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.

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.

The Generalized Shooting Arc-Length Method for Periodic Solutions and the Corresponding Periods of Nonlinear Dynamical Systems

2001 ◽  
Author(s):  
Dexin Li ◽  
Jianxue Xu
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Periodic Orbit ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Optimization Method ◽  
Arc Length ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Nonlinear Dynamical

Abstract In this paper, a generalized shooting/arc-length method for determining periodic orbit and its period of nonlinear dynamical system is presented. At first, by changing the time scale the period value of periodic orbit of the nonlinear system is drawn into the governing equation of this system. Then, by using the period value as a parameter, the shooting/arc-length procedure is taken for seeking such a periodic solution and its period simultaneously. The value of increment changed in iteration procedure is selected by using optimization method. The procedure involves the detennining of periodic orbit and its period value of the system. Thereby, the periodic orbit and period value of the system can be sought out rapidly and precisely. At last, the validity of such method is verified by determining the periodic orbit and period value for van der pol equation and nonlinear rotor-bear system.

scholarly journals Nonlinear Network Dynamics with Consensus–Dissensus Bifurcation

2021 ◽  
Vol 31 (1) ◽  
Author(s):  
Karel Devriendt ◽  
Renaud Lambiotte
Keyword(s):  
Dynamical System ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Global Bifurcation ◽  
Basins Of Attraction ◽  
Complete Graphs ◽  
Stationary States ◽  
Qualitative Properties ◽  
Nonlinear Dynamical ◽  
Tree Graphs

AbstractWe study a nonlinear dynamical system on networks inspired by the pitchfork bifurcation normal form. The system has several interesting interpretations: as an interconnection of several pitchfork systems, a gradient dynamical system and the dominating behaviour of a general class of nonlinear dynamical systems. The equilibrium behaviour of the system exhibits a global bifurcation with respect to the system parameter, with a transition from a single constant stationary state to a large range of possible stationary states. Our main result classifies the stability of (a subset of) these stationary states in terms of the effective resistances of the underlying graph; this classification clearly discerns the influence of the specific topology in which the local pitchfork systems are interconnected. We further describe exact solutions for graphs with external equitable partitions and characterize the basins of attraction on tree graphs. Our technical analysis is supplemented by a study of the system on a number of prototypical networks: tree graphs, complete graphs and barbell graphs. We describe a number of qualitative properties of the dynamics on these networks, with promising modelling consequences.

NONLINEAR DYNAMICAL SYSTEM AND CHAOS SYNCHRONIZATION

2008 ◽  
Vol 18 (05) ◽  
pp. 1531-1537 ◽  
Author(s):  
AYUB KHAN ◽  
PREMPAL SINGH
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Numerical Techniques ◽  
Nonlinear Dynamical ◽  
Chaotic Dynamical Systems ◽  
Response System ◽  
Identical Response

Chaos synchronization of nonlinear dynamical systems has been studied through theoretical and numerical techniques. For the synchronization of two identical nonlinear chaotic dynamical systems a theorem has been constructed based on the Lyapunov function, which requires a minimal knowledge of system's structure to synchronize with an identical response system. Numerical illustrations have been provided to verify the theorem.

scholarly journals The properties of children's language are caused by those of their mothers', and vice-versa

2020 ◽  
Author(s):  
Fermín Moscoso del Prado Martín ◽  
Jeremy Irvin ◽  
Daniel Spokoyny
Keyword(s):  
Dynamical System ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Related Question ◽  
Specific Response ◽  
Language Abilities ◽  
Causal Relations ◽  
Nonlinear Dynamical ◽  
Parent Child Interactions ◽  
Convergent Cross Mapping

Both the language used by young children (child language; CL) and the simplified language used by caretakers when talking to them (child-directed speech; CDS) become increasingly complex along development, eventually approaching regular adult language. Researchers disagree on whether children learn grammar from the input they receive (*usage-based theories*), or grammars are mostly innate, requiring only minimal input-based adjustments on the part of the children (*nativist theories*). A related question is whether parents adapt the complexity of CDS in specific response to their children's language abilities, or only in response to their level of general cognitive development. Parent-child interactions can be modelled by nonlinear dynamical systems. A technique recently developed in Ecology, Convergent Cross-Mapping (CCM), enables assessing causal relations between series from a nonlinear dynamical system. Here, we use CCM to reconstruct the network of *causal* relations between aspects of CL and CDS. This network supports a lexically-based syntax that is statistically learned. The results also indicate that mothers adapt CDS in response mainly to the *grammar* of CL. Our findings verify the strong causal predictions of usage-based theories, and are difficult for nativist theories to account for.

Amplitude-Dependent Stiffness Method for Studying Frequency and Damping Variations in Nonlinear Dynamical Systems

2017 ◽  
Author(s):  
Mohammad A. AL-Shudeifat
Keyword(s):  
Dynamical System ◽  
Nonlinear Dynamical Systems ◽  
Energy Content ◽  
Equations Of Motion ◽  
A Priori ◽  
Nonlinear Dynamical System ◽  
Nonlinear Coupling ◽  
Nonlinear Dynamical ◽  
Coupling Stiffness ◽  
Nonlinear Energy

A method is introduced here for extracting the fundamental backbone branches of the frequency energy plot in which the obtained nonlinear frequencies of the nonlinear dynamical system are plotted with respect to the nonlinear energy content. The proposed method is directly applied to the equations of motion where the solution is not required to be known a priori. The method is based on linearizing the nonlinear coupling force where a scaled amplitude-dependent coupling stiffness force is obtained to replace the original nonlinear coupling stiffness force. Accordingly, the backbone branches in the frequency-nonlinear-energy plot are extracted from the eigensolution of the mass-normalized amplitude-dependent global stiffness matrix of the nonlinear dynamical system. Moreover, the variations in the damping content under the effect of the nonlinear coupling stiffness are also studied. Interesting behavior of damping content under the effect of the amplitude-dependent stiffness has been observed and discussed.

On Periodic Motions in the First-Order Nonlinear Systems

2016 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yeyin Xu ◽  
Zhaobo Chen
Keyword(s):  
Dynamical System ◽  
Fourier Series ◽  
Dynamical Systems ◽  
Analytical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Nonlinear Dynamical System ◽  
Periodic Motions ◽  
Nonlinear Dynamical ◽  
First Order

In this paper, analytical solutions of periodic motions in the first-order nonlinear dynamical system are discussed from the finite Fourier series expression. The first-order nonlinear dynamical system is transformed to the dynamical system of coefficients in the Fourier series. From investigation of such dynamical system of coefficients, the analytical solutions of periodic motions are obtained, and the corresponding stability and bifurcation of periodic motions will be determined. In fact, this method provides a frequency-response analysis of periodic motions in nonlinear dynamical systems, which is alike the Laplace transformation of periodic motions for nonlinear dynamical systems. The harmonic frequency-amplitude curves are obtained for different-order harmonic terms in the Fourier series. Through such frequency-amplitude curves, the nonlinear characteristics of periodic motions in the first-order nonlinear system can be determined. From analytical solutions, the initial conditions are obtained for numerical simulations. From such initial conditions, numerical simulations are completed in comparison of the analytical solutions of periodic motions.

METRIC UNIVERSALITY OF ORDER IN ONE-DIMENSIONAL DYNAMICS

1993 ◽  
Vol 03 (03) ◽  
pp. 567-572
Author(s):  
MICHAEL FRAME ◽  
DAVID PEAK
Keyword(s):  
Dynamical System ◽  
Critical Point ◽  
Parameter Space ◽  
Nonlinear Dynamical System ◽  
Unimodal Maps ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
Wide Range ◽  
Parameter Values ◽  
Universal Representation

The orbit of the critical point of a nonlinear dynamical system defines a family of functions in the parameter space of the system. For unimodal maps a renormalization makes these functions indistinguishable over a wide range of parameter values. The universal representation of these functions leads directly to a number of interesting results: (1) the positions in the parameter space of the windows of order; (2) the sizes of the windows of order; (3) measures of distortion in the window structure; and (4) various generalized Feigenbaum numbers. We explicitly discuss the examples of the quadratic and sine maps.

scholarly journals On Algorithm of Integrability Classification of the Nonlinear Dynamical Systems via Computer Algebra Methods

2021 ◽  
pp. 7-12
Author(s):  
Bohdan Fil ◽  
Yaroslav Pelekh ◽  
Myroslava Vovk ◽  
Halyna Beregova ◽  
Tatiana Magerovska ◽  
...  
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Conservation Laws ◽  
Computer Algebra ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Nonlinear Dynamical ◽  
Wolfram Mathematica ◽  
De Vries

There is developed an algorithm to classify integrable nonlinear dynamical systems via Wolfram Mathematica. The hierarchy of conservation laws for the nonlinear dynamical system can be cal-culated by this algorithm. There are demonstrated some modifications of nonlinear Korteweg-de Vries equations integrated by inverse scatering method.

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