Dynamical Systems Techniques Research Articles

Dynamical study of ellipsoidal universes

A Universe containing uniform magnetic fields, strings, or domain walls is shown to have an ellipsoidal expansion. This case is motivated from observational cosmology, especially the anomaly concerning the low quadrupole amplitude compared to the best-fit ΛCDM prediction in WMAP data. It is shown that a Universe with eccentricity at decoupling of order 10−2 can reduce the quadrupole amplitude without affecting higher multipoles of the angular power spectrum of the temperature anisotropy. The evolution of ellipsoidal Universes is studied using dynamical system techniques for the first time. The determined critical points vary between saddle and past attractors depending on dark energy state equation parameter w Λ, with no future attractors. Numerical integrations of this dynamical system done using several initial conditions are showing new important results. For instance, a tendency for high expansion differences between planar and perpendicular axes is observed which contradicts previous assumptions on the evolution behaviour of ellipsoid Universes.

Kinematical and dynamical aspects of ghost-matter cosmologies

We consider the kinematical and dynamical evolution of Friedmann Universes with a mixture of non-interacting matter and a ghost-like field, in a scenario analogous to that advocated by the Quintom model. Assuming that the conventional matter dominates today, we find that the ghost component can bring the future expansion and the past contraction of the model to a finite halt. Moreover, at the moment the expansion or contraction stops, we find that the tendency of the Universe is to bounce back and re-collapse or re-expand. Therefore, the presence of a (never dominant) ghost-field with negative density could, in principle, drive the Universe into an eternal cycle of finite expansion, collapse, and re-expansion. Our study outlines the key features of such a scenario and provides a simple condition for it to occur. We also derive an autonomous set of differential equations and employ dynamical-system techniques to identify two families of fixed points, with and without spatial curvature respectively. The members of the first family correspond to coasting Universes and are stable in the Lyapunov sense. Those of the latter family are unstable repellers when their matter satisfies the strong energy condition and Lyapunov stable in the opposite case.

Unravelling the myths of R 0 in controlling the dynamics of COVID-19 outbreak: A modelling perspective.

COVID-19 is an emerging and rapidly evolving pandemic around the world, which causes severe acute respiratory syndrome and results in substantial morbidity and mortality. To examine the transmission dynamics of COVID-19, we investigate the spread of this pandemic using Malaysia as a case study and scrutinise its interactions with some exogenous factors such as limited medical resources and false detection problems. To do this, we employ a simple epidemiological model and analyse this system using modelling and dynamical systems techniques. We discover some contrasting findings with respect to the observations of basic reproduction number: while it is observed that R 0 seems to provide a good description of transmission dynamics in simple outbreak scenarios, this quantity might mislead the assessment on the severity of pandemic when certain complexities such as limited medical resources and false detection problems are incorporated into the model. In particular, we observe the possibility of a COVID-19 outbreak through bistable behaviour, even when the basic reproduction number is less than unity. Based on these findings, we caution policy makers not to make their decisions solely based on the guidance of the basic reproduction number only, which clearly could cause trouble.

Mathematical Model of ATM Activation and Chromatin Relaxation by Ionizing Radiation.

We propose a comprehensive mathematical model to study the dynamics of ionizing radiation induced Ataxia-telangiectasia mutated (ATM) activation that consists of ATM activation through dual mechanisms: the initiative activation pathway triggered by the DNA damage-induced local chromatin relaxation and the primary activation pathway consisting of a self-activation loop by interplay with chromatin relaxation. The model is expressed as a series of biochemical reactions, governed by a system of differential equations and analyzed by dynamical systems techniques. Radiation induced double strand breaks (DSBs) cause rapid local chromatin relaxation, which is independent of ATM but initiates ATM activation at damage sites. Key to the model description is how chromatin relaxation follows when active ATM phosphorylates KAP-1, which subsequently spreads throughout the chromatin and induces global chromatin relaxation. Additionally, the model describes how oxidative stress activation of ATM triggers a self-activation loop in which PP2A and ATF2 are released so that ATM can undergo autophosphorylation and acetylation for full activation in relaxed chromatin. In contrast, oxidative stress alone can partially activate ATM because phosphorylated ATM remains as a dimer. The model leads to predictions on ATM mediated responses to DSBs, oxidative stress, or both that can be tested by experiments.

Moment equations and cumulant-neglect closure techniques for non-linear dynamic systems under renewal impulse process excitations

The moment equations technique together with modified cumulant-neglect closure techniques is developed for a non-linear dynamic system subjected to a random train of impulses driven by an Erlang renewal counting process. The original non-Markov problem is converted into a Markov one by recasting the excitation process with the aid of an auxiliary, pure-jump stochastic process characterized by a Markov chain. Hence the conversion is carried out at the expense of augmentation of the state space of the dynamic system by Markov states of the auxiliary jump process. The problem is characterized by the set of joint probability density — discrete distribution function, which are the joint probabilities of the original state vector and of the Markov chain being in the particular jth state. Accordingly the statistical moments of the state variables are defined as integrals with respect to the mixed-type, probability density — discrete distribution function. The differential equations for moments are obtained with the aid of the forward integro-differential Chapman–Kolmogorov operator (Iwankiewicz, 2014). Two modified closure techniques are developed. The first one is the result of application of cumulant-neglect closure directly to unconditional moments. The second modified closure approximation technique is based on the representation of the joint probability density — discrete distribution function by conditioning it on two mutually exclusive and exhaustive events: that the Markov chain is in the jth state while the system is at rest (it is in the jth state for the first time) and that the Markov chain is in the jth state while the system is not at rest (it is in the jth state for any subsequent time). Thus the joint probability density function consists of a Dirac-delta spike and of the continuous part (cf Iwankiewicz et al. (1990)). The cumulant-neglect closure approximations are first formulated for the conditional moments resulting from the continuous part of the probability density function and next for the unconditional moments. As an example of a non-linear system the oscillator with cubic restoring force term is considered. The equations for moments up to the fourth-order are derived. Hence the modified cumulant-neglect closure approximations are derived for redundant fifth- and sixth order moments, both centralized and ordinary. The developed moment equations with modified cumulant-neglect closure techniques are verified against Monte Carlo simulations.

Finsler–Randers cosmology in the framework of a particle creation mechanism: a dynamical systems perspective

The present work deals with the dynamical evolution of Finsler–Randers (FR) cosmological model in the framework of particle creation mechanism. The particle creation mechanism presented the real non-equilibrium thermodynamics of the Universe. We investigate the phase plane analysis of the model. The evolution of cosmological solutions is studied by using dynamical system techniques. We find one additional critical point in FR cosmological model than Riemannian geometry. Stability and viability of the model are discussed. We find the solution of FR cosmological model which provide an acceleration phase of the Universe under some restriction. It is also shown that the evolution of the model at additional critical point is de Sitter Universe and can have stable late time attractor.

Global exponential stability of the Douglas-Rachford splitting dynamics

Many modern optimization problems admit a composite form in which the objective function is given by the sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via proximal methods and their variants, including the Douglas-Rachford (DR) splitting algorithm. In this paper, we view the DR splitting flow as a dynamical system and leverage techniques from control theory to study its global stability properties. In particular, for problems with strongly convex objective functions, we utilize the theory of integral quadratic constraints to prove global exponential stability of the ordinary differential equation that governs the evolution of the DR splitting flow. In our analysis, we use the fact that this algorithm can be interpreted as a variable-metric gradient method on the DR envelope and exploit structural properties of nonlinear terms that arise from composition of reflected proximal operators.

Analysis of Transport and Mixing Phenomenon to Invariant Manifolds Using LCS and KAM Theory Approach in Unsteady Dynamical Systems

The Lagrangian approach for the two-dimensional incompressible fluid flows has been studied with the help of dynamical systems techniques: Kolmogorov-Arnold-Moser (KAM) theory, stable, unstable manifold structures, and Lagrangian coherent structures (LCSs). For the time-dependent perturbation problems, we analyze in detail the development of transport barriers that play an important role in the transport process. Firstly, the analytical study of KAM theory is adopted to explain the transport and mixing phenomenon in measured and simulated airfoil flow. Then, the flow topology of unsteady flow behind an airfoil is investigated for the low Reynolds number problem. Simulations are carried out based on a particular Finite-Time Lyapunov Exponent (FTLE) technique for the detection of invariant manifolds of the hyperbolic trajectories. Besides, the Characteristic Base Split (CBS) scheme combined with a dual time stepping technique is utilized to simulate such transient flow problems. Thus, in the course of the current research, the role of the velocity phase plot during vortex formation is explored that is highly periodic and resulted in the formation of a stable pattern of manifolds and invariant tori. Hence, the proposed study encouraged the new picture of the vortex shedding and flow separation process. As a conclusion, our results give a better understanding of invariant tori control transport phenomena that will lead to a new heuristic for unsteady flows.

Scaling attractors in multi-field inflation

Multi-field inflation with a curved scalar geometry has been found to support background trajectories that violate the slow-roll, slow-turn conditions and thus have the potential to evade the swampland constraints. In order to understand how generic this novel behaviour is and what conditions lead to it, we perform a classification of dynamical attractors of two-field inflation that are of the scaling type. Scaling solutions form a one-parameter generalization of De Sitter solutions with a constant value of the first Hubble flow parameter ϵ and, as we argue and demonstrate, form a natural starting point for the study of non-slow-roll slow-turn behaviour. All scaling solutions can be classified as critical points of a specific dynamical system. We recover known multi-field inflationary attractors as approximate scaling solutions and classify their stability using dynamical system techniques. In particular, we discover that dynamical bifurcations play an integral role in the transition between geodesic and non-geodesic motion and discuss the ability of scaling solutions to describe realistic multi-field models. We revisit the criteria for background stability and show cases where the usual criteria found in the literature do not capture the background evolution of the system.

Can Signal Delay be Functional? Including Delay in Evolved Robot Controllers.

Engineers, control theorists, and neuroscientists often view the delay imposed by finite signal propagation velocities as a problem that needs to be compensated for or avoided. In this article, we consider the alternative possibility that in some cases, signal delay can be used functionally, that is, as an essential component of a cognitive system. To investigate this idea, we evolve a minimal robot controller to solve a basic stimulus-distinction task. The controller is constrained so that the solution must utilize a delayed recurrent signal. Different from previous evolutionary robotics studies, our controller is modeled using delay differential equations, which (unlike the ordinary differential equations of conventional continuous-time recurrent neural networks) can accurately capture delays in signal propagation. We analyze the evolved controller and its interaction with its environment using classical dynamical systems techniques. The analysis shows what kinds of invariant sets underlie the various successful and unsuccessful performances of the robot, and what kinds of bifurcations produce these invariant sets. In the second phase of our analysis, we turn our attention to the parameter θ, which describes the amount of signal delay included in the model. We show how the delay destabilizes certain attractors that would exist if there were no delay and creates other stable attractors, resulting in an agent that performs well at the target task.

D=11 cosmologies with teleparallel structure

The presence of additional compact dimensions in cosmological models is studied in the context of modified teleparallel theories of gravity. We focus the analysis on eleven dimensional spacetimes, where the seven dimensional extra dimensions are compactified. In particular, and due to the importance that global vector fields play within the conceptual body of teleparallel modified gravity models, we consider the additional dimensions to be products of parallelizable spheres. The global vector fields characterizing the different topologies are obtained, as well as the equations of motion associated to them. Using global dynamical system techniques, we discuss some physical consequences arising because of the existence of the extra dimensions. In particular, the possibility of having an early inflationary epoch driven by the presence of extra dimensions without other matter sources is discussed.

A new model order reduction method for the design of compensator by using moment matching algorithm

The aim of this paper is the construction of a new model reduction technique for large scale stable linear dynamic systems. It is principally focused on the dominant modes and time moments retention. This reduction implicates the translation of the overall important features confined in the large scale complete order model into the lower order system, allowing the computation of approximant denominator by using generalized pole clustering method. The approximant numerator is obtained by means of the factor division algorithm. As a result, a lower order system is obtained. To demonstrate its effectiveness, to highlight some fundamental of its features, and to accomplish its accuracy, a comparative study is done. Two standard numerical examples are taken, where approximant model computed by the proposed method is compared with the reduced order models computed from the recently proposed methods as well as well-known model reduction schemes. The paper is also emphasized on the design of compensator by using moment matching algorithm with the help of the reduced model. The design of compensator is validated and illustrated with the help of a standard numerical example taken from the literature.

Friedmann-like universes with weak torsion: a dynamical system approach

We consider Friedmann-like universes with torsion and take a step towards studying their stability. In so doing, we apply dynamical-system techniques to an autonomous system of differential equations, which monitors the evolution of these models via the associated density parameters. Assuming relatively weak torsion, we identify the system’s equilibrium points. These are found to represent homogeneous and isotropic spacetimes with nonzero torsion that undergo accelerated expansion. We then examine the linear stability of the aforementioned fixed points. Our results indicate that Friedmann-like cosmologies with weak torsion are generally stable attractors, either asymptotically or in the Lyapunov sense. In addition, depending on the equation of state of the matter, the equilibrium states can also act as intermediate saddle points, marking a transition from a torsional to a torsion-free universe.

Linear growth index of matter perturbations in Rastall gravity

Rastall gravity theory shows notable features consistent with physical observations in comparison to the standard Einstein theory. Recently, there has been a debate about the equivalence of Rastall gravity and general relativity. Motivated by this open issue, in the present work, we attempt to shed some light on this debate by analyzing the evolution of the Rastall based cosmological model at the background as well as perturbation level. Employing the dynamical system techniques, we found that at late times, the dynamics of the model resembles the ΛCDM model at the background level irrespective of the choice of Rastall's parameter. However, at the perturbation level, we found that the evolution of the growth index heavily depends on the Rastall's parameter and displays a significant deviation from the ΛCDM model.

The Swift\u2013Hohenberg Equation under Directional-Quenching: Finding Heteroclinic Connections Using Spatial and Spectral Decompositions

We study the existence of patterns (nontrivial, stationary solutions) in the one-dimensional Swift–Hohenberg Equation in a directional quenching scenario, that is, on xle 0 the energy potential associated to the equation is bistable, whereas on xge 0 it is monostable. This heterogeneity in the medium induces a symmetry break that makes the existence of heteroclinic orbits of the type point-to-periodic not only plausible but, as we prove here, true. In this search, we use an interesting result of [13] in order to understand the multiscale structure of the problem, namely, how multiple scales - fast/slow - interact with each other. In passing, we advocate for a new approach in finding connecting orbits, using what we call “far/near decompositions”, relying both on information about the spatial behavior of the solutions and on Fourier analysis. Our method is functional analytic and PDE based, relying minimally on dynamical system techniques and making no use of comparison principles whatsoever.

Continuous relaxations for the traveling salesman problem

In this work, we aim to explore connections between dynamical systems techniques and combinatorial optimization problems. In particular, we construct heuristic approaches for the traveling salesman problem (TSP) based on embedding the relaxed discrete optimization problem into appropriate manifolds. We explore multiple embedding techniques -- namely, the construction of new dynamical systems on the manifold of orthogonal matrices and associated Procrustes approximations of the TSP cost function. Using these dynamical systems, we analyze the local neighborhood around the optimal TSP solutions (which are equilibria) using computations to approximate the associated \emph{stable manifolds}. We find that these flows frequently converge to undesirable equilibria. However, the solutions of the dynamical systems and the associated Procrustes approximation provide an interesting biasing approach for the popular Lin--Kernighan heuristic which yields fast convergence. The Lin--Kernighan heuristic is typically based on the computation of edges that have a `high probability' of being in the shortest tour, thereby effectively pruning the search space. Our new approach, instead, relies on a natural relaxation of the combinatorial optimization problem to the manifold of orthogonal matrices and the subsequent use of this solution to bias the Lin--Kernighan heuristic. Although the initial cost of computing these edges using the Procrustes solution is higher than existing methods, we find that the Procrustes solution, when coupled with a homotopy computation, contains valuable information regarding the optimal edges. We explore the Procrustes based approach on several TSP instances and find that our approach often requires fewer $k$-opt moves than existing approaches. Broadly, we hope that this work initiates more work in the intersection of dynamical systems theory and combinatorial optimization.

Trajectory Design for the Wide Field Infrared Survey Telescope Mission

The Wide Field Infrared Survey Telescope (WFIRST) is an upcoming NASA-led observatory that will complete wide-field imaging and near-infrared sky surveys from a spacecraft in the Sun-Earth region. To identify a feasible mission trajectory—subject to geometric and maneuver constraints—an interactive trajectory design procedure, supported by dynamical systems techniques and early incorporation of the mission constraints, is developed. This rapid and informed approach is implemented as a module of Purdue University’s Adaptive Trajectory Design tool. In this paper, a feasible mission trajectory is constructed and output to a higher-fidelity modeling environment. Furthermore, station-keeping maneuvers are computed using a stable mode analysis strategy.

Local Bifurcations in the Cahn–Hilliard and Kuramoto–Sivashinsky Equations and in Their Generalizations

A periodic boundary value problem for a nonlinear evolution equation that takes the form of such well-known equations of mathematical physics as the Cahn–Hilliard, Kuramoto–Sivashinsky, and Kawahara equations for specific values of its coefficients is studied. Three bifurcation problems arising when the stability of the spatially homogeneous equilibrium states changes are studied. The analysis of these problems is based on the method of invariant manifolds, the normal form techniques for dynamic systems with an infinite-dimensional space of initial conditions, and asymptotic methods of analysis. Asymptotic formulas for the bifurcation solutions are found, and stability of these solutions is analyzed. For the Kuramoto–Sivashinsky and Kawahara equations, it is proved that a two-dimensional local attractor exists such that all solutions on it are unstable in Lyapunov’s sense.

Cosmological dynamics of the general non-canonical scalar field models

We extend the investigation of cosmological dynamics of the general non-canonical scalar field models by dynamical system techniques for a broad class of potentials and coupling functions. In other words, we do not restrict the analysis to exponential or power-law potentials and coupling functions. This type of investigation helps in understanding the general properties of a class of cosmological models. In order to better understand the phase space of the models, we investigate the various special cases and discuss the stability and viability issues. Performing a detailed stability analysis, we show that it is possible to describe the cosmic history of the universe at the background level namely the early radiation dominated era, intermediate matter dominated era and the late time dark energy domination. Moreover, we find that we can identify a broad class of potentials and coupling functions for which it is possible to get an appealing unified description of dark matter and dark energy. The results obtained here, therefore, enlarge the previous analyses wherein only a specific potential and coupling functions describe the unification of dark sectors. Further, we also observe that a specific scenario can also possibly explain the phenomenon of slow-roll inflationary exit.

Stability of the Einstein static Universe in Einstein\u2013Cartan\u2013Brans\u2013Dicke gravity

In the present work we consider the existence and stability of Einstein static ES Universe in Brans–Dicke (BD) theory with non-vanishing spacetime torsion. In this theory, torsion field can be generated by the BD scalar field as well as the intrinsic angular momentum (spin) of matter. Assuming the matter content of the Universe to be a Weyssenhoff fluid, which is a generalization of perfect fluid in general relativity (GR) in order to include the spin effects, we find that there exists a stable ES state for a suitable choice of the model parameters. We analyze the stability of the solution by considering linear homogeneous perturbations and discuss the conditions under which the solution can be stable against these type of perturbations. Moreover, using dynamical system techniques and numerical analysis, the stability regions of the ES Universe are parametrized by the BD coupling parameter and first and second derivatives of the BD scalar field potential, and it is explicitly shown that a large class of stable solutions exists within the respective parameter space. This allows for non-singular emergent cosmological scenarios where the Universe oscillates indefinitely about an initial ES solution and is thus past eternal.