scholarly journals On Using Curvature to Demonstrate Stability

2008 ◽  
Vol 2008 ◽  
pp. 1-7
Author(s):  
C. Connell McCluskey
Keyword(s):  
Differential Equations ◽  
Global Stability ◽  
Ordinary Differential Equations ◽  
Periodic Orbits ◽  
Large Amplitude ◽  
Minimum Distance ◽  
New Approach ◽  
Sudden Appearance ◽  
Globally Stable ◽  
Rule Out

A new approach for demonstrating the global stability of ordinary differential equations is given. It is shown that if the curvature of solutions is bounded on some set, then any nonconstant orbits that remain in the set, must contain points that lie some minimum distance apart from each other. This is used to establish a negative-criterion for periodic orbits. This is extended to give a method of proving an equilibrium to be globally stable. The approach can also be used to rule out the sudden appearance of large-amplitude periodic orbits.

scholarly journals Large Amplitude Vibrations of Thin Hyperelastic Plates: Neo-Hookean Model

2013 ◽  
Author(s):  
Ivan Breslavsky ◽  
Marco Amabili ◽  
Mathias Legrand
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Large Amplitude ◽  
Local Model ◽  
Static Deflection ◽  
Geometrical Nonlinearities ◽  
The Difference ◽  
Large Amplitude Vibrations ◽  
Quadratic And Cubic Nonlinearities ◽  
Rubber Plate

Static deflection and large amplitude vibrations of a rubber plate are analyzed. Both the geometrical and physical (material) nonlinearities are taken into account. The properties of the plate hyperelastic material are described by the Neo-Hookean law. A method for building a local model, which approximates the plate behavior around a deformed configuration, is proposed. This local model takes the form of a system of ordinary differential equations with quadratic and cubic nonlinearities. The results obtained with the help of this local model are compared to the solution of the exact model, and are found to be accurate. The difference between the model retaining both physical and geometrical non-inearities and a model with only geometrical nonlinearities is also analyzed. It is found that influence of physically induced nonlinearities at moderate strains is significant.

scholarly journals HARMONIC BALANCE FOR NON-PERIODIC HYPERBOLIC SOLUTIONS OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

2017 ◽  
Vol 22 (2) ◽  
pp. 140-156 ◽  
Author(s):  
Serge Bruno Yamgoue ◽  
Olivier Tiokeng Lekeufack ◽  
Timoleon Crepin Kofane
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Great Accuracy ◽  
Nonlinear Ordinary Differential Equations ◽  
New Approach ◽  
Heteroclinic Solutions ◽  
Analytical Approximations

In this paper, we propose a new approach for obtaining explicit analytical approximations to the homoclinic or heteroclinic solutions of a general class of strongly nonlinear ordinary differential equations describing conservative singledegree-of-freedom systems. Through a simple and explicit change of the independent variable that we introduce, these equations are transformed to others for which the original homoclinic or heteroclinic solutions are mapped into periodic solutions that satisfy some boundary conditions. Recent simplified methods of harmonic balance can then be exploited to construct highly accurate analytic approximations to these solutions. Here, we adopt the combination of Newton linearization with the harmonic balance to construct the approximates in incremental steps, thereby proposing both appropriate initial approximates and increments that together satisfy the required boundary conditions. Three examples including a septic Duffing oscillator, a controlled mechanical pendulum and a perturbed KdV equations are presented to illustrate the great accuracy and simplicity of the new approach.

Perturbations of Multipliers of Systems of Periodic Ordinary Differential Equations

2011 ◽  
Vol 3 (5) ◽  
pp. 562-571
Author(s):  
Leonid Berezansky ◽  
Michael Gil’ ◽  
Liora Troib
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Systems ◽  
New Approach ◽  
Stability And Instability ◽  
Perturbed Systems ◽  
The Stability

AbstractThe paper deals with periodic systems of ordinary differential equations (ODEs). A new approach to the investigation of variations of multipliers under perturbations is suggested. It enables us to establish explicit conditions for the stability and instability of perturbed systems.

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