scholarly journals Kam theory, Lindstedt series and the stability of the upside-down pendulum

2002 ◽  
Vol 9 (2) ◽  
pp. 413-426
Author(s):  
Kyriakos Georgiou ◽  
G. Gentile ◽  
Michele Bartuccelli
Keyword(s):  
Kam Theory ◽  
Lindstedt Series ◽  
The Stability

scholarly journals KAM Theory in Configuration Space and Cancellations in the Lindstedt Series

2010 ◽  
Vol 302 (2) ◽  
pp. 359-402 ◽  
Author(s):  
Livia Corsi ◽  
Guido Gentile ◽  
Michela Procesi
Keyword(s):  
Configuration Space ◽  
Kam Theory ◽  
Lindstedt Series

scholarly journals Nonlinear Stability of the Triangular Libration Points for Radiating and Oblate Primaries in CR3BP in Nonresonance Condition

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Nutan Singh ◽  
A. Narayan
Keyword(s):  
Resonance Frequency ◽  
Radiation Pressure ◽  
Nonlinear Stability ◽  
Equilibrium Points ◽  
Kam Theory ◽  
Boundary Region ◽  
Libration Points ◽  
Triangular Libration Points ◽  
Nonresonance Condition ◽  
The Stability

This paper investigates the existence of resonance and nonlinear stability of the triangular equilibrium points when both oblate primaries are luminous. The study is carried out near the resonance frequency, satisfying the conditionsω1=ω2,  ω1=2ω2, andω1=3ω2in circular cases by the application of Kolmogorov-Arnold-Moser (KAM) theory. The study is carried out for the various values of radiation pressure and oblateness parameters in general. It is noticed that the system experiences resonance atω1=2ω2,  ω1=3ω2for different values of radiation pressures and oblateness parameter. The caseω1=ω2corresponds to the boundary region of the stability for the system. It is found that, except for some values of the radiation pressure, and oblateness parameters and forμ≤μc=0.0385209, the triangular equilibrium points are stable.

SYNCHRONOUS/ASYNCHRONOUS PATTERNS IN THREE-CELL NETWORKS WITH MULTIPLE TIME DELAYS

2011 ◽  
Vol 21 (06) ◽  
pp. 1701-1718
Author(s):  
YUAN YUAN ◽  
LEI LI
Keyword(s):  
Time Delays ◽  
Fredholm Alternative ◽  
Alternative Theory ◽  
Multiple Time ◽  
Multiple Time Delays ◽  
Lindstedt Series ◽  
Leading Term ◽  
Coupled Cell Systems ◽  
The Stability ◽  
Cell Networks

Some patterns of synchrony/asynchrony in the dynamics of coupled cell systems can be predicted by symmetry. However, in the system without symmetry, the different patterns of periodic solutions may exist as well. We consider a general model including three cells with multiple time delays that connect in any possible manner. Our approach is based on the analytic construction by using a perturbation procedure together with the Fredholm alternative theory. Then we employ the Poincaré–Lindstedt series expansion to compute the Floquet exponents which determine the stability. Finally, we resort to numerical computation to get some insights about the leading term in the Floquet exponents, and the numerical simulations are given to confirm the theoretical results.

scholarly journals Kolmogorov-Arnold-Moser Theory and Symmetries for a Polynomial Quadratic Second Order Difference Equation

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 790
Author(s):  
Tarek F. Ibrahim ◽  
Zehra Nurkanović
Keyword(s):  
Difference Equation ◽  
Equilibrium Points ◽  
Kam Theory ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Elliptic Equilibrium ◽  
Order Difference Equation ◽  
The Difference ◽  
The Stability ◽  
Theoretical Results

By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of two elliptic equilibrium points (zero equilibrium and negative equilibrium) of the difference equation t n + 1 = α t n + β t n 2 − t n − 1 , n = 0 , 1 , 2 , … , where are t − 1 , t 0 , α ∈ R , α ≠ 0 , β > 0 . By using the symmetries we find the periodic solutions with some periods. Finally, some numerical examples are given to verify our theoretical results.

scholarly journals TWISTLESS KAM TORI, QUASI FLAT HOMOCLINIC INTERSECTIONS, AND OTHER CANCELLATIONS IN THE PERTURBATION SERIES OF CERTAIN COMPLETELY INTEGRABLE HAMILTONIAN SYSTEMS: A REVIEW

1994 ◽  
Vol 06 (03) ◽  
pp. 343-411 ◽  
Author(s):  
GIOVANNI GALLAVOTTI
Keyword(s):  
Perturbation Theory ◽  
Invariant Tori ◽  
Kam Theory ◽  
Perturbation Expansion ◽  
Rotation Velocity ◽  
Perturbation Series ◽  
Kam Tori ◽  
Stable And Unstable Manifolds ◽  
Rotation Numbers ◽  
The Stability

Rotators interacting with a pendulum via small, velocity independent, potentials are considered. If the interaction potential does not depend on the pendulum position then the pendulum and the rotators are decoupled and we study the invariant tori of the rotators system at fixed rotation numbers: we exhibit cancellations, to all orders of perturbation theory, that allow proving the stability and analyticity of the diophantine tori. We find in this way a proof of the KAM theorem by direct bounds of the k-th order coefficient of the perturbation expansion of the parametric equations of the tori in terms of their average anomalies: this extends Siegel's approach, from the linearization of analytic maps to the KAM theory; the convergence radius does not depend, in this case, on the twist strength, which could even vanish ("twistless KAM tori"). The same ideas apply to the case in which the potential couples the pendulum and the rotators: in this case the invariant tori with diophantine rotation numbers are unstable and have stable and unstable manifolds ("whiskers"): instead of studying the perturbation theory of the invariant tori we look for the cancellations that must be present because the homoclinic intersections of the whiskers are "quasi flat", if the rotation velocity of the quasi periodic motion on the tori is large. We rederive in this way the result that, under suitable conditions, the homoclinic splitting is smaller than any power in the period of the forcing and find the exact asymptotics in the two dimensional cases (e.g. in the case of a periodically forced pendulum). The technique can be applied to study other quantities: we mention, as another example, the homoclinic scattering phase shifts.

scholarly journals Birkhoff Normal Forms and KAM Theory for Gumowski-Mira Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
M. R. S. Kulenović ◽  
Z. Nurkanović ◽  
E. Pilav
Keyword(s):  
Normal Forms ◽  
Kam Theory ◽  
Equilibrium Solutions ◽  
Stability Of Equilibrium ◽  
The Stability

By using the KAM theory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation:xn+1=(2axn)/(1+xn2)-xn-1,  n=0,1,…, wherex-1,x0∈(-∞,∞), and we obtain the Birkhoff normal forms for this equation for different equilibrium solutions.

Stability of May’s Host–Parasitoid model with variable stocking upon parasitoids

2021 ◽  
pp. 2150072
Author(s):  
Senada Kalabušić ◽  
Esmir Pilav
Keyword(s):  
Growth Rate ◽  
Software Package ◽  
Equilibrium Points ◽  
Kam Theory ◽  
Intrinsic Growth Rate ◽  
Interior Equilibrium ◽  
Biological Meaning ◽  
Boundary Equilibrium ◽  
Parasitoid Population ◽  
The Stability

Using the Kolmogorov–Arnold–Mozer (KAM) theory, we investigate the stability of May’s host–parasitoid model’s solutions with proportional stocking upon the parasitoid population. We show the existence of the extinction, boundary, and interior equilibrium points. When the host population’s intrinsic growth rate and the releasement coefficient are less than one, both populations are extinct. There are an infinite number of boundary equilibrium points, which are nonhyperbolic and stable. Under certain conditions, there appear 1:1 nonisolated resonance fixed points for which we thoroughly described dynamics. Regarding the interior equilibrium point, we use the KAM theory to prove its stability. We give a biological meaning of obtained results. Using the software package Mathematica, we produce numerical simulations to support our findings.

Open discussion

1982 ◽  
Vol 99 ◽  
pp. 605-613
Author(s):  
P. S. Conti
Keyword(s):  
Buenos Aires ◽  
Large Mass ◽  
List Type ◽  
Common Phenomenon ◽  
Open Discussion ◽  
The Stability ◽  
Ring Nebulae

Conti: One of the main conclusions of the Wolf-Rayet symposium in Buenos Aires was that Wolf-Rayet stars are evolutionary products of massive objects. Some questions:–Do hot helium-rich stars, that are not Wolf-Rayet stars, exist?–What about the stability of helium rich stars of large mass? We know a helium rich star of ∼40 MO. Has the stability something to do with the wind?–Ring nebulae and bubbles : this seems to be a much more common phenomenon than we thought of some years age.–What is the origin of the subtypes? This is important to find a possible matching of scenarios to subtypes.

Symmetric Multistep Methods Revisited: II. Numerical Experiments

1999 ◽  
Vol 173 ◽  
pp. 309-314 ◽  
Author(s):  
T. Fukushima
Keyword(s):  
Multistep Methods ◽  
Computational Time ◽  
Integration Time ◽  
Implicit Methods ◽  
Symmetric Methods ◽  
Celestial Bodies ◽  
The Stability ◽  
Predictor Corrector ◽  
Symmetric Multistep Methods ◽  
Integration Errors

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

Uniformity of Magnification from Different Electron Microscopes

1967 ◽  
Vol 25 ◽  
pp. 32-33
Author(s):  
Godfrey C. Hoskins ◽  
V. Williams ◽  
V. Allison
Keyword(s):  
Diffraction Grating ◽  
The Other ◽  
Carbon Replica ◽  
Electron Microscopes ◽  
The Stability

The method demonstrated is an adaptation of a proven procedure for accurately determining the magnification of light photomicrographs. Because of the stability of modern electrical lenses, the method is shown to be directly applicable for providing precise reproducibility of magnification in various models of electron microscopes.A readily recognizable area of a carbon replica of a crossed-line diffraction grating is used as a standard. The same area of the standard was photographed in Phillips EM 200, Hitachi HU-11B2, and RCA EMU 3F electron microscopes at taps representative of the range of magnification of each. Negatives from one microscope were selected as guides and printed at convenient magnifications; then negatives from each of the other microscopes were projected to register with these prints. By deferring measurement to the print rather than comparing negatives, correspondence of magnification of the specimen in the three microscopes could be brought to within 2%.

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