scholarly journals Using local dynamics to explain analog forecasting of chaotic systems

2021 ◽  
Author(s):  
P. Platzer ◽  
P. Yiou ◽  
P. Naveau ◽  
P. Tandeo ◽  
Y. Zhen ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Chaotic Systems ◽  
Jacobian Matrix ◽  
Nearest Neighbors ◽  
Dynamical Equations ◽  
Local Approximations ◽  
Observational Noise ◽  
Flow Map ◽  
The Impact ◽  
Forecasting Errors

AbstractAnalogs are nearest neighbors of the state of a system. By using analogs and their successors in time, one is able to produce empirical forecasts. Several analog forecasting methods have been used in atmospheric applications and tested on well-known dynamical systems. Such methods are often used without reference to theoretical connections with dynamical systems. Yet, analog forecasting can be related to the dynamical equations of the system of interest. This study investigates the properties of different analog forecasting strategies by taking local approximations of the system’s dynamics. We find that analog forecasting performances are highly linked to the local Jacobian matrix of the flow map, and that analog forecasting combined with linear regression allows to capture projections of this Jacobian matrix. Additionally, the proposed methodology allows to efficiently estimate analog forecasting errors, an important component in many applications. Carrying out this analysis also allows to compare different analog forecasting operators, helping to choose which operator is best suited depending on the situation. These results are derived analytically and tested numerically on two simple chaotic dynamical systems. The impact of observational noise and of the number of analogs is evaluated theoretically and numerically.

A UNIFIED FRAMEWORK FOR SYNCHRONIZATION AND CONTROL OF DYNAMICAL SYSTEMS

1994 ◽  
Vol 04 (04) ◽  
pp. 979-998 ◽  
Author(s):  
CHAI WAH WU ◽  
LEON O. CHUA
Keyword(s):  
Dynamical Systems ◽  
Chaotic Systems ◽  
Main Tool ◽  
Lyapunov Stability Theory ◽  
Asymptotical Stability ◽  
Unified Framework ◽  
Partial Synchronization ◽  
Chua’S Oscillator ◽  
And Control

In this paper, we give a framework for synchronization of dynamical systems which unifies many results in synchronization and control of dynamical systems, in particular chaotic systems. We define concepts such as asymptotical synchronization, partial synchronization and synchronization error bounds. We show how asymptotical synchronization is related to asymptotical stability. The main tool we use to prove asymptotical stability and synchronization is Lyapunov stability theory. We illustrate how many previous results on synchronization and control of chaotic systems can be derived from this framework. We will also give a characterization of robustness of synchronization and show that master-slave asymptotical synchronization in Chua’s oscillator is robust.

scholarly journals A Twenty-First Century Guidebook for Applied Dynamical Systems

2006 ◽  
Vol 1 (4) ◽  
pp. 279-282 ◽  
Author(s):  
A. R. Champneys
Keyword(s):  
Dynamical Systems ◽  
21St Century ◽  
Vector Fields ◽  
Nonlinear Oscillations ◽  
First Century ◽  
Twenty First Century ◽  
The Impact ◽  
Young Researchers

This paper represents the author’s view on the impact of the book Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields by John Guckenheimer and Philip Holmes, first published in 1983 (Springer-Verlag, Berlin). In particular, the questions addressed are: if one were to write a similar book for the 21st century, which topics should be contained and what form should the book take in order to have a similar impact on the modern generation of young researchers in applied dynamical systems?

Noninvertible Piecewise Linear Maps Applied to Chaos Synchronization and Secure Communications

1997 ◽  
Vol 07 (07) ◽  
pp. 1617-1634 ◽  
Author(s):  
G. Millerioux ◽  
C. Mira
Keyword(s):  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Piecewise Linear ◽  
A Priori ◽  
Secure Communications ◽  
Technological Parameters ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Necessary And Sufficient

Recently, it was demonstrated that two chaotic dynamical systems can synchronize each other, leading to interesting applications as secure communications. We propose in this paper a special class of dynamical systems, noninvertible discrete piecewise linear, emphasizing on interesting advantages they present compared with continuous and differentiable nonlinear ones. The generic aspect of such systems, the simplicity of numerical implementation, and the robustness to mismatch of technological parameters make them good candidates. The classical concept of controllability in the control theory is presented and used in order to choose and predict the number of appropriate variables to be transmitted for synchronization. A necessary and sufficient condition of chaotic synchronization is established without computing numerical quantities, introducing a state affinity structure of chaotic systems which provides an a priori establishment of synchronization.

scholarly journals Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

2007 ◽  
Vol 14 (5) ◽  
pp. 615-620 ◽  
Author(s):  
Y. Saiki
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Phenomenon ◽  
Unstable Periodic Orbits ◽  
Newton Raphson ◽  
The Lorenz System

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

scholarly journals New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries

2020 ◽  
Vol 17 (06) ◽  
pp. 2050090 ◽  
Author(s):  
Jordi Gaset ◽  
Xavier Gràcia ◽  
Miguel C. Muñoz-Lecanda ◽  
Xavier Rivas ◽  
Narciso Román-Roy
Keyword(s):  
Dynamical Systems ◽  
Mechanical Systems ◽  
Dissipative Systems ◽  
Conserved Quantities ◽  
Dynamical Equations ◽  
Lagrangian Formulations ◽  
New Form

We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact dynamical equations, and we review two recently presented Lagrangian formalisms, studying their equivalence. We define several kinds of symmetries for contact dynamical systems, as well as the notion of dissipation laws, prove a dissipation theorem and give a way to construct conserved quantities. Some well-known examples of dissipative systems are discussed.

scholarly journals Synchronicity in predictive modelling: a new view of data assimilation

2006 ◽  
Vol 13 (6) ◽  
pp. 601-612 ◽  
Author(s):  
G. S. Duane ◽  
J. J. Tribbia ◽  
J. B. Weiss
Keyword(s):  
Dynamical Systems ◽  
Data Assimilation ◽  
Ad Hoc ◽  
Predictive Modelling ◽  
Dimensional System ◽  
General View ◽  
Machine Perception ◽  
Loosely Coupled ◽  
Unidirectional Flow ◽  
Observational Noise

Abstract. The problem of data assimilation can be viewed as one of synchronizing two dynamical systems, one representing "truth" and the other representing "model", with a unidirectional flow of information between the two. Synchronization of truth and model defines a general view of data assimilation, as machine perception, that is reminiscent of the Jung-Pauli notion of synchronicity between matter and mind. The dynamical systems paradigm of the synchronization of a pair of loosely coupled chaotic systems is expected to be useful because quasi-2D geophysical fluid models have been shown to synchronize when only medium-scale modes are coupled. The synchronization approach is equivalent to standard approaches based on least-squares optimization, including Kalman filtering, except in highly non-linear regions of state space where observational noise links regimes with qualitatively different dynamics. The synchronization approach is used to calculate covariance inflation factors from parameters describing the bimodality of a one-dimensional system. The factors agree in overall magnitude with those used in operational practice on an ad hoc basis. The calculation is robust against the introduction of stochastic model error arising from unresolved scales.

Dynamic Analysis of Mechanical Systems With Clearances—Part 1: Formation of Dynamic Model

1971 ◽  
Vol 93 (1) ◽  
pp. 305-309 ◽  
Author(s):  
S. Dubowsky ◽  
F. Freudenstein
Keyword(s):  
Equations Of Motion ◽  
Operating Conditions ◽  
Coupled Systems ◽  
Dynamical Response ◽  
Dynamic Force ◽  
Dynamical Equations ◽  
Mechanical Joint ◽  
The Impact ◽  
Insight Into

A mathematical model of an elastic mechanical joint with clearances has been formulated and the dynamical equations of motion derived (Part I). The model, which we have called an Impact Pair, is basic to the determination of the dynamical response of mechanical and electromechanical systems with clearances, including determination of dynamic force amplification, frequency response, time-displacement characteristics, and other dynamic characteristics. Whenever possible, the results for the impact pair under various operating conditions are illustrated by graphs, which may also offer some insight into the behavior of clearance-coupled systems.

scholarly journals Robust Adaptive Filter for Small Satellite Attitude Estimation Based on Magnetometer and Gyro

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhankui Zeng ◽  
Shijie Zhang ◽  
Yanjun Xing ◽  
Xibin Cao
Keyword(s):  
Kalman Filter ◽  
Attitude Estimation ◽  
Optimal Filtering ◽  
System State ◽  
Small Satellite ◽  
Adaptive Kalman Filter ◽  
Satellite Attitude ◽  
Observational Noise ◽  
Robust Adaptive ◽  
The Impact

Based on magnetometer and gyro measurement, a sequential scheme is proposed to determine the orbit and attitude of small satellite simultaneously. In order to reduce the impact of orbital errors on attitude estimation, a robust adaptive Kalman filter is developed. It uses a scale factor and an adaptive factor, which are constructed by Huber function and innovation sequence, respectively, to adjust the covariance matrix of system state and observational noise, change the weights of predicted and measured parameters, get suitable Kalman filter gain and approximate optimal filtering results. Numerical simulations are carried out and the proposed filter is approved to be robust for the noise disturbance and parameter uncertainty and can provide higher accuracy attitude estimation.

scholarly journals Multiple-criteria cash-management policies with particular liquidity terms

2019 ◽  
Author(s):  
Francisco Salas-Molina ◽  
David Pla-Santamaria ◽  
Ana Garcia-Bernabeu ◽  
Fernando Mayor-Vitoria
Keyword(s):  
Predictive Accuracy ◽  
Cash Flows ◽  
Expected Returns ◽  
Cash Management ◽  
Deterministic Equivalent ◽  
Multiple Assets ◽  
Management Policies ◽  
The Impact ◽  
Forecasting Errors ◽  
Near Future

Abstract Eliciting policies for cash management systems with multiple assets is by no means straightforward. Both the particular relationship between alternative assets and time delays from control decisions to availability of cash introduce additional difficulties. Here we propose a cash management model to derive short-term finance policies when considering multiple assets with different expected returns and particular liquidity terms for each alternative asset. In order to deal with the inherent uncertainty about the near future introduced by cash flows, we use forecasts as a key input to the model. We express uncertainty as lack of predictive accuracy and we derive a deterministic equivalent problem that depends on forecasting errors and preferences of cash managers. Since the assessment of the quality of forecasts is recommended, we describe a method to evaluate the impact of predictive accuracy in cash management policies. We illustrate this method through several numerical examples.

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