Dynamical Systems Point Research Articles

Ergodic Concepts for a Self‐Organizing Trivalent Spin Network: A Path to (2+1)$(2+1)$‐D Black Hole Entropy

AbstractFrom a dynamical systems point of view, a trivalent spin network model in Loop Quantum Gravity is considered, which presents self‐organized criticality (SOC), arising from a spin propagation dynamics. A partition function is obtained for the domains of stability connecting gauge non‐invariant avalanches, leading to an entropy formula for the asymptotic SOC state. The microscopic origin of this SOC entropy is therefore given by the excitation‐relaxation spin dynamics in the avalanche cycle. The puncturing of trivalent spin networks (TSN) edges participating in the avalanche are counted in terms of an ensemble perimeter over the implicit avalanches. By identifying this perimeter with that of an isolated ‐D black hole horizon, it is conjectured that the SOC entropy reduces to the Bekenstein‐Hawking perimeter‐entropy law for the Bañados, Teitelboim, and Zanelli (BTZ) black hole, by an appropriate adjustment of a potential function based on the thermodynamical formalism of Sinai, Ruelle, and Bowen.

Preperiodic points of polynomial dynamical systems over finite fields

For a prime p, positive integers [Formula: see text], and a polynomial f with coefficients in [Formula: see text], let [Formula: see text]. As n varies, the [Formula: see text] partition the set of strictly preperiodic points of the dynamical system induced by the action of f on [Formula: see text]. In this paper, we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of [Formula: see text] lying in a given [Formula: see text]. Moreover, when we generalize our definition of [Formula: see text], we obtain both upper and lower bounds for the resulting averages.

A predictive TDM-PON resource allocation using the ANN method based on equilibrium points of discrete dynamical systems

Resource allocation is one of the challenges in the time division multiple access (TDM) passive optical network (PON) area. Much research has been done to find an efficient resource allocation method. Using machine learning (ML) and finding proper ML algorithms in predictive dynamic bandwidth allocation remains an open research question. This paper aims to predict resource allocation accurately and efficiently based on historical requests. Artificial neural networks (ANN) with historically allocated bandwidth from different time series are used to build the prediction model. This model is constructed by the normal equation method. The proposed model needs a small dataset for its training phase, while its accuracy is higher than other models. The discrete dynamical system builds to find an equilibrium point, which is a stable point in the system. The proposed model can be implemented as a pre-training, or it can be dynamically refined when the model is running by using some of its output as feedback. The system evaluation verifies that the proposed DBA (EQ-ANN) elevates the quality of service (QoS) metrics, such as system throughput, packet drop probability, and packet means delay. The results show that the system throughput improves by up to 5%, and packet loss improves by up to 15%. Moreover, the prediction accuracy of the system reached 99.48%.

Requisites on material viscoelasticity for exceptional points in passive dynamical systems

Recent progress in non-Hermitian physics and the notion of exceptional point (EP) degeneracies in elastodynamics have led to the development of novel metamaterials for the control of elastic wave propagation, hypersensitive sensors, and actuators. The emergence of EPs in a parity-time symmetric system relies on judiciously engineered balanced gain and loss mechanisms. Creating gain requires complex circuits and amplification mechanisms, making engineering applications challenging. Here, we report strategies to achieve EPs in passive non-Hermitian elastodynamic systems with differential loss derived from viscoelastic materials. We compare different viscoelastic material models and show that the EP emerges only when the frequency-dependent loss-tangent of the viscoelastic material remains nearly constant in the frequency range of operation. This type of loss tangent occurs in materials that undergo stress-relaxation over a broad spectrum of relaxation times, for example, materials that follow the Kelvin–Voigt fractional derivative (KVFD) model. Using dynamic mechanical analysis, we show that a few common viscoelastic elastomers, such as polydimethylsiloxane and polyurethane rubber, follow the KVFD behavior such that the loss tangent becomes almost constant after a particular frequency. The material models we present and the demonstration of the potential of a widely available material system in creating EPs pave the way for developing non-Hermitian metamaterials with hypersensitivity to perturbations or enhanced emissivity.

Wada boundaries on a hyperbolic pair of pants

In this paper, the geodesics of an open multiply connected hyperbolic manifold are presented from the dynamical system point of view. The approach is completely numerical. Similar to the closed hyperbolic case there is a zero-measure set of periodic orbits. The difference is that now, most of the geodesics escape to infinity through one of the topologically distinct channels, and initial conditions are identified to which channel the orbit escapes. The initial condition mesh reveals basins of attraction to each channel. We verified that the basin boundary points are of the Wada type by two independent methods. We have also calculated the basin entropy of the system and verified that it depends on the opening angle of the escape channels. The manifold chosen is the pair of pants together with the leaves, so there are 3 distinct exit channels.

Gentlest Ascent Dynamics on Manifolds Defined by Adaptively Sampled Point-Clouds.

Finding saddle points of dynamical systems is an important problem in practical applications, such as the study of rare events of molecular systems. Gentlest ascent dynamics (GAD) (10.1088/0951-7715/24/6/008) is one of a number of algorithms in existence that attempt to find saddle points. It works by deriving a new dynamical system in which saddle points of the original system become stable equilibria. GAD has been recently generalized to the study of dynamical systems on manifolds (differential algebraic equations) described by equality constraints (10.1007/s10915-022-01838-3) and given in an extrinsic formulation. In this paper, we present an extension of GAD to manifolds defined by point-clouds, formulated by using an intrinsic viewpoint. These point-clouds are adaptively sampled during an iterative process that drives the system from the initial conformation (typically in the neighborhood of a stable equilibrium) to a saddle point. Our method requires the reactant (initial conformation), does not require the explicit constraint equations to be specified, and is purely data-driven.

A hyper-parameterization method for comprehensive ocean models: Advection of the image point

Idealized and comprehensive ocean models at low resolutions cannot reproduce nominally-resolved flow structures similar to those presented in the high-resolution solution. Although there are various underlying physical reasons for this, from the dynamical system point of view all these reasons manifest themselves as a low-resolution trajectory avoiding the phase space occupied by the reference solution (the high-resolution solution projected onto the coarse grid). In order to solve this problem, a set of hyper-parameterization methods has recently been proposed and successfully tested on idealized ocean models. In this work, for the first time we apply one of hyper-parameterization methods (Advection of the image point) to a comprehensive, rather than idealized, general circulation model of the North Atlantic.The results show that the hyper-parameterization method significantly outperforms the coarse-grid ocean model by reproducing both the large- and small-scale features of the Gulf Stream flow. The proposed method is much faster than even a single run of the coarse-grid ocean model, requires no modification of the model, and is easy to implement. Moreover, the method can take not only the reference solution as input data but also real measurements from different sources (drifters, weather stations, etc.), or combination of both. All this offers a great flexibility to ocean modellers working with mathematical models and/or measurements.

Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems

We analyse the stability of large, linear dynamical systems of variables that interact through a fully connected random matrix and have inhomogeneous growth rates. We show that in the absence of correlations between the coupling strengths, a system with interactions is always less stable than a system without interactions. Contrarily to the uncorrelated case, interactions that are antagonistic, i.e., characterised by negative correlations, can stabilise linear dynamical systems. In particular, when the strength of the interactions is not too strong, systems with antagonistic interactions are more stable than systems without interactions. These results are obtained with an exact theory for the spectral properties of fully connected random matrices with diagonal disorder.

Stability in mechanics and dynamics; a brief historical tour from Newton to modern tools, including Riemannian structures and entropy

AbstractThis article presents some historical developments in the theory of stability in mechanics from a dynamical systems point of view, focusing on mathematical methods from the earliest developments (Newton) and explaining some classical and some recent methods used, including Riemannian structures, entropy and notions of stability in dynamics.

Human Sway on a Balance Platform

In the body balance process control, a variety of postural strategies for balance recovery are used. Postural strategies are automatic stable muscular patterns. Such a mechanism reduces the degrees of freedom number, simplifies, speeds up the posture adjustment, and minimizes energy expenditure. They relate to the existence of some motor coordination sets and describe three mobility strategies: ankle, hip, and step strategy. The hip strategy occurs when interference is greater or the use of the ankle strategy is impossible, for example, due to the delay of the postural information cycle. From a dynamical systems point of view, a human attempting to balance upright on an unstable balance board represents the coupling of two dynamical systems, the human balance system with neuromuscular feedback supported on the balance board (an inverted pendulum). The coupling of these two dynamical systems, with time delay and nonlinearities, creates an ideal setting for the emergence of complex postural behaviour and unanticipated interactions between the individual, task, and the external dynamical system. The article presents an analysis of the pendulums concerning the issue of maintaining balance by man.

Detecting unstable periodic points of chaotic maps by stability transformation of reservoir

Detecting unstable periodic orbits in chaotic systems based on the time series is a fundamental problem in nonlinear dynamics, but it often becomes extremely challenging one. In this study, we propose a new approach for detecting unstable periodic points using reservoir computing and stability transformation method. We connects reservoir computing, which is a well-known machine learning technique, and stability transformation method, which can detect unstable periodic points in chaotic dynamical systems, to perform unstable periodic points detection in a data-driven and model-free process. In this paper, we use an example of the Hénon map to demonstrate detecting unstable fixed point and unstable 2-periodic points.

Markov Chain Monte Carlo for Gaussian: A Linear Control Perspective

Drawing samples from a given target probability distribution is a fundamental task in many science and engineering applications. A commonly used method for sampling is the Markov chain Monte Carlo (MCMC) which simulates a Markov chain whose stationary distribution coincides with the target one. In this work, we study the convergence and complexity of MCMC algorithms from a dynamic system point of view. We focus on the special cases with Gaussian target distributions and provide a Lyapunov perspective to them using tools from linear control theory. In particular, we systematically analyze two popular MCMC algorithms: Langevin Monte Carlo (LMC) and kinetic Langevin Monte Carlo (KLMC). By applying Lyapunov theory we derive impressive complexity bounds to these algorithms: for LMC, our result is better than all existing results, and for KLMC, ours matches the best known bound. Our analysis also highlights subtle differences between sampling and optimization that could inform the more challenging task to sample from general distributions. Overall, our findings offer valuable insights for improving MCMC algorithms.

A method for preserving nominally-resolved flow patterns in low-resolution ocean simulations: Constrained dynamics

Inability of low-resolution ocean models to simulate many important aspects of the large-scale general circulation is a common problem. In the view of physics, the main reason for this failure are the missed dynamical effects of the unresolved small scales of motion on the explicitly resolved large-scale circulation. Complimentary to this mainstream physics-based perspective, we propose to address this failure from the dynamical systems point of view, namely, as the persistent tendency of phase space trajectories representing the low-resolution solution to escape the right region of the corresponding phase space, which is occupied by the reference eddy-resolving solution. Based on this concept, we propose to use methods of constrained optimization to confine the low-resolution solution to remain within the correct phase space region, without attempting to amend the eddy physics by introducing a process-based parameterization. This approach is advocated as a novel framework for data-driven hyper-parameterizations of mesoscale oceanic eddies in non-eddy-resolving models. We tested the idea in the context of classical, baroclinic beta-plane turbulence model and showed that non-eddy-resolving solution can be substantially improved towards the reference eddy-resolving benchmark.

Detecting criticality in complex univariate time-series: A case study of the U.S. housing market crisis and other markets

The indices of asset markets are a key measure of an asset’s value and how this varies over time. However, these indices are known to switch abruptly between qualitatively different behavioural epochs underscoring the complexity of collective decision-making in large, interacting populations of economic agents. A central question in the study of index dynamics is how to represent these transitions in a way that allows for both equilibrium and non-equilibrium behaviour in a dynamical systems framework. With this in mind, we sought to detect non-equilibrium transitions between stable equilibria in housing market data across thirteen OECD countries from 1975 through to 2019. We are able to precisely locate transitions using the quadrant scan method that has been used in earlier studies to detect critical points in other dynamical systems. We then use stochastic Catastrophe Theory to reconstruct the multi-modal distributions near the U.S. housing crisis point and show mode-switching behaviour of markets transitioning from one equilibrium to another. Our results are in contrast to other recent findings for the U.S. market that were unable to find a critical point near the 2007 crisis.

On Bowen's entropy inequality and almost specification for flows

We study the Bowen topological entropy of generic and irregular points for certain dynamical systems. We define the topological entropy of noncompact sets for flows, analogous to Bowen's definition. We show that this entropy coincides with the Bowen topological entropy of the time-1 map on any set. We also show a Bowen's inequality for flows; namely, that the metric entropy with respect to every invariant measure for a continuous flow is an upper bound for the topological entropy of the set of generic points with respect to the same measure, and the equality is always true if the measure is ergodic. We propose a definition of almost specification property for flows and prove that a continuous flow has the almost specification property if the time-1 map satisfies this property. Using Bowen's inequality for flows, we show that every continuous flow with the almost specification property is saturated, extending a result of Meson and Vericat in [22]. Under the same hypotheses, we extend a result of Thompson on the entropy of irregular points in [29].

Finding Nontrivial Minimum Fixed Points in Discrete Dynamical Systems: Complexity, Special Case Algorithms and Heuristics

Networked discrete dynamical systems are often used to model the spread of contagions and decision-making by agents in coordination games. Fixed points of such dynamical systems represent configurations to which the system converges. In the dissemination of undesirable contagions (such as rumors and misinformation), convergence to fixed points with a small number of affected nodes is a desirable goal. Motivated by such considerations, we formulate a novel optimization problem of finding a nontrivial fixed point of the system with the minimum number of affected nodes. We establish that, unless P = NP, there is no polynomial-time algorithm for approximating a solution to this problem to within the factor n^(1 - epsilon) for any constant epsilon > 0. To cope with this computational intractability, we identify several special cases for which the problem can be solved efficiently. Further, we introduce an integer linear program to address the problem for networks of reasonable sizes. For solving the problem on larger networks, we propose a general heuristic framework along with greedy selection methods. Extensive experimental results on real-world networks demonstrate the effectiveness of the proposed heuristics. A full version of the manuscript, source code and data are available at: https://github.com/bridgelessqiu/NMIN-FPE

On Planar Curves with Position-Dependent Curvature

Motivated by homothetic solutions in curvature-driven flows of planar curves, as well as their many physical applications, this article carries out a systematic study of oriented smooth curves whose curvature $$\kappa $$ is a given function of position or direction. The analysis is informed by a dynamical systems point of view. Though focussed on situations where the prescribed curvature depends only on the distance r from one distinguished point, the basic dynamical concepts are seen to be applicable in other situations as well. As an application, a complete classification of all closed solutions of $$\kappa = ar^{b}$$ , with arbitrary real constants $$a,b$$ , is established.

Backward Asymptotics in S-Unimodal Maps

While the forward trajectory of a point in a discrete dynamical system is always unique, in general, a point can have infinitely many backward trajectories. The union of the limit points of all backward trajectories through [Formula: see text] was called by Hero the “special [Formula: see text]-limit” ([Formula: see text]-limit for short) of [Formula: see text]. In this article, we show that there is a hierarchy of [Formula: see text]-limits of points under iterations of a S-unimodal map: the size of the [Formula: see text]-limit of a point increases monotonically as the point gets closer and closer to the attractor. The [Formula: see text]-limit of any point of the attractor is the whole nonwandering set. This hierarchy reflects the structure of the graph of a S-unimodal map recently introduced jointly by Jim Yorke and the present author.

Geometry of transit orbits in the periodically-perturbed restricted three-body problem

In the circular restricted three-body problem, low energy transit orbits are revealed by linearizing the governing differential equations about the collinear Lagrange points. This procedure fails when time-periodic perturbations are considered, such as perturbation due to the sun (i.e., the bicircular problem) or orbital eccentricity of the primaries. For the case of a time-periodic perturbation, the Lagrange point is replaced by a periodic orbit, equivalently viewed as a hyperbolic-elliptic fixed point of a symplectic map (the stroboscopic Poincaré map). Transit and non-transit orbits can be identified in the discrete map about the fixed point, in analogy with the geometric construction of Conley and McGehee about the index-1 saddle equilibrium point in the continuous dynamical system. Furthermore, though the continuous time system does not conserve the Hamiltonian energy (which is time-varying), the linearized map locally conserves a time-independent effective Hamiltonian function. We demonstrate that the phase space geometry of transit and non-transit orbits is preserved in going from the unperturbed to a periodically-perturbed situation, which carries over to the full nonlinear equations.

Power spectrum scaling as a measure of critical slowing down and precursor to tipping points in dynamical systems

Many dynamical systems experience sudden shifts in behaviour known as tipping points or critical transitions, often preceded by the ‘critical slowing down’ (CSD) phenomenon whereby the recovery times of a system increase as the tipping point is approached. Many attempts have been made to find a tipping point indicator: a proxy for CSD, such that a change in the indicator acts as an early warning signal. Several generic tipping point indicators have been suggested, these include the power spectrum (PS) scaling exponent whose use as an indicator has previously been justified by its relationship to the well-established detrended fluctuation analysis (DFA) exponent. In this paper we justify the use of the PS indicator analytically, by considering a mathematical formulation of the CSD phenomenon. We assess the usefulness of estimating the PS scaling exponent in a tipping point context when the PS does not exhibit power-law scaling, or changes over time. In addition we show that this method is robust against trends and oscillations in the time series, making it a good candidate for studying resilience of systems with periodic oscillations which are observed in ecology and geophysics.