VISUALIZING BIFURCATIONS IN HIGH DIMENSIONAL SYSTEMS: THE SPECTRAL BIFURCATION DIAGRAM

2003 ◽  
Vol 13 (10) ◽  
pp. 3015-3027 ◽  
Author(s):  
DAVID ORRELL ◽  
LEONARD A. SMITH
Keyword(s):  
Control Parameter ◽  
Nonlinear Dynamical Systems ◽  
The Other ◽  
High Dimensional ◽  
Behavior Changes ◽  
Bifurcation Diagrams ◽  
Nonlinear Dynamical ◽  
The Third ◽  
Alternative Approach ◽  
Lorenz 96

This paper presents methods to visualize bifurcations in flows of nonlinear dynamical systems, using the Lorenz '96 systems as examples. Three techniques are considered; the first two, density and max/min diagrams, are analagous to the bifurcation diagrams used for maps, which indicate how the system's behavior changes with a control parameter. However the diagrams are generally harder to interpret than the corresponding diagrams of maps, due to the continuous nature of the flow. The third technique takes an alternative approach: by calculating the power spectrum at each value of the control parameter, a plot is produced which clearly shows the changes between periodic, quasi-periodic, and chaotic states, and reveals structure not shown by the other methods.

scholarly journals Ensemble Riemannian Data Assimilation: Towards High-dimensional Implementation

2021 ◽  
Author(s):  
Sagar Kumar Tamang ◽  
Ardeshir Ebtehaj ◽  
Peter Jan van Leeuwen ◽  
Gilad Lerman ◽  
Efi Foufoula-Georgiou
Keyword(s):  
Dynamical Systems ◽  
Data Assimilation ◽  
Mean Squared Error ◽  
Likelihood Function ◽  
Nonlinear Dynamical Systems ◽  
High Dimensional ◽  
Nonlinear Dynamical ◽  
Systematic Biases ◽  
Background Probability ◽  
Lorenz 96

Abstract. This paper presents the results of the Ensemble Riemannian Data Assimilation for relatively high-dimensional nonlinear dynamical systems, focusing on the chaotic Lorenz-96 model and a two-layer quasi-geostrophic (QG) model of atmospheric circulation. The analysis state in this approach is inferred from a joint distribution that optimally couples the background probability distribution and the likelihood function, enabling formal treatment of systematic biases without any Gaussian assumptions. Despite the risk of the curse of dimensionality in the computation of the coupling distribution, comparisons with the classic implementation of the particle filter and the stochastic ensemble Kalman filter demonstrate that with the same ensemble size, the presented methodology could improve the predictability of dynamical systems. In particular, under systematic errors, the root mean squared error of the analysis state can be reduced by 20 % (30 %) in Lorenz-96 (QG) model.

HIGH-DIMENSIONAL CHAOS IN DISSIPATIVE AND DRIVEN DYNAMICAL SYSTEMS

2009 ◽  
Vol 19 (09) ◽  
pp. 2823-2869 ◽  
Author(s):  
Z. E. MUSIELAK ◽  
D. E. MUSIELAK
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
High Dimensional ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Onset Of Chaos ◽  
General Rules ◽  
Van Der Pol Oscillators ◽  
Control And Synchronization

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.

scholarly journals Multidimensional Approximation of Nonlinear Dynamical Systems

2019 ◽  
Vol 14 (6) ◽  
Author(s):  
Patrick Gelß ◽  
Stefan Klus ◽  
Jens Eisert ◽  
Christof Schütte
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Measurement Data ◽  
Data Driven ◽  
High Dimensional ◽  
Data Sets ◽  
Nonlinear Dynamical ◽  
Tensor Network ◽  
Wide Range ◽  
Multidimensional Approximation

A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method multidimensional approximation of nonlinear dynamical systems (MANDy) which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems.

QUENCHING LORENZIAN CHAOS

2004 ◽  
Vol 14 (08) ◽  
pp. 2875-2884 ◽  
Author(s):  
RAYMOND HIDE ◽  
PATRICK E. McSHARRY ◽  
CHRISTOPHER C. FINLAY ◽  
GUY D. PESKETT
Keyword(s):  
Parameter Space ◽  
Control Parameter ◽  
Nonlinear Dynamical Systems ◽  
General Interest ◽  
Nonlinear Ordinary Differential Equations ◽  
The Novel ◽  
Nonlinear Dynamical ◽  
F Region ◽  
Dimensionless Variables ◽  
Autonomous Set

How fluctuations can be eliminated or attenuated is a matter of general interest in the study of steadily-forced dissipative nonlinear dynamical systems. Here, we extend previous work on "nonlinear quenching" [Hide, 1997] by investigating the phenomenon in systems governed by the novel autonomous set of nonlinear ordinary differential equations (ODE's) [Formula: see text], ẏ=-xzq+bx-y and ż=xyq-cz (where (x, y, z) are time(t)-dependent dimensionless variables and [Formula: see text], etc.) in representative cases when q, the "quenching function", satisfies q=1-e+ey with 0≤e≤1. Control parameter space based on a,b and c can be divided into two "regions", an S-region where the persistent solutions that remain after initial transients have died away are steady, and an F-region where persistent solutions fluctuate indefinitely. The "Hopf boundary" between the two regions is located where b=bH(a, c; e) (say), with the much studied point (a, b, c)=(10, 28, 8/3), where the persistent "Lorenzian" chaos that arises in the case when e=0 was first found lying close to b=bH(a, c; 0). As e increases from zero the S-region expands in total "volume" at the expense of F-region, which disappears altogether when e=1 leaving persistent solutions that are steady throughout the entire parameter space.

scholarly journals Normal Form for High-Dimensional Nonlinear System and Its Application to a Viscoelastic Moving Belt

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
S. P. Chen ◽  
Y. H. Qian
Keyword(s):  
Normal Form ◽  
Nonlinear Dynamical Systems ◽  
High Dimensional ◽  
Computation Method ◽  
Double Zero ◽  
Nonlinear Dynamical ◽  
Novel Method ◽  
Explicit Formulae ◽  
The Stability ◽  
Pure Imaginary Eigenvalues

This paper is concerned with the computation of the normal form and its application to a viscoelastic moving belt. First, a new computation method is proposed for significantly refining the normal forms for high-dimensional nonlinear systems. The improved method is described in detail by analyzing the four-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain three different cases, that are, (i) two pairs of pure imaginary eigenvalues, (ii) one nonsemisimple double zero and a pair of pure imaginary eigenvalues, and (iii) two nonsemisimple double zero eigenvalues. Then, three explicit formulae are derived, herein, which can be used to compute the coefficients of the normal form and the associated nonlinear transformation. Finally, employing the present method, we study the nonlinear oscillation of the viscoelastic moving belt under parametric excitations. The stability and bifurcation of the nonlinear vibration system are studied. Through the illustrative example, the feasibility and merit of this novel method are also demonstrated and discussed.

scholarly journals Chaos and Complexity Dynamics of Evolutionary Systems

2020 ◽  
Author(s):  
Lal Mohan Saha
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Chaotic Motion ◽  
Correlation Dimension ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Attractors ◽  
Tabular Form ◽  
Bifurcation Diagrams ◽  
Nonlinear Dynamical

Chaotic phenomena and presence of complexity in various nonlinear dynamical systems extensively discussed in the context of recent researches. Discrete as well as continuous dynamical systems both considered here. Visualization of regularity and chaotic motion presented through bifurcation diagrams by varying a parameter of the system while keeping other parameters constant. In the processes, some perfect indicator of regularity and chaos discussed with appropriate examples. Measure of chaos in terms of Lyapunov exponents and that of complexity as increase in topological entropies discussed. The methodology to calculate these explained in details with exciting examples. Regular and chaotic attractors emerging during the study are drawn and analyzed. Correlation dimension, which provides the dimensionality of a chaotic attractor discussed in detail and calculated for different systems. Results obtained presented through graphics and in tabular form. Two techniques of chaos control, pulsive feedback control and asymptotic stability analysis, discussed and applied to control chaotic motion for certain cases. Finally, a brief discussion held for the concluded investigation.

Modeling and System Identification using Extended Kalman Filter for a Quadrotor System

2013 ◽  
Vol 313-314 ◽  
pp. 976-981 ◽  
Author(s):  
Norafizah Abas ◽  
Ari Legowo ◽  
Zulkiflie Ibrahim ◽  
Norhidayah Rahim ◽  
Anuar M. Kassim
Keyword(s):  
Kalman Filter ◽  
System Identification ◽  
Extended Kalman Filter ◽  
Nonlinear Dynamical Systems ◽  
Flight Test ◽  
The Other ◽  
Estimation Technique ◽  
Nonlinear Dynamical ◽  
Aerial Vehicle ◽  
Quadrotor System

Quadrotor has emerged as a popular testbed for Unmanned Aerial Vehicle (UAV) research due to its simplicity in construction and maintenance, and its vertical take-off, landing and hovering capabilities. It is a flying rotorcraft that has four lift-generating propellers; two of the propellers rotate clockwise and the other two rotate counter-clockwise. This paper presents modeling and system identification for auto-stabilization of a quadrotor system through the implementation of Extended Kalman Filter (EKF). EKF has known to be typical estimation technique used to estimate the state vectors and parameters of nonlinear dynamical systems. In this paper, two main processes are highlighted; dynamic modeling of the quadrotor and the implementation of EKF algorithms. The aim is to obtain a more accurate dynamic modelby identify and estimate the needed parameters for thequadrotor. The obtained results demonstrate the performances of EKF based on the flight test applied to the quadrotor system.

Overview of Methodologic Issues for Pharmacologic Trials in Mild, Moderate, and Severe Alzheimer's Disease

1996 ◽  
Vol 8 (2) ◽  
pp. 159-193 ◽  
Author(s):  
Barry Reisberg ◽  
Emile H. Franssen ◽  
Maciej Bobinski ◽  
Stefanie Auer ◽  
Isabel Monteiro ◽  
...  
Keyword(s):  
Alzheimer’S Disease ◽  
Alzheimer's Disease ◽  
The Other ◽  
Aging Brain ◽  
Behavior Changes ◽  
Alternative Approach ◽  
Early Aging ◽  
Brain Behavior ◽  
State Examination

To address the issue of mild, moderate, and severe Alzheimer's disease (AD), it is necessary to initially establish some agreement on terminology. In recent decades, these terms have frequently been defined using screening instrument scores with measures such as the Mini-Menal State Examination (MMSE). There are many problems with this approach, perhaps the most salient of which is that it has contributed to the total and tragic neglect of patients with severe AD. An alternative approach to the classification of AD severity is staging. This approach has advanced to the point where moderately severe and severe AD can be described in detail. Procedures for describing this previously neglected latter portion of AD have recently been extensively validated. Staging is also uniquely useful at the other end of the severity spectrum, in differentiating early aging brain/behavior changes, incipient AD, and mild AD. Temporally, with staging procedures, it is possible to track the course of AD approximately three times more accurately than with the MMSE. The net result of the advances in AD delineation is that issues such as prophylaxis, modification of course, treatment of behavioral distrubances, loss of ambulation, progressive rigidity, and the development of contractures in AD patients can now be addressed in a scientifically meaningful way that will hopefully bestow much benefit in AD patients and those who care for them.

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