Low-dimensional Dynamical Systems Research Articles

Reduced Markovian models of dynamical systems

Leveraging recent work on data-driven methods for constructing a finite state space Markov process from dynamical systems, we address two problems for obtaining further reduced statistical representations. The first problem is to extract the most salient reduced-order dynamics for a given timescale by using a modified clustering algorithm from network theory. The second problem is to provide an alternative construction for the infinitesimal generator of a Markov process that respects statistical features over a large range of time scales. We demonstrate the methodology on three low-dimensional dynamical systems with stochastic and chaotic dynamics. We then apply the method to two high-dimensional dynamical systems, the Kuramoto-Sivashinky equations and data sampled from fluid-flow experiments via Particle Image Velocimetry. We show that the methodology presented herein provides a robust reduced-order statistical representation of the underlying system.

Posterior comparison of model dynamics in several hybrid turbulence model forms

Hybrid turbulence models that can accurately reproduce unsteady three-dimensional flow physics across the entire range of grid scales and turbulence dynamics from Reynolds-averaged Navier–Stokes (RANS), through large-eddy simulation (LES), down to direct numerical simulations (DNS) are of increasing interest to the turbulence modeling community. However, despite decades of research and development, the basic tasks of eliminating poor-performing hybrid RANS-LES models and accelerating adoption of superior models through well-designed validation and verification have yet to occur. As a step in this direction, in this work we evaluate thirteen different hybrid RANS-LES models via systematic grid refinement of decaying homogeneous isotropic turbulence. We further derive a novel mathematical framework for assessing the energy partitioning dynamics of each Hybrid RANS-LES model, wherein model-to-model variations in energy partitioning can be interpreted as different feedback mechanisms operating on a low-dimensional nonlinear dynamical system. We found that model forms similar to the flow simulation methodology—also often termed very-large eddy simulation—are dynamically inconsistent with DNS at all resolutions. Additionally, we found a strong dynamical similarity in the feedback mechanisms of all models related to detached eddy simulation and partially averaged Navier–Stokes that is inherent to their general model forms.

Adjoint-Based Design Optimization of Stability Constrained Systems

A partial differential equation (PDE) constrained design optimization problem usually optimizes a characteristic of a dynamical system around an equilibrium point. However, a commonly omitted constraint is the linear stability constraint at the equilibrium point, which undermines the optimized solution’s applicability. To enforce the linear stability constraint in practical gradient-based optimization, the derivatives must be computed accurately, and their computational cost must scale favorably with the number of design variables. In this paper, we propose an algorithm based on the coupled adjoint method and the algorithmic differentiation method that can compute the derivative of such constraint accurately and efficiently. We verify the proposed method using several simple low-dimensional dynamical systems. The relative difference between the adjoint method and the finite differences is between 10−6 to 10−8. The proposed method is demonstrated through several optimizations, including a nonlinear aeroelastic optimization. The proposed algorithm has the potential to be applied to more complex problems involving large-scale nonlinear PDEs, such as aircraft flutter and buffet suppression.

Single droplet or bubble and its stability: Kinetic theory and dynamical system approaches.

Steady solutions of a single droplet or bubble in the van der Waals fluid are investigated on the basis of the kinetic model equationthat has been recently proposed by Miyauchi etal. [Gas Dynamics with Applications in Industry and Life Sciences (Springer, Cham, 2023), pp. 19-39]. Under the thermal equilibrium condition and isotropic assumption with respect to the origin of the coordinates, the kinetic equationis reduced to an ordinary differential equationfor the density, which can be regarded as a low-dimensional dynamical system. The possible density distribution is studied as a flow in the low-dimensional phase space. It is clarified that a single droplet or bubble can be understood as a flow that goes into a fixed point and that the flow is qualitatively different in the unstable and metastable parameter regions. The features of the obtained density distributions in individual regions are also clarified. Finally, the stability of those solutions is studied by direct numerical experiments of the kinetic equation.

Existence and stability of a quasi-periodic two-dimensional motion of a self-propelled particle

The mechanism of self-propelled particle motion has attracted much interest in mathematical and physical understanding of the locomotion of living organisms. In a top-down approach, simple time-evolution equations are suitable for qualitatively analyzing the transition between the different types of solutions and the influence of the intrinsic symmetry of systems despite failing to quantitatively reproduce the phenomena. We aim to rigorously show the existence of the rotational, oscillatory, and quasi-periodic solutions and determine their stabilities regarding a canonical equation proposed by Koyano et al. (J Chem Phys 143(1):014117, 2015) for a self-propelled particle confined by a parabolic potential. In the proof, the original equation is reduced to a lower dimensional dynamical system by applying Fenichel’s theorem on the persistence of normally hyperbolic invariant manifolds and the averaging method. Furthermore, the averaged system is identified with essentially a one-dimensional equation because the original equation is O(2)-symmetric.

The Determining Role of Covariances in Large Networks of Stochastic Neurons.

Biological neural networks are notoriously hard to model due to their stochastic behavior and high dimensionality. We tackle this problem by constructing a dynamical model of both the expectations and covariances of the fractions of active and refractory neurons in the network's populations. We do so by describing the evolution of the states of individual neurons with a continuous-time Markov chain, from which we formally derive a low-dimensional dynamical system. This is done by solving a moment closure problem in a way that is compatible with the nonlinearity and boundedness of the activation function. Our dynamical system captures the behavior of the high-dimensional stochastic model even in cases where the mean-field approximation fails to do so. Taking into account the second-order moments modifies the solutions that would be obtained with the mean-field approximation and can lead to the appearance or disappearance of fixed points and limit cycles. We moreover perform numerical experiments where the mean-field approximation leads to periodically oscillating solutions, while the solutions of the second-order model can be interpreted as an average taken over many realizations of the stochastic model. Altogether, our results highlight the importance of including higher moments when studying stochastic networks and deepen our understanding of correlated neuronal activity.

Hydrological records can be used to reconstruct the resilience of watersheds to climatic extremes

Hydrologic resilience modeling is used in public watershed management to assess watershed ability to supply life-supporting ecoservices under extreme climatic and environmental conditions. Literature surveys criticize resilience models for failing to capture watershed dynamics and undergo adequate testing. Both shortcomings compromise their ability to provide management options reliably protecting water security under real-world conditions. We formulate an empirical protocol to establish real-world correspondence. The protocol applies empirical nonlinear dynamics to reconstruct hydrologic dynamics from watershed records, and analyze the response of reconstructed dynamics to extreme regional climatic conditions. We devise an AI-based early-warning system to forecast (out-of-sample) reconstructed hydrologic resilience dynamics. Application to the La Tejería (Spain) experimental watershed finds it to be a low dimensional nonlinear deterministic dynamic system responding to internal stressors by irregularly oscillating along a watershed attractor. Reconstructed and forecasted hydrologic resilience behavior faithfully captures monthly wet-cold/dry-warm weather patterns characterizing the Mediterranean region.

Tipping in a low-dimensional model of a tropical cyclone

A presumed impact of global climate change is the increase in frequency and intensity of tropical cyclones. Due to the possible destruction that occurs when tropical cyclones make landfall, understanding their formation should be of mass interest. In 2017, Kerry Emanuel modeled tropical cyclone formation by developing a low-dimensional dynamical system which couples tangential wind speed of the eye-wall with the inner-core moisture. For physically relevant parameters, this dynamical system always contains three fixed points: a stable fixed point at the origin corresponding to a non-storm state, an additional asymptotically stable fixed point corresponding to a stable storm state, and a saddle corresponding to an unstable storm state. The goal of this work is to provide insight into the underlying mechanisms that govern the formation and suppression of tropical cyclones through both analytical arguments and numerical experiments. We present a case study of both rate and noise-induced tipping between the stable states, relating to the destabilization or formation of a tropical cyclone. While the stochastic system exhibits transitions both to and from the non-storm state, noise-induced tipping is more likely to form a storm, whereas rate-induced tipping is more likely to cause the storm to destabilize. In fact, rate-induced tipping can never lead to the formation of a storm when acting alone. When rate-induced tipping causes the storm to destabilize, a striking result is that both wind shear and maximal potential velocity have to increase at a substantial rate in order to affect tipping away from the active hurricane state. For storm formation through noise-induced tipping, we identify a specific direction along which the non-storm state is most likely to get activated.

Multiscale model reduction for incompressible flows

We introduce a projection-based model reduction method that systematically accounts for nonlinear interactions between the resolved and unresolved scales of the flow in a low-dimensional dynamical systems model. The proposed method uses a separation of time scales between the resolved and subscale variables to derive a reduced-order model with cubic closure terms for the truncated modes, generalizing the classic Stuart–Landau equation. The leading-order cubic terms are determined by averaging out fast variables through a perturbation series approximation of the action of a stochastic Koopman operator. We show analytically that this multiscale closure model can capture both the effects of mean-flow deformation and the energy cascade before demonstrating improved stability and accuracy in models of chaotic lid-driven cavity flow and vortex pairing in a mixing layer. This approach to closure modelling establishes a general theory for the origin and role of cubic nonlinearities in low-dimensional models of incompressible flows.

Detection of separatrices and chaotic seas based on orbit amplitudes

The Maximum Eccentricity Method (MEM) is a standard tool for the analysis of planetary systems and their stability. The method amounts to estimating the maximal stretch of orbits over sampled domains of initial conditions. The present paper leverages on the MEM to introduce a sharp detector of separatrices and chaotic seas. After introducing the MEM analogue for nearly-integrable action-angle Hamiltonians, i.e., diameters, we use low-dimensional dynamical systems with multi-resonant modes and junctions, supporting chaotic motions, to recognise the drivers of the diameter metric. Once this is appreciated, we present a second-derivative based index measuring the regularity of this application. This quantity turns to be a sensitive and robust indicator to detect separatrices, resonant webs and chaotic seas. We discuss practical applications of this framework in the context of $N$-body simulations for the planetary case affected by mean-motion resonances, and demonstrate the ability of the index to distinguish minute structures of the phase space, otherwise undetected with the original MEM.

Optimal Transport for Parameter Identification of Chaotic Dynamics via Invariant Measures

We study an optimal transportation approach for recovering parameters in dynamical systems with a single smoothly varying attractor. We assume that the data are not sufficient for estimating time derivatives of state variables but enough to approximate the long-time behavior of the system through an approximation of its physical measure. Thus, we fit physical measures by taking the Wasserstein distance from optimal transportation as a misfit function between two probability distributions. In particular, we analyze the regularity of the resulting loss function for general transportation costs and derive gradient formulas. Physical measures are approximated as fixed points of suitable PDE-based Perron–Frobenius operators. Test cases discussed in the paper include common low-dimensional dynamical systems.

Self-tuned criticality: Controlling a neuron near its bifurcation point via temporal correlations.

Previous work showed that the collective activity of large neuronal networks can be tamed to remain near its critical point by a feedback control that maximizes the temporal correlations of the mean-field fluctuations. Since such correlations behave similarly near instabilities across nonlinear dynamical systems, it is expected that the principle should control also low-dimensional dynamical systems exhibiting continuous or discontinuous bifurcations from fixed points to limit cycles. Here we present numerical evidence that the dynamics of a single neuron can be controlled in the vicinity of its bifurcation point. The approach is tested in two models: a two-dimensional generic excitable map and the paradigmatic FitzHugh-Nagumo neuron model. The results show that in both cases, the system can be self-tuned to its bifurcation point by modifying the control parameter according to the first coefficient of the autocorrelation function.

Quantifying chaos using Lagrangian descriptors.

We present and validate simple and efficient methods to estimate the chaoticity of orbits in low-dimensional conservative dynamical systems, namely, autonomous Hamiltonian systems and area-preserving symplectic maps, from computations of Lagrangian descriptors (LDs) on short time scales. Two quantities are proposed for determining the chaotic or regular nature of orbits in a system's phase space, which are based on the values of the LDs of these orbits and of nearby ones: The difference and ratio of neighboring orbits' LDs. Using as generic test models the prototypical two degree of freedom Hénon-Heiles system and the two-dimensional standard map, we find that these indicators are able to correctly characterize the chaotic or regular nature of orbits to better than 90% agreement with results obtained by implementing the Smaller Alignment Index (SALI) method, which is a well-established chaos detection technique. Further investigating the performance of the two introduced quantities, we discuss the effects of the total integration time and of the spacing between the used neighboring orbits on the accuracy of the methods, finding that even typical short time, coarse-grid LD computations are sufficient to provide reliable quantification of the systems' chaotic component, using less CPU time than the SALI. In addition to quantifying chaos, the introduced indicators have the ability to reveal details about the systems' local and global chaotic phase space structure. Our findings clearly suggest that LDs can also be used to quantify and investigate chaos in continuous and discrete low-dimensional conservative dynamical systems.

A dimensionality reduction method for computing reachable tubes based on piecewise pseudo-time dependent Hamilton–Jacobi equation

Reachability analysis is a powerful tool for studying the safety of nonlinear systems, in which one of the key points is the computation of reachable tubes. As a common method in engineering, the Hamiltonian Jacobi technique often faces the “curse of dimensionality”. Its computational complexity grows exponentially with the dimensionality of the system state space. This paper proposes a dimensionality reduction method for the computation of reachable tubes that can be used for problems with dynamical systems of a particular form and a columnar target set. In the proposed method, one state variable is considered a pseudo-time variable, and the remaining state variables are contained within a low-dimensional dynamical system. Multiple slices of the original reachable tube are obtained by solving the Hamilton–Jacobi equation constructed based on this low-dimensional dynamical system and then stacking these slices to reconstruct the original reachable tube. Since the solved Hamilton–Jacobi equation is one dimension lower than the Hamilton–Jacobi equation in the original problem, the complexity of the computation is significantly reduced. Furthermore, the proposed method can be combined with existing methods to further reduce the dimensionality of the reachability problem. The computational accuracy and efficiency of the proposed method are demonstrated by some examples.

On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set

The second part of Hilbert’s sixteenth problem consists in determining the upper bound H(n) for the number of limit cycles that planar polynomial vector fields of degree n can have. For n≥2, it is still unknown whether H(n) is finite or not. The main achievements obtained so far establish lower bounds for H(n). Regarding asymptotic behavior, the best result says that H(n) grows as fast as n2log(n). Better lower bounds for small values of n are known in the research literature. In the recent paper “Some open problems in low dimensional dynamical systems”, by A. Gasull, Problem 18 proposes a different Hilbert’s sixteenth type problem, namely improving the lower bounds for L(n), n∈N, which is defined as the maximum number of limit cycles that planar piecewise linear differential systems with two zones separated by a branch of an algebraic curve of degree n can have. So far, L(n)≥[n/2],n∈N, is the best known general lower bound. Again, better lower bounds for small values of n are known in the research literature. Here, by using a recently developed second order Melnikov method for nonsmooth systems with nonlinear discontinuity manifold, it is shown that L(n) grows as fast as n2. This will be achieved by providing lower bounds for L(n), which improve previous estimates for n≥4.

The first International Conference on Mechanical System Dynamics grandly convened in Nanjing, China

The first International Conference on Mechanical System Dynamics grandly convened in Nanjing, China

Dimension reduction of dynamical systems on networks with leading and non-leading eigenvectors of adjacency matrices

Dimension reduction techniques for dynamical systems on networks are considered to promote our understanding of the original high-dimensional dynamics. One strategy of dimension reduction is to derive a low-dimensional dynamical system whose behavior approximates the observables of the original dynamical system that are weighted linear summations of the state variables at the different nodes. Recently proposed methods use the leading eigenvector of the adjacency matrix of the network as the mixture weights to obtain such observables. In the present study, we explore performances of this type of one-dimensional reductions of dynamical systems on networks when we use non-leading eigenvectors of the adjacency matrix as the mixture weights. Our theory predicts that non-leading eigenvectors can be more efficient than the leading eigenvector and enables us to select the eigenvector minimizing the error. We numerically verify that the optimal non-leading eigenvector outperforms the leading eigenvector for some dynamical systems and networks. We also argue that, despite our theory, it is practically better to use the leading eigenvector as the mixture weights to avoid misplacing the bifurcation point too distantly and to be resistant against dynamical noise.

The Use of Reduced Models to Generate Irregular, Broad-Band Signals That Resemble Brain Rhythms.

The brain produces rhythms in a variety of frequency bands. Some are likely by-products of neuronal processes; others are thought to be top-down. Produced entirely naturally, these rhythms have clearly recognizable beats, but they are very far from periodic in the sense of mathematics. The signals are broad-band, episodic, wandering in amplitude and frequency; the rhythm comes and goes, degrading and regenerating. Gamma rhythms, in particular, have been studied by many authors in computational neuroscience, using reduced models as well as networks of hundreds to thousands of integrate-and-fire neurons. All of these models captured successfully the oscillatory nature of gamma rhythms, but the irregular character of gamma in reduced models has not been investigated thoroughly. In this article, we tackle the mathematical question of whether signals with the properties of brain rhythms can be generated from low dimensional dynamical systems. We found that while adding white noise to single periodic cycles can to some degree simulate gamma dynamics, such models tend to be limited in their ability to capture the range of behaviors observed. Using an ODE with two variables inspired by the FitzHugh-Nagumo and Leslie-Gower models, with stochastically varying coefficients designed to control independently amplitude, frequency, and degree of degeneracy, we were able to replicate the qualitative characteristics of natural brain rhythms. To demonstrate model versatility, we simulate the power spectral densities of gamma rhythms in various brain states recorded in experiments.

Machine-learning potential of a single pendulum.

Reservoir computing offers a great computational framework where a physical system can directly be used as computational substrate. Typically a "reservoir" is comprised of a large number of dynamical systems, and is consequently high dimensional. In this work, we use just a single simple low-dimensional dynamical system, namely, a driven pendulum, as a potential reservoir to implement reservoir computing. Remarkably we demonstrate, through numerical simulations as well as a proof-of-principle experimental realization, that one can successfully perform learning tasks using this single system. The underlying idea is to utilize the rich intrinsic dynamical patterns of the driven pendulum, especially the transient dynamics which has so far been an untapped resource. This allows even a single system to serve as a suitable candidate for a reservoir. Specifically, we analyze the performance of the single pendulum reservoir for two classes of tasks: temporal and nontemporal data processing. The accuracy and robustness of the performance exhibited by this minimal one-node reservoir in implementing these tasks strongly suggest an alternative direction in designing the reservoir layer from the point of view of efficient applications. Further, the simplicity of our learning system offers an opportunity to better understand the framework of reservoir computing in general and indicates the remarkable machine-learning potential of even a single simple nonlinear system.

Detecting chaos in lineage-trees: A deep learning approach

Many complex phenomena, from weather systems to heartbeat rhythm patterns, are effectively modeled as low-dimensional dynamical systems. Such systems may behave chaotically under certain conditions, and so the ability to detect chaos based on empirical measurement is an important step in understanding these processes. Classifying a system as chaotic entails estimating its largest Lyapunov exponent (LLE), which quantifies the average rate of convergence or divergence of initially close trajectories in state space. Estimating the largest Lyapunov exponent from observations of a process is especially challenging in low-dimensional systems affected by a small amount of dynamical noise, which are used to model many real-world processes, and in particular biological systems. We describe a method for accurately estimating the largest Lyapunov exponent from noisy data, based on training deep learning models on synthetically generated trajectories. Once the deep learning models are trained, they can be used to detect the LLE from empirical data without any assumption on the underlying dynamics. Our method naturally extends to different input topologies, allowing us to analyze tree-shaped data, characteristic of inheritance dynamics, where near-identical replication offers a unique and hitherto unstudied avenue to detect chaos. We also characterize the input information extracted by our models for their predictions, allowing for a deeper understanding into the different ways by which chaos can be analyzed in different topologies.