scholarly journals A New Approach in Deterministic Global Optimisation of Problems with Ordinary Differential Equations

2004 ◽  
pp. 83-108 ◽  
Author(s):  
B. Chachuat ◽  
M. A. Latifi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Global Optimisation ◽  
New Approach

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 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.

On a New Approach to the Solution of the Nonlinear Filtering Equation of Jump Processes

1996 ◽  
Vol 10 (1) ◽  
pp. 153-163 ◽  
Author(s):  
Kaisheng Fan
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Filtering ◽  
Jump Processes ◽  
Algebraic Equations ◽  
Normalization Procedure ◽  
New Approach ◽  
Partially Observed ◽  
On Line

An implementable on-line approach to solve the nonlinear filtering equation for a partially observed system in which both the state and observation processes are jump processes is presented. By making use of the special structure of jump processes, the new method allows us to obtain the solution of the resulting nonlinear filtering equation from solving a linear system of ordinary differential equations and a linear system of algebraic equations recursively and via a simple normalization procedure.

A New Approach to Solving Periodic Differential Systems

2021 ◽  
Vol 1 ◽  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Oscillatory Motion ◽  
Differential Systems ◽  
New Approach

Mathematicians and physicists are well acquainted with second-order ordinary differential equations (ODE), the most prominent of them being the class of equations that govern oscillatory motion.

Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments

2020 ◽  
Vol 70 (2) ◽  
pp. 389-400
Author(s):  
Simona Fišnarová ◽  
Robert Mařík
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
New Approach ◽  
Oscillation Criteria ◽  
Neutral Differential Equations ◽  
Oscillatory Properties

Abstract Neutral differential equations are one of the most important extensions of classical ordinary differential equations and aim to give a better explanation for modeling phenomena where ordinary differential equations are insufficient. Naturally, all the questions studied in the scope of ordinary differential equations attracted the attention also for neutral differential equations. In this paper we study the oscillatory properties of second order half-linear neutral differential equations. We present oscillation criteria derived using a new approach. This approach allows us to reduce common restrictions on the deviations in arguments which are present in the currently known results of this type.

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