COMPUTATION AND VISUALIZATION OF BIFURCATION SURFACES

2008 ◽  
Vol 18 (08) ◽  
pp. 2191-2206 ◽  
Author(s):  
DIRK STIEFS ◽  
THILO GROSS ◽  
RALF STEUER ◽  
ULRIKE FEUDEL
Keyword(s):  
Dynamical System ◽  
Critical Parameter ◽  
Nonlinear Dynamical System ◽  
Three Dimensional ◽  
Global Dynamics ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
Nonlinear Dynamical ◽  
Qualitative Understanding ◽  
Local And Global Dynamics

The localization of critical parameter sets called bifurcations is often a central task of the analysis of a nonlinear dynamical system. Bifurcations of codimension 1 that can be directly observed in nature and experiments form surfaces in three-dimensional parameter spaces. In this paper, we propose an algorithm that combines adaptive triangulation with the theory of complex systems to compute and visualize such bifurcation surfaces in a very efficient way. The visualization can enhance the qualitative understanding of a system. Moreover, it can help to quickly locate more complex bifurcation situations corresponding to bifurcations of higher codimension at the intersections of bifurcation surfaces. Together with the approach of generalized models the proposed algorithm enables us to gain extensive insights in the local and global dynamics not only in one special system but in whole classes of systems. To illustrate this ability we analyze three examples from different fields of science.

INTERACTIVE GRAPHICAL EXPLORATION OF MULTIDIMENSIONAL NONLINEAR DYNAMICAL SYSTEMS

1992 ◽  
Vol 02 (02) ◽  
pp. 251-261 ◽  
Author(s):  
JUDY CHALLINGER
Keyword(s):  
Dynamical System ◽  
Parameter Space ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Three Dimensional ◽  
Initial Point ◽  
Unit Cube ◽  
Dimensional Parameter ◽  
Data Set ◽  
Nonlinear Dynamical

This paper discusses the application of an inherently three-dimensional graphical representation tool, isosurfaces, as a means to interactively explore and visualize the attractors of a nonlinear dynamical system with a fifteen-dimensional parameter space. A program has been written which allows the scientist to interactively select and visualize three-dimensional sub-spaces of the fifteen-dimensional parameter space. The dynamical system used to illustrate these concepts is a discrete-time, nonlinear, three-nation Richardson model with economic constraints. This dynamical system, which models the shifting alliances of nations in an arms race, maps an initial point in the unit cube to another point in the unit cube after multiple iterations of the model functions. Using an isosurface function on the resulting volumetric data set, surfaces indicating the changing alliances of nations are generated and rendered.

Prediction of Solutions of the Duffing System with Tuned Mass Damper

2020 ◽  
Vol 22 (4) ◽  
pp. 983-990
Author(s):  
Konrad Mnich
Keyword(s):  
Dynamical System ◽  
Duffing Oscillator ◽  
Probabilistic Approach ◽  
Nonlinear Dynamical System ◽  
Tuned Mass Damper ◽  
Nonlinear Dynamical ◽  
Duffing System ◽  
Mass Damper ◽  
Basin Stability ◽  
Stability Concept

AbstractIn this work we analyze the behavior of a nonlinear dynamical system using a probabilistic approach. We focus on the coexistence of solutions and we check how the changes in the parameters of excitation influence the dynamics of the system. For the demonstration we use the Duffing oscillator with the tuned mass absorber. We mention the numerous attractors present in such a system and describe how they were found with the method based on the basin stability concept.

DYNAMICAL ESTIMATION OF CALENDAR EFFECTS

2006 ◽  
Vol 06 (01) ◽  
pp. L7-L15
Author(s):  
ALEXANDROS LEONTITSIS
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Noise Level ◽  
Nonlinear Dynamical System ◽  
Main Tool ◽  
Experimental Results ◽  
Accurate Estimation ◽  
Nonlinear Dynamical ◽  
Correlation Integral ◽  
Calendar Effects

The paper introduces a method for estimation and reduction of calendar effects from time series, which their fluctuations are governed by a nonlinear dynamical system and additive normal noise. Calendar effects can be considered deviations of the distribution(s) of particular group(s) of observations that have a common characteristic related to the calendar. The concept of this method is the following: since the calendar effects are not related to the dynamics of the time series, the accurate estimation and reduction will result a time series with a smaller amount of noise level (i.e. more accurate dynamics). The main tool of this method is the correlation integral, due to its inherit capability of modeling both the dynamics and the additive normal noise. Experimental results are presented on the Nasdaq Cmp. index.

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