On the Understanding of Chaos in Duffings Equation Including a Comparison With Experiment

1986 ◽  
Vol 53 (1) ◽  
pp. 5-9 ◽  
Author(s):  
E. H. Dowell ◽  
C. Pezeshki
Keyword(s):  
Theoretical Model ◽  
Initial Value Problem ◽  
Physical Experiment ◽  
Chaotic Oscillations ◽  
External Excitation ◽  
Initial Value ◽  
Comparison Of Results ◽  
Buckled Beam ◽  
Comparison With Experiment ◽  
Forced Excitation

The dynamics of a buckled beam are studied for both the initial value problem and forced external excitation. The principal focus is on chaotic oscillations due to forced excitation. In particular, a discussion of their relationship to the initial value problem and a comparison of results from a theoretical model with those from a physical experiment are presented.

On the Threshold Force for Chaotic Motions for a Forced Buckled Beam

1988 ◽  
Vol 55 (1) ◽  
pp. 190-196 ◽  
Author(s):  
D. M. Tang ◽  
E. H. Dowell
Keyword(s):  
Experimental Data ◽  
Numerical Simulations ◽  
System Response ◽  
Chaotic Oscillations ◽  
External Excitation ◽  
Chaotic Motions ◽  
Higher Modes ◽  
Threshold Force ◽  
Buckled Beam

The effects of higher modes on the chaotic oscillations of a buckled beam under forced external excitation are studied. Of principal interest are the threshold force required for chaotic motions and the influence of damping on the system response. A comparison is also presented of results from numerical simulations with experimental data.

An evaluation of a model equation for water waves

1981 ◽  
Vol 302 (1471) ◽  
pp. 457-510 ◽  
Keyword(s):  
Theoretical Model ◽  
Initial Value Problem ◽  
Water Waves ◽  
Laboratory Experiments ◽  
Initial Conditions ◽  
Initial Value ◽  
Dispersive Waves ◽  
Weakly Nonlinear ◽  
Spatial Periodicity ◽  
Long Channel

This study assesses a particular model for the unidirectional propagation of water waves, comparing its predictions with the results of a set of laboratory experiments. The equation to be tested is a one-dimensional representation of weakly nonlinear, dispersive waves in shallow water. A model for such flows was proposed by Korteweg & de Vries (1895) and this has provided the theoretical basis for a number of laboratory experiments. Some recent studies that have been made in the area are those of Zabusky & Galvin (1971), Hammack (1973) and Hammack & Segur (1974). In each case the theoretical model gave a good qualitative account of the experiments, but the quantitative comparisons were not very extensive. One of the purposes of this paper is to provide a more detailed quantitative assessment of a particular model than has been given to date. An important aspect of the formulation of the theoretical model is the specification of the initial conditions and the boundary conditions for the equation. Zabusky & Galvin (1971) considered an initial-value problem having spatial periodicity, whereas Hammack (1973) and Hammack & Segur (1974) considered an initial-value problem posed on the real line. In contrast, we shall consider an initial-value problem posed on the half line with boundary data specified at the origin. This problem was chosen to correspond with an experiment in which waves were generated at one end of a long channel, and obviates certain difficulties inherent in the other formulations.

scholarly journals Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

2018 ◽  
Vol 5 (1) ◽  
pp. 102-112 ◽  
Author(s):  
Shekhar Singh Negi ◽  
Syed Abbas ◽  
Muslim Malik
Keyword(s):  
Time Scales ◽  
Initial Value Problem ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Type Inequality ◽  
Second Order ◽  
Oscillation Criterion ◽  
Initial Value ◽  
Nonlinear Dynamic Equation ◽  
Singular Initial Value Problem

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

scholarly journals A Neural Network Technique for the Derivation of Runge-Kutta Pairs Adjusted for Scalar Autonomous Problems

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1842
Author(s):  
Vladislav N. Kovalnogov ◽  
Ruslan V. Fedorov ◽  
Yuri A. Khakhalev ◽  
Theodore E. Simos ◽  
Charalampos Tsitouras
Keyword(s):  
Neural Network ◽  
Differential Evolution ◽  
Initial Value Problem ◽  
Fitness Function ◽  
Initial Value ◽  
Runge Kutta

We consider the scalar autonomous initial value problem as solved by an explicit Runge-Kutta pair of orders 6 and 5. We focus on an efficient family of such pairs, which were studied extensively in previous decades. This family comes with 5 coefficients that one is able to select arbitrarily. We set, as a fitness function, a certain measure, which is evaluated after running the pair in a couple of relevant problems. Thus, we may adjust the coefficients of the pair, minimizing this fitness function using the differential evolution technique. We conclude with a method (i.e. a Runge-Kutta pair) which outperforms other pairs of the same two orders in a variety of scalar autonomous problems.

scholarly journals On the sub–diffusion fractional initial value problem with time variable order

2021 ◽  
Vol 10 (1) ◽  
pp. 1301-1315
Author(s):  
Eduardo Cuesta ◽  
Mokhtar Kirane ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Regularity Result ◽  
Time Variable ◽  
Variable Order ◽  
Initial Value ◽  
Integration By Parts ◽  
Integration By Parts Formula ◽  
Variable Order Fractional Derivative

Abstract We consider a fractional derivative with order varying in time. Then, we derive for it a Leibniz' inequality and an integration by parts formula. We also study an initial value problem with our time variable order fractional derivative and present a regularity result for it, and a study on the asymptotic behavior.

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