.

At least 19 recordsLinked to original sources
arXiv2026

Signature of glassy dynamics in dynamic modes decompositions

Glasses are traditionally characterized by their rugged landscape of disordered low-energy states and their slow relaxation towards thermodynamic equilibrium. Far from equilibrium, dynamical forms of glassy behavior with anomalous algebraic relaxation have also been noted, for example, in networks of coupled oscillators. Due to their disordered and high-dimensional nature, such systems have been difficult to study theoretically, but data-driven methods are emerging as a promising alternative that may aid in their analysis. Here, we characterize glassy dynamics using the dynamic mode decomposition, a data-driven spectral computation that approximates the Koopman spectrum. We show that the gap between oscillatory and decaying modes in the Koopman spectrum vanishes in systems exhibiting algebraic relaxation, and thus, we propose a model-agnostic signature for robustly detecting and analyzing glassy dynamics. We demonstrate the utility of our approach through both a minimal example of a one-dimensional ODE and a high-dimensional example of coupled oscillators.

cond-mat.dis-nn↗
arXiv2024

Data-Driven Discovery of a New Ginzburg-Landau Reduced-Order Model for Vortex Shedding

Vortex shedding is an important physical phenomenon observed across many spatial and temporal scales in fluids. Previous experimental and theoretical studies have established a hierarchy of local and global reduced-order models for vortex shedding based on the Stuart-Landau and Ginzburg-Landau equations. In this work, we employ data-driven methods to discover a new Landau variable for the complex Ginzburg-Landau equation (CGLE) for periodic two-dimensional vortex shedding past a cylinder. We first coarse grain vorticity field data from direct numerical simulations of the fluid by integrating over the vertical spatial dimension, and then time-delay embed this new variable to obtain phase information necessary for CGLE analysis. Finally, we use the sparse identification of nonlinear dynamics (SINDy) to learn an interpretable system of coupled real partial differential equations that balance model complexity and prediction error. We develop local homogeneous models in key regions of the wake, as well as global heterogeneous models that are parameterized by the streamwise coordinate across the entire wake. After computing the normal-form parameters from the discovered equations, we classify the different models and determine behavior and stability in the different regions of the wake, demonstrating the validity of these models across a range of spatiotemporal subdomains. Different dynamics are found to dominate at different stations within the wake. In addition to introducing a novel Landau variable for vortex shedding, this work may also help characterize other vortex shedding systems and inform the design of control schemes.

physics.flu-dyn↗
arXiv2024

Complex localization mechanisms in networks of coupled oscillators: two case studies

Localized phenomena abound in nature and throughout the physical sciences. Some universal mechanisms for localization have been characterized, such as in the snaking bifurcations of localized steady states in pattern-forming partial differential equations. While much of this understanding has been targeted at steady states, recent studies have noted complex dynamical localization phenomena in systems of coupled oscillators. These localized states can come in the form of symmetry-breaking chimera patterns that exhibit a coexistence of coherence and incoherence in symmetric networks of coupled oscillators and gap solitons emerging in the band gap of parametrically driven networks of oscillators. Here, we report detailed numerical continuations of localized time-periodic states in systems of coupled oscillators, while also documenting the numerous bifurcations they give way to. We find novel routes to localization involving bifurcations of heteroclinic cycles in networks of Janus oscillators and strange bifurcation diagrams resembling chaotic tangles in a parametrically driven array of coupled pendula. We highlight the important role of discrete symmetries and the symmetric branch points that emerge in symmetric models.

nlin.PS↗
arXiv2023

Data-driven discovery and extrapolation of parameterized pattern-forming dynamics

Pattern-forming systems can exhibit a diverse array of complex behaviors as external parameters are varied, enabling a variety of useful functions in biological and engineered systems. First-principles derivations of the underlying transitions can be characterized using bifurcation theory on model systems whose governing equations are known. In contrast, data-driven methods for more complicated and realistic systems whose governing evolution dynamics are unknown have only recently been developed. Here we develop a data-driven approach, the {\em sparse identification for nonlinear dynamics with control parameters} (SINDyCP), to discover dynamics for systems with adjustable control parameters, such as an external driving strength. We demonstrate the method on systems of varying complexity, ranging from discrete maps to systems of partial differential equations. To mitigate the impact of measurement noise, we also develop a weak formulation of SINDyCP and assess its performance on noisy data. We demonstrate applications including the discovery of universal pattern-formation equations, and their bifurcation dependencies, directly from data accessible from experiments and the extrapolation of predictions beyond the weakly nonlinear regime near the onset of an instability.

nlin.PS↗
arXiv2023

Prevalence of multistability and nonstationarity in driven chemical networks

External flows of energy, entropy, and matter can cause sudden transitions in the stability of biological and industrial systems, fundamentally altering their dynamical function. How might we control and design these transitions in chemical reaction networks? Here, we analyze transitions giving rise to complex behavior in random reaction networks subject to external driving forces. In the absence of driving, we characterize the uniqueness of the steady state and identify the percolation of a giant connected component in these networks as the number of reactions increases. When subject to chemical driving (influx and outflux of chemical species), the steady state can undergo bifurcations, leading to multistability or oscillatory dynamics. By quantifying the prevalence of these bifurcations, we show how chemical driving and network sparsity tend to promote the emergence of these complex dynamics and increased rates of entropy production. We show that catalysis also plays an important role in the emergence of complexity, strongly correlating with the prevalence of bifurcations. Our results suggest that coupling a minimal number of chemical signatures with external driving can lead to features present in biochemical processes and abiogenesis.

q-bio.MN↗
arXiv2023

Benchmarking sparse system identification with low-dimensional chaos

Sparse system identification is the data-driven process of obtaining parsimonious differential equations that describe the evolution of a dynamical system, balancing model complexity and accuracy. There has been rapid innovation in system identification across scientific domains, but there remains a gap in the literature for large-scale methodological comparisons that are evaluated on a variety of dynamical systems. In this work, we systematically benchmark sparse regression variants by utilizing the dysts standardized database of chaotic systems. In particular, we demonstrate how this open-source tool can be used to quantitatively compare different methods of system identification. To illustrate how this benchmark can be utilized, we perform a large comparison of four algorithms for solving the sparse identification of nonlinear dynamics (SINDy) optimization problem, finding strong performance of the original algorithm and a recent mixed-integer discrete algorithm. In all cases, we used ensembling to improve the noise robustness of SINDy and provide statistical comparisons. In addition, we show very compelling evidence that the weak SINDy formulation provides significant improvements over the traditional method, even on clean data. Lastly, we investigate how Pareto-optimal models generated from SINDy algorithms depend on the properties of the equations, finding that the performance shows no significant dependence on a set of dynamical properties that quantify the amount of chaos, scale separation, degree of nonlinearity, and the syntactic complexity.

cs.LG↗
arXiv2022

PySINDy: A comprehensive Python package for robust sparse system identification

Automated data-driven modeling, the process of directly discovering the governing equations of a system from data, is increasingly being used across the scientific community. PySINDy is a Python package that provides tools for applying the sparse identification of nonlinear dynamics (SINDy) approach to data-driven model discovery. In this major update to PySINDy, we implement several advanced features that enable the discovery of more general differential equations from noisy and limited data. The library of candidate terms is extended for the identification of actuated systems, partial differential equations (PDEs), and implicit differential equations. Robust formulations, including the integral form of SINDy and ensembling techniques, are also implemented to improve performance for real-world data. Finally, we provide a range of new optimization algorithms, including several sparse regression techniques and algorithms to enforce and promote inequality constraints and stability. Together, these updates enable entirely new SINDy model discovery capabilities that have not been reported in the literature, such as constrained PDE identification and ensembling with different sparse regression optimizers.

eess.SY↗
arXiv2021

Heterogeneity-stabilized homogeneous states in driven media

Understanding the relationship between symmetry breaking, system properties, and instabilities has been a problem of longstanding scientific interest. Symmetry-breaking instabilities underlie the formation of important patterns in driven systems, but there are many instances in which such instabilities are undesirable. Using parametric resonance as a model process, here we show that a range of states that would be destabilized by symmetry-breaking instabilities can be preserved and stabilized by the introduction of suitable system asymmetry. Because symmetric states are spatially homogeneous and asymmetric systems are spatially heterogeneous, we refer to this effect as heterogeneity-stabilized homogeneity. We illustrate this effect theoretically using driven pendulum array models and demonstrate it experimentally using Faraday wave instabilities. Our results have potential implications for the mitigation of instabilities in engineered systems and the emergence of homogeneous states in natural systems with inherent heterogeneities.

cond-mat.dis-nn↗
arXiv2021

Coherent Dynamics Enhanced by Uncorrelated Noise

Synchronization is a widespread phenomenon observed in physical, biological, and social networks, which persists even under the influence of strong noise. Previous research on oscillators subject to common noise has shown that noise can actually facilitate synchronization, as correlations in the dynamics can be inherited from the noise itself. However, in many spatially distributed networks, such as the mammalian circadian system, the noise that different oscillators experience can be effectively uncorrelated. Here, we show that uncorrelated noise can in fact enhance synchronization when the oscillators are coupled. Strikingly, our analysis also shows that uncorrelated noise can be more effective than common noise in enhancing synchronization. We first establish these results theoretically for phase and phase-amplitude oscillators subject to either or both additive and multiplicative noise. We then confirm the predictions through experiments on coupled electrochemical oscillators. Our findings suggest that uncorrelated noise can promote rather than inhibit coherence in natural systems and that the same effect can be harnessed in engineered systems.

nlin.AO↗
arXiv2021

Anharmonic classical time crystals: A coresonance pattern formation mechanism

Driven many-body systems have been shown to exhibit discrete time crystal phases characterized by broken discrete time-translational symmetry. This has been achieved generally through a subharmonic response, in which the system undergoes one oscillation every other driving period. Here, we demonstrate that classical time crystals do not need to resonate in a subharmonic fashion but instead can also exhibit a continuously tunable anharmonic response to driving, which we show can emerge through a coresonance between modes in different branches of the dispersion relation in a parametrically driven medium. This response, characterized by a typically incommensurate ratio between the resonant frequencies and the driving frequency, is demonstrated by introducing a time crystal model consisting of an array of coupled pendula with alternating lengths. Importantly, the coresonance mechanism is the result of a bifurcation involving a fixed point and an invariant torus, with no intermediate limit cycles. This bifurcation thus gives rise to a many-body symmetry-breaking phenomenon directly connecting the symmetry-unbroken phase with a previously uncharacterized phase of matter, which we call an anharmonic time crystal phase. The mechanism is shown to generalize to driven media with any number of coupled fields and is expected to give rise to anharmonic responses in a range of weakly damped pattern-forming systems, with potential applications to the study of nonequilibrium phases, frequency conversion, and acoustic cloaking.

nlin.PS↗
arXiv2020

Missing links as a source of seemingly variable constants in complex reaction networks

A major challenge in network science is to determine parameters governing complex network dynamics from experimental observations and theoretical models. In complex chemical reaction networks, for example, such as those describing processes in internal combustion engines and power generators, rate constant estimates vary significantly across studies despite substantial experimental efforts. Here, we examine the possibility that variability in measured constants can be largely attributed to the impact of missing network information on parameter estimation. Through the numerical simulation of measurements in incomplete chemical reaction networks, we show that unaccountability of network links presumed unimportant (with local sensitivity amounting to less than two percent of that of a measured link) can create apparent rate constant variations as large as one order of magnitude even if no experimental errors are present in the data. Furthermore, the correlation coefficient between the logarithmic deviation of the rate constant estimate and the cumulative relative sensitivity of the neglected reactions was less than $0.5$ in all cases. Thus, for dynamical processes on complex networks, iteratively expanding a model by determining new parameters from data collected under specific conditions is unlikely to produce reliable results.

cond-mat.dis-nn↗
arXiv2020

Non-normality and non-monotonic dynamics in complex reaction networks

Complex chemical reaction networks, which underlie many industrial and biological processes, often exhibit non-monotonic changes in chemical species concentrations, typically described using nonlinear models. Such non-monotonic dynamics are in principle possible even in linear models if the matrices defining the models are non-normal, as characterized by a necessarily non-orthogonal set of eigenvectors. However, the extent to which non-normality is responsible for non-monotonic behavior remains an open question. Here, using a master equation to model the reaction dynamics, we derive a general condition for observing non-monotonic dynamics of individual species, establishing that non-normality promotes non-monotonicity but is not a requirement for it. In contrast, we show that non-normality is a requirement for non-monotonic dynamics to be observed in the Rényi entropy. Using hydrogen combustion as an example application, we demonstrate that non-monotonic dynamics under experimental conditions are supported by a linear chain of connected components, in contrast with the dominance of a single giant component observed in typical random reaction networks. The exact linearity of the master equation enables development of rigorous theory and simulations for dynamical networks of unprecedented size (approaching $10^5$ dynamical variables, even for a network of only 20 reactions and involving less than 100 atoms). Our conclusions are expected to hold for other combustion processes, and the general theory we develop is applicable to all chemical reaction networks, including biological ones.

nlin.AO↗
arXiv2020

Critical Switching in Globally Attractive Chimeras

We report on a new type of chimera state that attracts almost all initial conditions and exhibits power-law switching behavior in networks of coupled oscillators. Such switching chimeras consist of two symmetric configurations, which we refer to as subchimeras, in which one cluster is synchronized and the other is incoherent. Despite each subchimera being linearly stable, switching chimeras are extremely sensitive to noise: arbitrarily small noise triggers and sustains persistent switching between the two symmetric subchimeras. The average switching frequency scales as a power law with the noise intensity, which is in contrast with the exponential scaling observed in typical stochastic transitions. Rigorous numerical analysis reveals that the power-law switching behavior originates from intermingled basins of attraction associated with the two subchimeras, which in turn are induced by chaos and symmetry in the system. The theoretical results are supported by experiments on coupled optoelectronic oscillators, which demonstrate the generality and robustness of switching chimeras.

cond-mat.dis-nn↗
arXiv2019

Multifaceted dynamics of Janus oscillator networks

Recent research has led to the discovery of fundamental new phenomena in network synchronization, including chimera states, explosive synchronization, and asymmetry-induced synchronization. Each of these phenomena has thus far been observed only in systems designed to exhibit that one phenomenon, which raises the questions of whether they are mutually compatible and, if so, under what conditions they co-occur. Here, we introduce a class of remarkably simple oscillator networks that concurrently exhibit all of these phenomena. The dynamical units consist of pairs of nonidentical phase oscillators, which we refer to as Janus oscillators by analogy with Janus particles and the mythological figure from which their name is derived. In contrast to previous studies, these networks exhibit (i) explosive synchronization with identical oscillators; (ii) extreme multistability of chimera states, including traveling, intermittent, and bouncing chimeras; and (iii) asymmetry-induced synchronization in which synchronization is promoted by random oscillator heterogeneity. These networks also exhibit the previously unobserved possibility of inverted synchronization transitions, in which a transition to a more synchronous state is induced by a reduction rather than an increase in the coupling strength. These various phenomena are shown to emerge under rather parsimonious conditions, and even in locally connected ring topologies, which has the potential to facilitate their use to control and manipulate synchronization in experiments.

nlin.AO↗
arXiv2019

Minimal scattering entanglement in one-dimensional trapped gases

The prospect of controlling entanglement in interacting quantum systems offers a myriad of technological and scientific promises, given the progress in experimental studies in systems such as ultracold trapped gases. This control is often challenging because of decoherence, the process by which environmental interactions create spurious entanglements that can destroy the desired entanglement. Considering the collisional decoherence that is relevant for quantum measurements utilizing scattering in one-dimensional trapped gases, here we derive a relationship between particle masses and wave packet widths that minimizes the entanglement created during scattering. We assess the relevance of our results by directly observing this relationship in the emergent scales of a master equation for a particle undergoing nonthermal scattering. Our relationship is independent of the details of the particle interactions and sheds light on how to design scattering processes that minimize decoherence.

cond-mat.quant-gas↗
arXiv2017

Chimera States in Continuous Media: Existence and Distinctness

The defining property of chimera states is the coexistence of coherent and incoherent domains in systems that are structurally and spatially homogeneous. The recent realization that such states might be common in oscillator networks raises the question of whether an analogous phenomenon can occur in continuous media. Here, we show that chimera states can exist in continuous systems even when the coupling is strictly local, as in many fluid and pattern forming media. Using the complex Ginzburg-Landau equation as a model system, we characterize chimera states consisting of a coherent domain of a frozen spiral structure and an incoherent domain of amplitude turbulence. We show that in this case, in contrast with discrete network systems, fluctuations in the local coupling field play a crucial role in limiting the coherent regions. We suggest these findings shed light on new possible forms of coexisting order and disorder in fluid systems.

nlin.PS↗
arXiv2013

Longitudinal inverted compressibility in super-strained metamaterials

We develop a statistical physics theory for solid-solid phase transitions in which a metamaterial undergoes longitudinal contraction in response to increase in external tension. Such transitions, which are forbidden in thermodynamic equilibrium, have recently been shown to be possible during the decay of metastable, super-strained states. We present a first-principles model to predict these transitions and validate it using molecular dynamics simulations. Aside from its immediate mechanical implications, our theory points to a wealth of analogous inverted responses, such as inverted susceptibility or heat-capacity transitions, allowed when considering realistic scales.

cond-mat.stat-mech↗