scholarly journals Prescribing transient and asymptotic behaviour to deterministic systems with stochastic initial conditions

2020 ◽  
pp. 1-14 ◽  
Author(s):  
Martijn Dresscher ◽  
Bayu Jayawardhana
Keyword(s):  
Asymptotic Behaviour ◽  
Initial Conditions ◽  
Deterministic Systems

Chairman’s introduction

1987 ◽  
Vol 413 (1844) ◽  
pp. 3-4
Keyword(s):  
Asymptotic Behaviour ◽  
Frequency Spectrum ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Strange Attractors ◽  
Qualitative Properties ◽  
Differential Topology ◽  
Deterministic Systems ◽  
Sensitive Dependence ◽  
Quantitative Properties

The understanding of chaos and strange attractors is one of the most exciting areas of mathematics today. It is the question of how the asymptotic behaviour of deterministic systems can exhibit unpredictability and apparent chaos, due to sensitive dependence upon initial conditions, and yet at the same time preserve a coherent global structure. The field represents a remarkable confluence of several different strands of thought. 1. Firstly came the influence of differential topology, giving global geometric insight and emphasis on qualitative properties. By qualitative properties I mean invariants under differentiable changes of coordinates, as opposed to quantitative properties which are invariant only under linear changes of coordinates. To give an example of this influence, I recall a year-long symposium at Warwick in 1979/80, which involved sustained interaction between pure mathematicians and experimentalists, and one of the most striking consequences of that interaction was a transformation in the way that experimentalists now present their data. It is generally in a much more translucent form: instead of merely plotting a frequency spectrum and calling the incomprehensible part ‘noise’, they began to draw computer pictures of underlying three-dimensional strange attractors.

scholarly journals Advancing from Convection-Allowing NWP to Warn-on-Forecast: Evidence of Progress

2018 ◽  
Vol 33 (2) ◽  
pp. 599-607 ◽  
Author(s):  
John R. Lawson ◽  
John S. Kain ◽  
Nusrat Yussouf ◽  
David C. Dowell ◽  
Dustan M. Wheatley ◽  
...  
Keyword(s):  
Data Assimilation ◽  
Real Time ◽  
Initial Conditions ◽  
Spatial Scales ◽  
Weather Prediction ◽  
Radar Data ◽  
Model Systems ◽  
Deterministic Systems ◽  
Forecast Lead Time ◽  
The Impact

Abstract The Warn-on-Forecast (WoF) program, driven by advanced data assimilation and ensemble design of numerical weather prediction (NWP) systems, seeks to advance 0–3-h NWP to aid National Weather Service warnings for thunderstorm-induced hazards. An early prototype of the WoF prediction system is the National Severe Storms Laboratory (NSSL) Experimental WoF System for ensembles (NEWSe), which comprises 36 ensemble members with varied initial conditions and parameterization suites. In the present study, real-time 3-h quantitative precipitation forecasts (QPFs) during spring 2016 from NEWSe members are compared against those from two real-time deterministic systems: the operational High Resolution Rapid Refresh (HRRR, version 1) and an upgraded, experimental configuration of the HRRR. All three model systems were run at 3-km horizontal grid spacing and differ in initialization, particularly in the radar data assimilation methods. It is the impact of this difference that is evaluated herein using both traditional and scale-aware verification schemes. NEWSe, evaluated deterministically for each member, shows marked improvement over the two HRRR versions for 0–3-h QPFs, especially at higher thresholds and smaller spatial scales. This improvement diminishes with forecast lead time. The experimental HRRR model, which became operational as HRRR version 2 in August 2016, also provides added skill over HRRR version 1.

scholarly journals Sources of uncertainty in deterministic dynamics: an informal overview

2011 ◽  
Vol 369 (1956) ◽  
pp. 4705-4729 ◽  
Author(s):  
Ian Stewart
Keyword(s):  
Chaotic Dynamics ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Sources Of Uncertainty ◽  
Deterministic Systems ◽  
Coin Tossing ◽  
Random Switching ◽  
Butterfly Effect ◽  
Basin Boundaries ◽  
Sensitivity To Initial Conditions

The discovery of chaotic dynamics implies that deterministic systems may not be predictable in any meaningful sense. The best-known source of unpredictability is sensitivity to initial conditions (popularly known as the butterfly effect), in which small errors or disturbances grow exponentially. However, there are many other sources of uncertainty in nonlinear dynamics. We provide an informal overview of some of these, with an emphasis on the underlying geometry in phase space. The main topics are the butterfly effect, uncertainty in initial conditions in non-chaotic systems, such as coin tossing, heteroclinic connections leading to apparently random switching between states, topological complexity of basin boundaries, bifurcations (popularly known as tipping points) and collisions of chaotic attractors. We briefly discuss possible ways to detect, exploit or mitigate these effects. The paper is intended for non-specialists.

A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations

1966 ◽  
Vol 25 ◽  
pp. 281-287 ◽  
Author(s):  
P. E. Zadunaisky
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Difference Method ◽  
Initial Conditions ◽  
Current Theory ◽  
Numerical Process

Let bex′=f(t,x) a system of ordinary differential equations, with initial conditionsx(a) =s, which is integrated numerically by a finite difference method of orderpand constant steph.To estimate the truncation and round-off errors accumulated during the numerical process it is established a method based on the current theory of the asymptotic behaviour (whenh→0) of errors. This method should avoid the main difficulties that arise when the results of the theory must be applied to practical cases. The method has been successfully tested and applied to estimate the errors accumulated in a numerical computation of planetary perturbations on the orbit of a comet.

scholarly journals Imperfect Automata: Effects of “mutation” on the evolution of automaton 01101110 (Rule 110)

2019 ◽  
Author(s):  
Andrei Popa
Keyword(s):  
Artificial Intelligence ◽  
Initial Conditions ◽  
Biological Systems ◽  
Mutation Rates ◽  
Small Probability ◽  
Deterministic Systems ◽  
Simple Rules ◽  
Computational Work ◽  
A Cell ◽  
Key Topics

Cellular automata are discreet mathematical models. They consist of cells; each cell can exist in a limited number of mutually exclusive states, like 0 or 1. The state of each cell at time t is determined by simple rules, based on the states of its neighboring cells. While exploring their relevance to behavioral sciences (McDowell & Popa, 2009) one aspect caught my attention: it seemed that all emerging structures and patterns could be traced back to the first few generations; patterns were evolving, colliding, changing, and disappearing, but no new patterns or structures were emerging from non-patterns (i.e., "uniformity"). In biological systems, novelty is made possible by mutation (Thomas, 1974) – a concept central to my computational work on learning (Popa & McDowell, 2016) and on the emergence of psychological objects and phenomena from changes in neuronal activation states (Popa, 2019). Unlike biological systems, automata are deterministic systems, governed by precise rules. The question examined here was: what if these rules weren’t precise? What if every time a cell is created, there’s a small probability to make a mistake, to write 0 instead of 1 and vice-versa? I explored this idea in the context of Rule 110 (01101110), an elementary CA notorious for its fascinating properties (Wolfram, 2002). A small amount of mutation – e.g., 0.00005 probability to make mistakes – facilitated the emergence of new patterns and structures, disconnected from the initial conditions. Mutation rates of 0.0001 – 0.0005 produced an abundance of irregular, interacting structures. Mutation rates higher than 1% prevented the emergence of discernible patterns, producing instead an amalgam of ill-defined, organic-looking structures. These results suggested that imperfect automata may provide useful insight on the evolution of non-deterministic systems and on the emergence of novelty – two key topics in machine learning and artificial intelligence.

On hydrodynamic equations and their relation to kinetic theory and statistical mechanics

2017 ◽  
Author(s):  
◽  
Jianfei Xue
Keyword(s):  
Initial Conditions ◽  
Hamiltonian Dynamics ◽  
Euler System ◽  
Initial Time ◽  
Continuous Systems ◽  
Hydrodynamic Equations ◽  
University Of Missouri ◽  
Deterministic Systems ◽  
Pair Potentials ◽  
Hard Rods

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] We establish conditions on the Hamiltonian evolution of interacting molecules that imply hydrodynamic equations at the limit of infinitely many molecules and show that these conditions are satisfied whenever the solutions of the classical equations for N interacting molecules obey uniform in N bounds. We show that this holds when the initial conditions are bounded and the molecule interaction is weak enough at the initial time. We then obtain hydrodynamic equations that coincide with Maxwell's. We then construct explicit examples of spontaneous energy generation and nonuniqueness for the standard compressible Euler system, with and without pressure, agagin by taking limits of Hamiltonian dynamics as the number of molecules increases to infinity. The examples come from rescaling of well-posed, deterministic systems of molecules that either collide elastically or interact via singular pair potentials. We also obtain Percus macroscopic equation as the limit of a sequence of single systems of N hard rods with the number of hard rods going to infinity. Finally, we establish the strict convexity of the pressure as a thermodynamic limit for continuous systems. As a result we show the existence of a local bijection between macroscopic density, velocity, and energy on one hand and thermodynamic parameters on the other, for continuous systems.

scholarly journals Predictability of extreme values in geophysical models

2012 ◽  
Vol 19 (5) ◽  
pp. 529-539 ◽  
Author(s):  
A. E. Sterk ◽  
M. P. Holland ◽  
P. Rabassa ◽  
H. W. Broer ◽  
R. Vitolo
Keyword(s):  
Extreme Value Theory ◽  
Finite Time ◽  
Lead Time ◽  
Extreme Values ◽  
Initial Conditions ◽  
Value Theory ◽  
Wind Speeds ◽  
Finite Time Lyapunov Exponents ◽  
Deterministic Systems ◽  
Time Predictability

Abstract. Extreme value theory in deterministic systems is concerned with unlikely large (or small) values of an observable evaluated along evolutions of the system. In this paper we study the finite-time predictability of extreme values, such as convection, energy, and wind speeds, in three geophysical models. We study whether finite-time Lyapunov exponents are larger or smaller for initial conditions leading to extremes. General statements on whether extreme values are better or less predictable are not possible: the predictability of extreme values depends on the observable, the attractor of the system, and the prediction lead time.

scholarly journals Asymptotic behaviour of orbit determination for hyperbolic maps

2021 ◽  
Vol 133 (6) ◽  
Author(s):  
Stefano Marò ◽  
Claudio Bonanno
Keyword(s):  
Asymptotic Behaviour ◽  
Orbit Determination ◽  
Initial Conditions ◽  
Confidence Region ◽  
Time Span ◽  
Simultaneous Increase ◽  
Dynamical Parameter ◽  
Rate Of Decay

AbstractWe deal with the orbit determination problem for hyperbolic maps. The problem consists in determining the initial conditions of an orbit and, eventually, other parameters of the model from some observations. We study the behaviour of the confidence region in the case of simultaneous increase in the number of observations and the time span over which they are performed. More precisely, we describe the geometry of the confidence region for the solution, distinguishing whether a parameter is added to the estimate of the initial conditions or not. We prove that the inclusion of a dynamical parameter causes a change in the rate of decay of the uncertainties, as suggested by some known numerical evidences.

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