scholarly journals Climate Change in a Differential Equations Course: Using Bifurcation Diagrams to Explore Small Changes with Big Effects

CODEE Journal ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 1-10
Author(s):  
Justin Dunmyre ◽  
Nicholas Fortune ◽  
Tianna Bogart ◽  
Chris Rasmussen ◽  
Karen Keene
Keyword(s):  
Climate Change ◽  
Differential Equations ◽  
Bifurcation Diagrams

THE THEORY OF CONFINORS IN CHUA’S CIRCUIT: ACCURATE ANALYSIS OF BIFURCATIONS AND ATTRACTORS

1993 ◽  
Vol 03 (02) ◽  
pp. 333-361 ◽  
Author(s):  
RENÉ LOZI ◽  
SHIGEHIRO USHIKI
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Phase Portrait ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Bifurcation Diagrams ◽  
Accurate Analysis ◽  
Chua's Circuit

We apply the new concept of confinors and anti-confinors, initially defined for ordinary differential equations constrained on a cusp manifold, to the equations governing the circuit dynamics of Chua’s circuit. We especially emphasize some properties of the confinors of Chua’s equation with respect to the patterns in the time waveforms. Some of these properties lead to a very accurate numerical method for the computation of the half-Poincaré maps which reveal the precise structures of Chua’s strange attractors and the exact bifurcation diagrams with the help of a special sequence of change of coordinates. We also recall how such accurate methods allow the reliable numerical observation of the coexistence of three distinct chaotic attractors for at least one choice of the parameters. Chua’s equation seemssurprisingly rich in very new behaviors not yet reported even in other dynamical systems. The application of the theory of confinors to Chua’s equation and the use of sequences of Taylor’s coordinates could give new perspectives to the study of dynamical systems by uncovering very unusual behaviors not yet reported in the literature. The main paradox here is that the theory of confinors, which could appear as a theory of rough analysis of the phase portrait of Chua’s equation, leads instead to a very accurate analysis of this phase portrait.

scholarly journals A Study on the Complexity of a New Chaotic Financial System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Yi Liao ◽  
Yiran Zhou ◽  
Fei Xu ◽  
Xiao-Bao Shu
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Financial System ◽  
System Of Differential Equations ◽  
Bifurcation Diagrams ◽  
Complex Dynamic ◽  
System Parameters ◽  
Wide Range

The interaction of elements in a financial system can exhibit complex dynamic behaviours. In this article, we use a system of differential equations to model the evolution of a financial system and study its complexity. Numerical simulations show that the system exhibits a variety of rich dynamic behaviours, including chaos. Bifurcation diagrams show that the system behaves chaotically over a wide range of system parameters.

Studies in Chaotic Vibrations of Buckled Beams

1989 ◽  
Vol 42 (11S) ◽  
pp. S100-S107 ◽  
Author(s):  
S. Hanagud ◽  
N. S. Abhyankar ◽  
R. Chander
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Phase Plane ◽  
Neutral Axis ◽  
Bifurcation Diagrams ◽  
Partial Differential ◽  
Buckled Beams ◽  
Chaotic Vibrations

We have studied chaotic vibrations of buckled beams with base excitations. Nonlinearities due to the stretching of the neutral axis, transverse stops and their combination have been studied. The resulting ordinary and partial differential equations have been integrated numerically. Results have been presented in the form of phase plane plots and bifurcation diagrams.

Parametric Resonance of Axially Accelerating Unidirectional Plates Partially Immersed in Fluid

2020 ◽  
Author(s):  
Hongying Li ◽  
Shumeng Zhang ◽  
Jian Li ◽  
Xibo Wang
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Chaotic Behavior ◽  
Plate Theory ◽  
Three Dimension ◽  
Bifurcation Diagrams ◽  
Moving Plate ◽  
Fluid Structure ◽  
Coupled Equations ◽  
Motion State

Abstract This paper investigates the nonlinear vibration of an axially accelerating moving plate considering fluid-structure interaction. Nonlinear coupled equations of motion are derived by means of Kármán plate theory, the Galerkin method is then applied to transform the nonlinear partial differential equations into nonlinear ordinary differential equations. The steady-state response, various bifurcations and chaotic behavior of the system are studied by the multiple scales method and Runge-Kutta method. The dynamical characteristics of the system are examined via response curves and bifurcation diagrams of Poincaré maps. By three-dimension bifurcation diagrams, change of motion state can be easily observed along with the variation of system parameters during the whole parametric space, meanwhile, it is found that fluctuation amplitude plays a most significant role in the change of motion state for the fluid-structure coupling system.

A method for the complete qualitative analysis of two coupled ordinary differential equations dependent on three parameters

1988 ◽  
Vol 416 (1851) ◽  
pp. 361-389 ◽  
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Ordinary Differential Equations ◽  
Parameter Space ◽  
Bifurcation Theory ◽  
Parameter Plane ◽  
Phase Portraits ◽  
Relevant Parameter ◽  
Bifurcation Diagrams ◽  
Independent Parameters

In this paper, recent advances in bifurcation theory are specialized to systems describable by two coupled ordinary differential equations (ODEs) containing at most three independent parameters. For such systems, by plotting in the relevant parameter plane the locus of successively degenerate singular points, a complete description of all the qualitatively distinct behaviour of the system can be obtained. The description is in terms of phase portraits and bifurcation diagrams. Even though much use is made of existing results obtained via local analyses, the results of this technique cover the entire parameter space. Furthermore, because the information is built up in successive stages the question of whether the parameters universally unfold a given degeneracy does not arise. This can mean a major saving in effort, particularly for degenerate Hopf points. Finally if, as is often the case, the parameters appear in the system in a simple way, the procedure can be applied analytically because the variables (which will appear non-linearly) can be used to parametrize the relevant loci.

CHAOS IN A THREE-DIMENSIONAL GENERAL MODEL OF NEURAL NETWORK

2002 ◽  
Vol 12 (10) ◽  
pp. 2271-2281 ◽  
Author(s):  
A. DAS ◽  
PRITHA DAS ◽  
A. B. ROY
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
General Model ◽  
Three Dimensional ◽  
Stationary Regime ◽  
Decay Rates ◽  
Phase Portraits ◽  
Study Phase ◽  
Bifurcation Diagrams ◽  
Parameter Values

The dynamics of a network of three neurons with all possible connections is studied here. The equations of control are given by three differential equations with nonlinear, positive and bounded sigmoidal response function of the neurons. The system passes from stable to periodic and then to chaotic regimes and returns to stationary regime with change in parameter values of synaptic weights and decay rates. We have developed programs and used Locbif package to study phase portraits, bifurcation diagrams which confirm the result. Lyapunov Exponents have been calculated to confirm chaos.

Stability and Bifurcations in Three-Dimensional Analysis of Axially Moving Beams

2013 ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Marco Amabili ◽  
Hamed Farokhi
Keyword(s):  
Differential Equations ◽  
Fourier Transforms ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Time Integration ◽  
Three Dimensional ◽  
Axially Moving Beam ◽  
Bifurcation Diagrams ◽  
Response Curves ◽  
Axially Moving

The geometrically nonlinear dynamics of a three-dimensional axially moving beam is investigated numerically for both sub and supercritical regimes. Hamilton’s principle is employed to derive the equations of motion for in-plane and out-of plane displacements. The Galerkin scheme is applied to the nonlinear partial differential equations of motion yielding a set of second-order nonlinear ordinary differential equations with coupled terms. The pseudo-arclength continuation technique is employed to solve the discretized equations numerically so as to obtain the nonlinear resonant responses; direct time integration is conducted to obtain the bifurcation diagrams of the system. The results are presented in the form of the frequency-response curves, bifurcation diagrams, time histories, phase-plane portraits, and fast Fourier transforms for different sets of system parameters.

Chaos in Robot Control Equations

1997 ◽  
Vol 07 (03) ◽  
pp. 707-720 ◽  
Author(s):  
Shrinivas Lankalapalli ◽  
Ashitava Ghosal
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
System Of Nonlinear Equations ◽  
Nonlinear Ordinary Differential Equations ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Repetitive Motion ◽  
Route To Chaos

The motion of a feedback controlled robot can be described by a set of nonlinear ordinary differential equations. In this paper, we examine the system of two second-order, nonlinear ordinary differential equations which model a simple two-degree-of-freedom planar robot, undergoing repetitive motion in a plane in the absence of gravity, and under two well-known robot controllers, namely a proportional and derivative controller and a model-based controller. We show that these differential equations exhibit chaotic behavior for certain ranges of the proportional and derivative gains of the controller and for certain values of a parameter which quantifies the mismatch between the model and the actual robot. The system of nonlinear equations are non-autonomous and the phase space is four-dimensional. Hence, it is difficult to obtain significant analytical results. In this paper, we use the Lyapunov exponent to test for chaos and present numerically obtained chaos maps giving ranges of gains and mismatch parameters which result in chaotic motions. We also present plots of the chaotic attractor and bifurcation diagrams for certain values of the gains and mismatch parameters. From the bifurcation diagrams, it appears that the route to chaos is through period doubling.

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