scholarly journals Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs

2015 ◽  
Vol 36 (2) ◽  
pp. 763-784 ◽  
Author(s):  
Xijun Hu ◽  
Penghui Wang
Keyword(s):  
Periodic Solutions ◽  
Trace Formula ◽  
Type Formula

scholarly journals Trace formulas for a class of non-Fredholm operators: A review

2016 ◽  
Vol 28 (10) ◽  
pp. 1630002 ◽  
Author(s):  
Alan Carey ◽  
Fritz Gesztesy ◽  
Harald Grosse ◽  
Galina Levitina ◽  
Denis Potapov ◽  
...  
Keyword(s):  
Essential Spectrum ◽  
Trace Formula ◽  
Differential Operators ◽  
Trace Class ◽  
Spectral Shift ◽  
General Situation ◽  
Spectral Flow ◽  
Fredholm Operators ◽  
Type Formula ◽  
Non Fredholm Operators

Take a one-parameter family of self-adjoint Fredholm operators [Formula: see text] on a Hilbert space [Formula: see text], joining endpoints [Formula: see text]. There is a long history of work on the question of whether the spectral flow along this path is given by the index of the operator [Formula: see text] acting in [Formula: see text], where [Formula: see text] denotes the multiplication operator [Formula: see text] for [Formula: see text]. Most results are about the case where the operators [Formula: see text] have compact resolvent. In this article, we review what is known when these operators have some essential spectrum and describe some new results. Using the operators [Formula: see text], [Formula: see text], an abstract trace formula for Fredholm operators with essential spectrum was proved in [23], extending a result of Pushnitski [35], although, still under strong hypotheses on [Formula: see text]: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text]. Associated to the pairs [Formula: see text] and [Formula: see text] are Krein spectral shift functions [Formula: see text] and [Formula: see text], respectively. From the trace formula, it was shown that there is a second, Pushnitski-type, formula: [Formula: see text] This can be employed to establish the desired equality, [Formula: see text] This equality was generalized to non-Fredholm operators in [14] in the form [Formula: see text] replacing the Fredholm index on the left-hand side by the Witten index of [Formula: see text] and [Formula: see text] on the right-hand side by an appropriate arithmetic mean (assuming [Formula: see text] is a right and left Lebesgue point for [Formula: see text] denoted by [Formula: see text] and [Formula: see text], respectively). But this applies only under the restrictive assumption that the endpoint [Formula: see text] is a relatively trace class perturbation of [Formula: see text] (ruling out general differential operators). In addition to reviewing this previous work, we describe in this article some extensions using a [Formula: see text]-dimensional setup, where [Formula: see text] are non-Fredholm differential operators. By a careful analysis we prove, for a class of examples, that the preceding trace formula still holds in this more general situation. Then we prove that the Pushnitski-type formula for spectral shift functions also holds and this then gives the equality of spectral shift functions in the form [Formula: see text] for the [Formula: see text]-dimensional model operator at hand. This shows that neither the relatively trace class perturbation assumption nor the Fredholm assumption are required if one works with spectral shift functions. The results support the view that the spectral shift function should be a replacement for the spectral flow in certain non-Fredholm situations and also point the way to the study of higher-dimensional cases. We discuss the connection with summability questions in Fredholm modules in an appendix.

On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

Sign in / Sign up

Export Citation Format

Share Document