scholarly journals RECONSIDERING TRIGONOMETRIC INTEGRATORS

2009 ◽  
Vol 50 (3) ◽  
pp. 320-332 ◽  
Author(s):  
DION R. J. O’NEALE ◽  
ROBERT I. MCLACHLAN
Keyword(s):  
Hamiltonian System ◽  
Differential Equations ◽  
Nonlinear Stability ◽  
Higher Order ◽  
Statistical Properties ◽  
Highly Oscillatory ◽  
Time Averages ◽  
Highly Oscillatory Differential Equations ◽  
Low Order

AbstractIn this paper we look at the performance of trigonometric integrators applied to highly oscillatory differential equations. It is widely known that some of the trigonometric integrators suffer from low-order resonances for particular step sizes. We show here that, in general, trigonometric integrators also suffer from higher-order resonances which can lead to loss of nonlinear stability. We illustrate this with the Fermi–Pasta–Ulam problem, a highly oscillatory Hamiltonian system. We also show that in some cases trigonometric integrators preserve invariant or adiabatic quantities but at the wrong values. We use statistical properties such as time averages to further evaluate the performance of the trigonometric methods and compare the performance with that of the mid-point rule.

scholarly journals Multi-revolution composition methods for highly oscillatory differential equations

2014 ◽  
Vol 128 (1) ◽  
pp. 167-192 ◽  
Author(s):  
Philippe Chartier ◽  
Joseba Makazaga ◽  
Ander Murua ◽  
Gilles Vilmart
Keyword(s):  
Differential Equations ◽  
Composition Methods ◽  
Highly Oscillatory ◽  
Highly Oscillatory Differential Equations

Interaction of Lower and Higher Order Hamiltonian Resonances

2018 ◽  
Vol 28 (08) ◽  
pp. 1850097 ◽  
Author(s):  
Ferdinand Verhulst
Keyword(s):  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Nonlinear Stability ◽  
Normal Forms ◽  
Double Resonance ◽  
Higher Order ◽  
Order Resonance ◽  
Resonance Zone ◽  
Stability Results ◽  
Low Order

The tools of normal forms and recurrence are used to analyze the interaction of low and higher order resonances in Hamiltonian systems. The resonance zones where the short-periodic solutions of the low order resonances exist are characterized by small variations of the corresponding actions that match the variations of the higher order resonance; this yields cases of embedded double resonance. The resulting interaction produces periodic solutions that in some cases destabilize a resonance zone. Applications are given to the three dof [Formula: see text] resonance and to periodic FPU-chains producing unexpected nonlinear stability results and quasi-trapping phenomena.

scholarly journals Word series high-order averaging of highly oscillatory differential equations with delay

2019 ◽  
Vol 4 (2) ◽  
pp. 445-454 ◽  
Author(s):  
J. M. Sanz-Serna ◽  
Beibei Zhu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
High Order ◽  
Constant Delay ◽  
Integer Multiple ◽  
Averaged Equations ◽  
Highly Oscillatory ◽  
Highly Oscillatory Differential Equations ◽  
Delay Differential ◽  
Equations With Delay

AbstractWe show that, when the delay is an integer multiple of the forcing period, it is possible to obtain easily high-order averaged versions of periodically forced systems of delay differential equations with constant delay. Our approach is based on the use of word series techniques to obtain high-order averaged equations for differential equations without delay.

scholarly journals Oscillations of a higher order nonlinear differential equations with impulses

2010 ◽  
Vol 43 (3) ◽  
Author(s):  
Kun-wen Wen ◽  
Gen-Qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Impulsive Differential Equations ◽  
Higher Order ◽  
Jump Conditions ◽  
Oscillation Criteria ◽  
Oscillation Properties ◽  
Low Order

AbstractImpulsive differential equations are important in simulation of processes with jump conditions. Although there are quite a few studies on the oscillation properties of low order equations, there are not too many studies of higher order equations. In this paper, we derive several oscillation criteria which are either new or improve several recent results in the literature. In addition, we provide several examples to illustrate the use of our results.

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