Averaging and integral manifolds

1970 ◽  
Vol 2 (2) ◽  
pp. 197-222 ◽  
Author(s):  
W. A. Coppel ◽  
K. J. Palmer
Keyword(s):  
Autonomous System ◽  
Method Of Characteristics ◽  
Integral Manifold ◽  
Artificial Viscosity ◽  
Original Method ◽  
Method Of Averaging ◽  
System Of Differential Equations ◽  
Viscosity Term ◽  
Integral Manifolds ◽  
The Individual

An integral manifold for a system of differential equations is a manifold such that any solution of the equations which has a point on it is entirely contained on it. The method of averaging establishes the existence of such a manifold for a system which is a perturbation of an autonomous system with a periodic orbit. The existence of the manifold is established here under more general hypotheses, namely for perturbations which are ‘integrally small’. The method differs from the original method of Bogolyubov and Mitropolskii and operates directly with the individual solutions. This is made possibly by the use of an appropriate norm, and is equivalent to solving the partial differential equation which occurs in work by Moser and Sacker by the method of characteristics rather than by the introduction of an artificial viscosity term. Moreover, detailed smoothness properties of the manifold are obtained. For periodic perturbations the integral manifold is a torus and these smoothness properties are just sufficient to permit the application of Denjoy's theorem.

scholarly journals Two Conjectures of G.D. Birkhoff

1999 ◽  
Vol 172 ◽  
pp. 439-440
Author(s):  
Christopher K. Mccord ◽  
Kenneth R. Meyer
Keyword(s):  
Angular Momentum ◽  
Integral Manifold ◽  
Center Of Mass ◽  
Angular Momentum Vector ◽  
Momentum Vector ◽  
Algebraic Set ◽  
Special Values ◽  
Integral Manifolds ◽  
Energy Center ◽  
Three Body

The spatial (planar) three-body problem admits the ten (six) integrals of energy, center of mass, linear momentum and angular momentum. Fixing these integrals defines an eight (six) dimensional algebraic set called the integral manifold, 𝔐(c, h) (m(c, h)), which depends on the energy level h and the magnitude c of the angular momentum vector. The seven (five) dimensional reduced integral manifold, 𝔐R(c, h) (mR(c, h)), is the quotient space 𝔐(c, h)/SO2 (m(c, h)/SO2) where the SO2 action is rotation about the angular momentum vector. We want to determine how the geometry or topology of these sets depends on c and h. It turns out that there is one bifurcation parameter, ν = −c2h, and nme (six) special values of this parameter, νi, i = 1, …, 9.At each of the special values the geometric restrictions imposed by the integrals change, but one of these values, ν5, does not give rise to a change in the topology of the integral manifolds 𝔐(c, h) and 𝔐R(c, h). The other eight special values give rise to nine different topologically distinct cases. We give a complete description of the geometry of these sets along with their homology. These results confirm some conjectures and refutes several others.

scholarly journals Spiral Shocks in Accretion Disks with SPH

1997 ◽  
Vol 163 ◽  
pp. 770-770
Author(s):  
James Rhys Murray
Keyword(s):  
Accretion Disks ◽  
Equations Of Motion ◽  
Fixed Ratio ◽  
Artificial Viscosity ◽  
Spiral Density Waves ◽  
Viscosity Term ◽  
Particle Hydrodynamics ◽  
Smoothed Particle ◽  
Low Mass ◽  
Continued Use

AbstractSmoothed Particle Hydrodynamics (SPH) is now seen as a numerical scheme well suited to the study of accretion disks. SPH simulations have been conducted of cataclysmic variable disks (Lubow 1991, Murray 1996, Armitage and Livio 1996), galactic disks (Artymowicz and Lubow 1989), and protostellar disks (Artymowicz and Lubow 1994). It is therefore important to test the technique against theory and other numerical results to obtain an estimate of the accuracy and reliability of SPH in this context. Previously SPH has been tested against standard stationary and time-dependent results of viscous thin disk theory (Murray 1996). Strictly these tests relate to disks where ‘viscous’ terms dominate pressure terms in the equations of motion.In this paper we describe tests of the code more appropriate for hot disks where pressure forces are relatively more important than viscosity. Specifically we consider the form of the spiral density waves that can be excited in a disk by a perturbing gravitational potential. Very low mass perturbing bodies excite linear spiral waves which redistribute angular momentum in the disk. For increasingly massive perturbers, the disk response becomes nonlinear and eventually shocks form. In the standard formulation of SPH, an artificial viscosity term is added to the SPH equations to improve shock capture. This is equivalent to introducing a fixed ratio of shear to bulk viscosity into the equations of motion. In Eulerian schemes, artificial viscosity has been discarded in favour of other more accurate, less dissipative schemes for resolving shocks. The continued use of artificial viscosity in SPH has become a source of ‘friction’ between numericists. The simulations described here demonstrate the scheme’s ability to resolve spiral shocks, and show that SPH is a valuable tool for probing the structure of tidally perturbed accretion disks.

scholarly journals Parameters affecting water hammer in metal pipelines

2018 ◽  
Vol 44 ◽  
pp. 00183
Author(s):  
Kamil Urbanowicz ◽  
Mateusz Firkowski
Keyword(s):  
Numerical Modelling ◽  
Method Of Characteristics ◽  
Water Hammer ◽  
Supply System ◽  
Hydraulic Systems ◽  
Wave Pressure ◽  
Hydraulic Machinery ◽  
Metal Pipes ◽  
The Individual ◽  
Pressure Changes

The water hammer related to rapid wave pressure changes in hydraulic systems have been subjected to intensive research for more than a hundred years. Nevertheless, a large number of new papers appear each year. Current literature indicates model differences resulting from the used material of the pipe. In the hydraulic machinery, elastic (metal) pipes are usually used, while water transport in water supply system is currently realized with pipes whose deformation of the walls is viscoelastic. In this paper, the individual and group impact of all parameters influencing the results of numerical modelling of the water hammer occurring in the pipes will be analysed. The method of characteristics will be used to solve partial differential equations describing the flow.

scholarly journals Robust Control of Underactuated Systems: Higher Order Integral Sliding Mode Approach

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Sami ud Din ◽  
Qudrat Khan ◽  
Fazal ur Rehman ◽  
Rini Akmeliawati
Keyword(s):  
Robust Control ◽  
Closed Loop ◽  
Sliding Mode ◽  
Integral Manifold ◽  
Control Design ◽  
Underactuated Systems ◽  
Control Laws ◽  
Beam System ◽  
Integral Manifolds ◽  
Robust Control Design

This paper presents a robust control design for the class of underactuated uncertain nonlinear systems. Either the nonlinear model of the underactuated systems is transformed into an input output form and then an integral manifold is devised for the control design purpose or an integral manifold is defined directly for the concerned class. Having defined the integral manifolds discontinuous control laws are designed which are capable of maintaining sliding mode from the very beginning. The closed loop stability of these systems is presented in an impressive way. The effectiveness and demand of the designed control laws are verified via the simulation and experimental results of ball and beam system.

ON DYNAMICAL SYSTEMS GENERATED BY TWO ALTERNATING VECTOR FIELDS

1996 ◽  
Vol 06 (11) ◽  
pp. 2015-2030 ◽  
Author(s):  
A. KLÍČ ◽  
P. POKORNÝ
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Time Evolution ◽  
Autonomous System ◽  
Vector Fields ◽  
Numerical Study ◽  
Method Of Averaging ◽  
Controlling Of Chaos

Dynamical systems with time evolution determined by two alternating vector fields are investigated both analytically and numerically. When the two vector fields are related by an involutory diffeomorphism G then the fixed points of G (either isolated or non-isolated) are shown to give rise to branches of periodic solutions of the resulting non-autonomous system. The method of averaging is used for small switching periods. Detailed numerical study of both conservative (“blinking vortex”) and dissipative (“blinking nodes”, “blinking cycles” and “blinking Lorenz”) systems shows that the technique of blinking can be used to initiating and controlling of chaos.

scholarly journals The method of the exophthalmos value predicted calculation when planning the orbital decompression procedure in patients with endocrine ophthalmopathy

2021 ◽  
Vol 14 (3) ◽  
pp. 41-48
Author(s):  
Konstantin A. Konovalov ◽  
Dmitrii V. Davydov ◽  
Dmitrii Anatolevich Lezhnev
Keyword(s):  
Statistical Significance ◽  
Original Method ◽  
Ophthalmological Examination ◽  
Orbital Decompression ◽  
Orbital Fat ◽  
Additional Equipment ◽  
Endocrine Ophthalmopathy ◽  
Group A ◽  
Decompression Procedure ◽  
The Individual

BACKGROUND: The most effective method of surgical treatment of lipogenic and mixed forms of edematous exophthalmos is currently the internal orbital decompression. During this surgical procedure, the excessive pathologically altered adipose tissue is removed from the external and the internal surgical spaces of the orbit. Many scientists are developing methods for calculating the volume of orbital fat, but the question on developing a method for predicted exophthalmos after internal orbital decompression, which could be used without attracting additional equipment and software, is easy to learn and does not require a long calculation time, remains actual. This method has to take into account the individual features of the patients orbital structure and be used for calculations in the bilateral proptosis correction. AIM: To develop and evaluate the effectiveness of a new method for calculating the eyeball position after orbital decompression. MATERIALS AND METHODS: 64 patients (126 orbits) with lipogenic and mixed forms of endocrine ophthalmopathy were examined. All patients underwent internal orbital decompression, during which the orbital fat was removed, the volume of which was calculated according to the developed original method. Patients underwent ophthalmological examination and MSCT before surgery and 6 months after it. . RESULTS: As a result of orbital decompression in the examined group, a decrease in proptosis was observed in all patients, and the exophthalmos calculated by the method corresponded to the eyeball position in patients in 6 months after surgery. The level of statistical significance of the planned postoperative eyeball position in relation to the actual postoperative exophthalmos calculated according to the Students t-test was 0.98 (p 0.05), that is, it can be argued that the groups do not differ, and no statistically significant differences were found. CONCLUSIONS: The developed method for calculating the estimated postoperative exophthalmos is effective without using additional software. This technique allows you to achieve a symmetrical eyeball position in the postoperative period and to reduce the risk of complications.

scholarly journals Averaging and integral manifolds (II)

1970 ◽  
Vol 2 (3) ◽  
pp. 369-399 ◽  
Author(s):  
K. J. Palmer
Keyword(s):  
Real Line ◽  
Integral Manifold ◽  
The Real ◽  
Stability Properties ◽  
Particular Solutions ◽  
Integral Manifolds ◽  
The Stability ◽  
Positive Half

In the first part of this paper (written jointly with W.A. Coppel) the existence and properties of an integral manifold were established for the systemx′ = f(t, x, y)y′ = A(t)y + g(t, x, y)where f and g are “integrally small”. In this second part of the paper the stability properties of the integral manifold are investigated. Solutions are found which are bounded on the positive half of the real line and it is shown that these solutions approach the manifold exponentially and, moreover, that they are asymptotic to particular solutions on the manifold.

Sign in / Sign up

Export Citation Format

Share Document