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ADVANCES in MATHEMATICS and COMPUTER SCIENCE and THEIR
APPLICATIONS
Proceedings of the 18th International Conference on Mathematical Methods, Computational Techniques and Intelligent Systems (MAMECTIS '16)
Proceedings of the 9th International Conference on Manufacturing
Engineering, Quality and Production Systems (MEQAPS '16)
Proceedings of the 9th International Conference on Urban Planning and Transportation (UPT '16)
Proceedings of the 7th European Conference on Applied Mathematics and
Informatics (AMATHI '16)
Venice, Italy January 29-31, 2016
Mathematics and Computers in Science and Engineering Series | 57
ISSN: 2227-4588 ISBN: 978-1-61804-360-3
ADVANCES in MATHEMATICS and COMPUTER SCIENCE and THEIR APPLICATIONS Proceedings of the 18th International Conference on Mathematical Methods, Computational Techniques and Intelligent Systems (MAMECTIS '16) Proceedings of the 9th International Conference on Manufacturing Engineering, Quality and Production Systems (MEQAPS '16) Proceedings of the 9th International Conference on Urban Planning and Transportation (UPT '16) Proceedings of the 7th European Conference on Applied Mathematics and Informatics (AMATHI '16) Venice, Italy January 29-31, 2016 Published by WSEAS Press www.wseas.org Copyright © 2016, by WSEAS Press All the copyright of the present book belongs to the World Scientific and Engineering Academy and Society Press. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Editor of World Scientific and Engineering Academy and Society Press. All papers of the present volume were peer reviewed by no less that two independent reviewers. Acceptance was granted when both reviewers' recommendations were positive.
ISSN: 2227-4588 ISBN: 978-1-61804-360-3
ADVANCES in MATHEMATICS and COMPUTER SCIENCE and THEIR
APPLICATIONS
Proceedings of the 18th International Conference on Mathematical Methods, Computational Techniques and Intelligent Systems (MAMECTIS '16)
Proceedings of the 9th International Conference on Manufacturing
Engineering, Quality and Production Systems (MEQAPS '16)
Proceedings of the 9th International Conference on Urban Planning and Transportation (UPT '16)
Proceedings of the 7th European Conference on Applied Mathematics and
Informatics (AMATHI '16)
Venice, Italy January 29-31, 2016
Editor: Prof. Imre J. Rudas, Obuda University, Hungary Associate Editor: Prof. Prof. Sang C. Lee, Daegu Gyeongbuk Institute of Science and Technology, Korea Committee Members-Reviewers: Anca Croitoru Sk. Sarif Hassan Gabriela Mircea Isaac Yeboah Dana Anderson Elena Scutelnicu Daniela Litan Ibrahim Canak Vasile Cojocaru Hamideh Eskandari Claudia-Georgeta Carstea Elena Zaitseva Mihai Tiberiu Lates Yixin Bao Gabriella Bognar Muhammet Koksal Md. Shamim Akhter Vassos Vassiliou Yuqing Zhou Maria Dobritoiu U. C. Jha Bhagwati Prasad Emmanuel Lopez-Neri Carla Pinto Jose Manuel Mesa Fernández Zanariah Abdul Majid David Nicoleta Snezhana Georgieva Gocheva-Ilieva Al Emran Ismail Robert L. Bishop Glenn Loury Fernando Alvarez Morris Adelman Mark J. Perry Reinhard Neck Ricardo Gouveia Rodrigues Ehab Bayoumi Igor Kuzle Maria do Rosario Alves Calado Gheorghe-Daniel Andreescu Bharat Doshi Gang Yao Biswa Nath Datta Gamal Elnagar Goricanec Darko Lu Peng Pavel Loskot Shuliang Li Panos Pardalos Ronald Yager Stephen Anco
Adrian Constantin Ying Fan Juergen Garloff Y. Jiang Vitale Cardone Leonardo Cascini Domenico Guida Joseph Quartieri Stefano Riemma Gianfranco Rizzo Mario Vento Hyung Hee Cho Kumar Tamma Igor Sevostianov Cho W. Solomon To Jiin-Yuh Jang Robert Reuben Ali K. El Wahed Yury A. Rossikhin Pierre-Yves Manach Ramanarayanan Balachandran Sorinel Adrian Oprisan Yoshihiro Tomita J. Quartieri Marcin Kaminski ZhuangJian Liu Abdullatif Ben-Nakhi Ottavia Corbi Xianwen Kong Ahmet Selim Dalkilic Essam Eldin Khalil Jose Alberto Duarte Moller Seung-Bok Choi Marina Shitikova Junwu Wang Jia-Jang Wu Moran Wang Gilbert-Rainer Gillich Thomas Panagopoulos Jon Bryan Burley Cyril Fleaurant Biagio Guccione Jose Beltrao Ioannis Ispikoudis Giuseppe Genon Jose Luis Miralles Carlos Guerrero Inga Straupe Giuseppe Luigi Cirelli Francesco Ferrini Sarma Cakula
Claudio Guarnaccia Haziq Jeelani Ivan G. Avramidi Michel Chipot Xiaodong Yan Ravi P. Agarwal Yushun Wang Patricia J. Y. Wong Jim Zhu H. M. Srivastava Martin Bohner Martin Schechter Ferhan M. Atici Marco Sabatini Gerd Teschke Meirong Zhang Lucio Boccardo Shanhe Wu Natig M. Atakishiyev Jianming Zhan Narcisa C. Apreutesei Chun-Gang Zhu Naseer Shahzad Sining Zheng Satit Saejung Juan J. Trujillo Tiecheng Xia Stevo Stevic Lucas Jodar Noemi Wolanski Zhenya Yan Juan Carlos Cortes Lopez Wei-Shih Du Kailash C. Patidar Janusz Brzdek Jinde Cao Josef Diblik Jianqing Chen Abdelghani Bellouquid
Table of Contents
Mathematical Aspects of Risk Management in Interconnected Utility Networks 11 Stefan Schauer, Sandra König, Stefan Rass Interval Methods of Optimization in Condition of Uncertainty 18 Vitaly Levin Hybrid Reformulation Based on a New Hybrid Ohm’s Law for an Electrical Energy Hybrid Systems
27
Sang C. Lee, Doo Seok Lee Effect of First Order Chemical Reaction on Magneto Convection of Immiscible Fluids in a Vertical Channel
32
J. C. Umavathi, Maurizio Ssaso, J. Prathap Kumar A Stochastic Model for Real Estate Pricing Estimation and Description of Its Dynamics Based on the Pseudo-Poisson Type Processes
40
Oleg Rusakov, Michael Laskin How Embedding Nanoparticles Effects Semiconductor Thermal Conductivity 44 Giovanni Mascali Integrability in Elementary Functions of Certain Classes of Nonconservative Systems 50 Maxim V. Shamolin Verb-Object Selectional Preferences in Chinese Based on Distributional Semantic Model 59 Chao Li, Fuji Ren, Xin Kang A Conceptual Design of Next Generation Computer : Hybrid Spectral Computing and Analog Wave Computing
63
Sang C. Lee, Hyun Lee A Fresh Look at Lefebvre’s Spatial Triad and Differential Space: A Central Place in Planning Theory?
68
Michael Leary-Owhin Cataloguing of Historical Masonry Bridges in the Campania Region for Damage Evaluation 74 Alessandro Baratta, Ileana Corbi, Ottavia Corbi, Francesca Tropeano Turning Brownfield Redevelopment-the Case of Summerset at Frick Park Neighborhood 78 Xiaodan Li, Hao Yang Integrating Form Defects of Mechanical Joints into the Tolerance Studies 84 Yann Ledoux, Serge Samper, Julien Grandjean Experimental Wind Tunnel Testing of a New Multidisciplinary Morphing Wing Model 90 Michel Joel Tchatchueng Kammegne, Ruxandra Mihaela Botez, Mahmoud Mamou, Youssef Mebarki, Andreea Koreanschi, Oliviu Sugar Gabor, Teodor Lucian Grigorie
Deformations and Elements of Deformation Theory 98 Nikolaj Glazunov On Approximations by Polynomial and Trigonometrical Integro-Differential Splines of Two Variables
103
I. G. Burova, S. V. Poluyanov Recent Development of Control Theory from the Exterior Differential System and First Integral Theoretical Point of View
110
Heungju Ahn Embracing the Unknown in Intelligent Systems 115 Rodolfo A. Fiorini Medium Access Control (MAC) Enabling Parallelism to Mitigate Exposed Terminals Issues in Multihop Wireless Mesh Network (WMN)
125
Vigneswara Rao Gannapathy, Ahamed Fayeez Bin Tuani Ibrahim, Lim Kim Chuan, Mohamad Kadim Bin Suaidi
Structural Damage Index Identification for Historical Masonry Bridges 135 Alessandro Baratta, Ileana Corbi, Ottavia Corbi, Francesca Tropeano Relativistic and Hereditary Viewpoints on the Dimensionless Hybrid Transformation 139 Sang C. Lee, Heung Ju Ahn A New Theorem on the Localization of Factored Fourier Series 144 Hikmet Seyhan Ӧzarslan, Sebnem Yildiz Euler/Quasi-Wavelet Method for the Variable Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term
147
Leijie Qiao, Xu Da Hybrid Modelling by Projective Geometry 159 Doo Seok Lee, Sang C. Lee Statistics of k-µ Random Variable 163 Dragana Krstić, Zoran Jovanović, Radmila Gerov, Dragan Radenković, Vladeta Milenković Application of Synthetic Pseudo Panels to Investigate the Out-of-Home Discretionary Activity Duration
172
Naznin Sultana Daisy, Mohammad Hesam Hafezi, Muhammad Ahsanul Habib Interpolation of B1-Spline Curve 191 Hyung Bae Jung, Doo-Yeoun Cho, Ju-Hwan Cha Signific Knowlede in Mathematical Science 196 Al Mutairi Alya O. An Improved Harris-Affine Invariant Interest Point Detector 199 Karina Perez-Daniel, Enrique Escamilla, Mariko Nakano, Hector Perez-Meana
Classifier-Assisted Optimization Frameworks 209 Yoel Tenne Characterizations of Feebly Totally Open Functions 217 Raja Mohammad Latif, Mohammad Rafiq Raja, Muhammad Razaq Convergence Analysis of an Extended Newton-Type Method for Implicit Functions and their Solution Mappings
224
Mohammed Harunor Rashid Infrastructure Development and Financing Approaches in Developing Nations: Constraints and Opportunities
235
Olufemi A. Oyedele Probabilistic Method of Data Computation, Interpolation and Modeling 252 Dariusz Jacek Jakóbczak Wireless RF Communication Based on TMS320c6713 DSP 258 Neelima Kolhare, Rajshree Sarvadnya Applications of Orthogonal Matrices in 2D Data Retrieval 264 Dariusz Jacek Jakóbczak Comparative Study of Thermally Induced Vibrations in Orthotropic Rectangular Plates: The ANOVA Approach
268
Tripti Johri, Indu Johri Comparison of Individual Charts to Monitor Peroxide Index of Olive Oil 272 Luís M. Grilo, Helena L. Grilo Nonlinear Parabolic PDE-Based Image Filtering 276 Tudor Barbu Authors Index 281
Integrability in Elementary Functions of Certain Classesof Nonconservative Systems
MAXIM V. SHAMOLINLomonosov Moscow State University
Institute of MechanicsMichurinskii Ave., 1, 119192 Moscow
RUSSIAN [email protected], [email protected]
Abstract: We study the non-conservative systems for which the methods for studying, for example, Hamiltoniansystems is not applicable in general. Therefore, for such systems, it is necessary, in some sense, to “directly”integrate the main equation of dynamics. Herewith, we offer more universal interpretation of both obtained casesand new ones of complete integrability in transcendental functions in dynamics in a non-conservative force field.
Key–Words:Dynamical Systems With Variable Dissipation, Integrability
The results of the proposed work are a develop-ment of the previous studies, including a certain ap-plied problem from rigid body dynamics [1], wherecomplete lists of transcendental first integrals ex-pressed through a finite combination of elementaryfunctions were obtained. Later on, this circumstanceallows us to perform a complete analysis of all phasetrajectories and show those their properties whichhave a roughness and are preserved for systems of amore general form. The complete integrability of suchsystems are related to symmetries of latent type.
As is known, the concept of integrability is suffi-ciently fuzzy in general. In its construction, it is nec-essary to take into account the meaning in which it isunderstood(we mean a certain criterion with respectto which one makes a conclusion that the structure oftrajectories of the dynamical system considered is es-pecially simple and “attractive and simple”, in whichclass of functions, we seek for first integrals, etc. (seealso [2]).
In this work, we accept the approach that as theclass of functions for first integrals, takes transcenden-tal functions, and, moreover, elementary ones. Here,the transcendence is understood not in the sense of el-ementary function theory (for example, trigonometricfunctions), but in the sense of existence of essentiallysingular points for them (according to the classifica-tion accepted in the theory of function of one complexvariable). In this case, we need to formally continuethe function considered to the complex domain (seealso [3]).
1 Preliminary Arguments and Re-sults
Of course, in the general case, it is sufficiently difficultto construct a certain theory of integrability for non-conservative systems (even of low dimension). But ina number of cases where the system studied have addi-tional symmetries, we succeed in finding first integralsthrough finite combinations of elementary functions[4].
Proposed work appeared from the plane problemof motion of a rigid body in a resisting medium whosecontact surface with the medium is a part of its ex-terior surface. In this case, the force field is con-structed from the reasons of the medium action on thebody under streamline (or separation) flow around un-der the quasi-stationarity conditions. It turns out thatthe study of motion of such classes of bodies reducesto systems with energy scattering ((purely) dissipativesystems or systems in a dissipative force field) or sys-tems with energy pumping (the so-called systems withanti-dissipation, or systems with dispersing forces).Note that such problems were already appeared in ap-plied aerodynamics (see also [5]).
The previously considered problems have stimu-lated (and stimulate) the development of quality toolsfor studying, which essentially complement the quali-tative theory of nonconservative systems with dissipa-tion of both signs (see also [6]).
Also, we have qualitatively studied nonlinear ef-fects of motion in plane and spatial rigid body dynam-ics and have justified on the qualitative level the neces-sity of introducing the definitions of relative rough-
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ness and relative non-roughness of various degrees(see also [7]).
The following series of results allowed us to pre-pare this work in a very definite style.
i) We have elaborated the methods for qualita-tive studying dissipative systems and systems withanti-dissipation that allo us to obtain the conditionfor bifurcation of birth of stable and unstable auto-oscillations and also the conditions for absence of anysingular trajectories. We succeeded in generalizingthe method for studying plane topographical Poincaresystems to higher dimensions. We have obtained suf-ficient Poisson stability conditions(density near itself)of certain classes of non-closed trajectories of dynam-ical systems [8].
ii) In two- and three-dimensional rigid body dy-namics, we have discovered complete lists of first in-tegrals of dissipative systems and systems with anti-dissipation that are transcendental (in the sense ofclassification of their singularities) functions that areexpressed through elementary functions in a numberof cases. We have introduced new definitions of rela-tive roughness and relative non-roughness of variousdegrees, which have the integrated systems [9].
iii) We have obtained multiparameter families oftopologically non-equivalent phase portraits arising inpurely dissipative systems (i.e., systems with variabledissipation and nonzero (positive) mean). Almost ev-ery portrait of such families is (absolutely) rough [10].
iv) We have discovered new qualitative analogsbetween the properties of motion of free bodies in aresisting medium that is fixed at infinity, and bodies inthe run-of medium flow [11].
Many results of this work were regularly reportedat numerous workshops, including the workshop “Ac-tual Problems of Geometry and Mechanics” named af-ter professor V. V. Trofimov led by D. V. Georgievskiiand M. V. Shamolin.
2 Variable Dissipation DynamicalSystems as a Class of Systems Ad-mitting a Complete Integration
2.1 Visual characteristic of variable dissipa-tion dynamical systems
Since, in the initial modelling of the medium actionon a rigid body, we have used the experimental infor-mation about the properties of streamline flow, it be-comes necessary to study the class of dynamical sys-tems that has the property of (relative) structural sta-bility (relative roughness). Therefore, it is quite natu-ral to introduce these definitions for such systems. In
this case, many of the systems considered are (abso-lutely) rough in the Andronov-Pontryagin sense [12,13].
After some simplifications (for example, in thetwo-dimensional dynamics), the dynamical part of thegeneral system of equations of plane-parallel motioncan be reduced to a pendulum-like second-order sys-tem in which there is a linear nonconservative (sign-alternating) force with coefficients that have differentsigns for different values of the periodic phase coordi-nate in the system.
Therefore, in this case, we will speak of systemswith the so-called variable dissipation, where the term“variable” refers not to the value of the dissipationcoefficient, but to the possible alternation of its sign(therefore, it is more reasonable to use the term “sign-alternating”).
On the average, for the period of the existing peri-odic coordinate, the dissipation can be either positive(“purely” dissipative systems), or negative (systemswith dispersing forces), and also it can be equal to zero(but not identically in this case). In the latter case, wespeak of the zero mean variable dissipation systems(such systems can be associated with “almost” con-servative systems).
As was already mentioned previously, we havedetected important mechanical analogs arising in thecomparison of qualitative properties of the stationarymotion of a free body and the equilibrium of a pen-dulum in a medium flow. Such analogs have a deepsupport meaning since they allow us to extend theproperties of nonlinear dynamical systems for a pen-dulum to dynamical systems for a free body. Boththese systems belong to the class of the so-called zeromean variable dissipation pendulum-like dynamicalsystems.
Under additional conditions, the equivalence de-scribed above is extended to the case of spatial mo-tion, which allows us to speak of the common charac-ter of symmetries in a zero mean variable dissipationsystem under the plane-parallel and spatial motions(for the plane and spatial variants of a pendulum inthe medium flow, see also [1–3]).
In this connection, previously, we have intro-duced the definitions of relative structural stability(relative roughness), and relative structural instability(relative nonroughness) of various degrees. The latterproperties were proved for systems that arise, e.g., in[4, 5].
The purely dissipative systems (as well as(purely) anti-dissipative ones), which in our case canbelong to nonzero zero mean variable dissipation sys-tems, are, as a rule, structurally stable ((absolutely)rough), whereas zero mean variable dissipation sys-tems (which, as a rule, have additional symmetries)
Advances in Mathematics and Computer Science and their Applications
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are either structurally unstable (nonrough) or only rel-atively structurally stable (relatively nonrough). In thegeneral case, it is difficult to prove the latter assertion.
For example, a dynamical system of the form
α = Ω + β sinα,
Ω = −β sinα cosα(1)
is relatively structurally stable (relatively rough) andis topologically equivalent to the system describing aclamped pendulum in an over-run medium flow [6, 7].
Below we present its first integral, which is a tran-scendental (in the sense of the theory of functionsof one complex variable having essentially singularpoints after its continuation to the complex domain)function of phase variables expressed through a fi-nite combination of elementary functions. The phasecylinder R2α, Ω of quasi-velocities of the systemconsidered has an interesting topological structure ofpartition into trajectories (for more detail, see [9]).
Although the dynamical system considered is notconservative, in the rotation domain (and only in it) ofthe phase planeR2α, Ω, it admits the preservationof an invariant measure with variable density. Thisproperty characterizes the system considered as a zeromean variable dissipation system.
2.2 One of the definitions of a zero meanvariable dissipation system
We will study systems of ordinary differential equa-tions having a periodic phase coordinate. The systemsunder study have those symmetries under which, onthe average, for the period in the periodic coordinates,their phase volume is preserved. So, for example, thefollowing pendulum-like system with smooth and pe-riodic in α of periodT right-hand sideV(α, ω) of theform
α = −ω + f(α),ω = g(α),
f(α + T ) = f(α), g(α + T ) = g(α),(2)
preserves its phase area on the phase cylinder over theperiodT : ∫ T
0divV(α, ω)dα =
=∫ T
0
(∂
∂α(−ω + f(α)) +
∂
∂ωg(α)
)dα = (3)
=∫ T
0f ′(α)dα = 0.
The system considered is equivalent to the pendu-lum equation
α− f ′(α)α + g(α) = 0, (4)
in which the integral of the coefficientf ′(α) standingby the dissipative termα is equal to zero on the aver-age for the period.
It is seen that the system considered has thosesymmetries under which it becomes the so-calledzeromean variable dissipation systemin the sense of thefollowing definition.
Definition 1 Consider a smooth autonomous systemof the (n + 1)th order of normal form given on thecylinderRnx × S1α mod2π, whereα is a pe-riodic coordinate of periodT > 0. The divergenceof the right-hand sideV(x, α) (which, in general, is afunction of all phase variables and is not identicallyequal to zero) of this system is denoted by divV(x, α).This system is called a zero (nonzero) mean variabledissipation system if the function
∫ T
0divV(x, α)dα (5)
is equal (not equal) to zero identically. Moreover, insome cases (for example, when singularities arise atseparate points of the circleS1α mod2π), this in-tegral is understood in the sense of principal value.
It should be noted that giving a general definitionof a zero (nonzero) mean variable dissipation systemis not simple. The definition just presented uses theconcept of divergence (as is known, the divergence ofthe right-hand side of a system in normal form charac-terizes the variation of the phase volume in the phasespace of this system).
3 Systems with Symmetries and ZeroMean Variable Dissipation
Let us consider systems of the form (the dot denotesthe derivative in time)
α = fα(ω, sinα, cosα),ωk = fk(ω, sinα, cosα), k = 1, . . . , n,
(6)
given on the set
S1α mod2π\K ×Rnω, (7)
ω = (ω1, . . . , ωn), where the smooth functionsfλ(u1, u2, u3), λ = α, 1, . . . , n, of three variablesu1, u2, u3 are as follows:
fλ(−u1,−u2, u3) = −fλ(u1, u2, u3),fα(u1, u2,−u3) = fα(u1, u2, u3),fk(u1, u2,−u3) = −fk(u1, u2, u3),
(8)
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herewith, the functionsfk(u1, u2, u3) are definedforu3 = 0 for anyk = 1, . . . , n.
The setK is either empty or consists of finitelymany points of the circleS1α mod2π.
The latter two variablesu2 andu3 in the functionsfλ(u1, u2, u3) depend on one parameterα, but theyare allocated into different groups because of the fol-lowing reasons. First, not in the whole domain, theyare one-to-one expressed through each other, and, sec-ond, the first of them is an odd function and the secondis an even function ofα, which influences the symme-tries of system (6) in different ways.
To the system (6), let us put in correspondence thefollowing nonautonomous system
dωk
dα=
fk(ω, sinα, cosα)fα(ω, sinα, cosα)
, k = 1, . . . , n, (9)
which via the substitutionτ = sinα, reduces to theform
dωk
dτ=
fk(ω, τ, ϕk(τ))fα(ω, τ, ϕα(τ))
, k = 1, . . . , n, (10)
ϕλ(−τ) = ϕλ(τ), λ = α, 1, . . . , n.
The latter system can have in particular an alge-braic right-hand side (i.e., it can be the ratio of twopolynomials); sometimes this helps us to find its firstintegrals in explicit form.
The following assertion embeds the class of sys-tems (6) in the class of zero mean variable dissipationsystems. The inverse embedding does not hold in gen-eral.
Theorem 2 Systems of the form (6) are zero meanvariable dissipation dynamical systems.
This theorem is proved by using the certain sym-metries (8) of system (6) only, listed above, and usesthe periodicity of the right-hand side of the system onα.
Indeed, let calculate the specified divergence ofthe vector field of system (6). It equals to
∂fα(ω, sinα, cosα)∂u2
cosα−
−∂fα(ω, sinα, cosα)∂u3
sinα+
+n∑
k=1
∂fk(ω, sinα, cosα)∂u1
. (11)
The following integral on two first summands(11) is equal to zero:
∫ 2π
0
∂fα(ω, sinα, cosα)
∂u2d sinα+
+∂fα(ω, sinα, cosα)
∂u3d cosα
=
=∫ 2π
0
∂fα(ω, sinα, cosα)∂α
dα = hα(ω) ≡ 0, (12)
since the functionfα(ω, sinα, cosα) is periodic oneonα.
So, by virtue of the third equation (8), the follow-ing property holds for anyk = 1, . . . , n:
∂fk(ω, sinα, cosα)∂u1
= cosα · ∂gk(ω, sinα)∂u1
, (13)
herewith, the functiongk(u1, u2) is rather smooth forany numberk = 1, . . . , n.
Then the integral on period2π from the right-hand side of the equation (13) gives
∫ 2π
0
∂gk(ω, sinα)∂u1
d sinα = hk(ω) ≡ 0 (14)
for anyk = 1, . . . , n. And we get the assertion of thetheorem 2 from the equalities (12), (14).
The converse assertion is not true in general sincewe can present a set of dynamical systems on the two-dimensional cylinder, being zero mean variable dissi-pation systems that have none of the above-listed sym-metries.
In this work, we are mainly concerned withthe case where the functionsfλ(ω, τ, ϕk(τ)) (λ =α, 1, . . . , n) are polynomials inω, τ .
Example 1. We consider, in particular,pendulum-like systems on the two-dimensional cylin-derS1α mod 2π × R1ω with parameterb > 0from rigid body dynamics:
α = −ω + b sinα,ω = sinα cosα,
(15)
and
α = −ω + b sinα cos2 α + bω2 sinα,ω = sinα cosα− bω sin2 α cosα+
+bω3 cosα,(16)
in the variables(ω, τ), to these systems we can putin correspondence the following equations with alge-braic right-hand sides:
dω
dτ=
τ
−ω + bτ, (17)
anddω
dτ=
τ + bω[ω2 − τ2]−ω + bτ + bτ [ω2 − τ2]
(18)
which have the form (4.2). Moreover, these systemsare zero mean variable dissipation dynamical systems,which is easily directly verified.
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Indeed, the divergences of its right-hand sides areequal to
b cosα
andb cosα[4ω2 + cos2 α− 3 sin2 α],
respectively. It is easy to verify that they belong to theclass of systems (6).
Moreover, each of these systems has a first inte-gral that is a transcendental (in the sense of the the-ory of functions of one complex variable) functionexpressed through a finite combination of elementaryfunctions.
Let us present one more important example of ahigher-order system having the listed properties.
Example 2.To the following system
α = −z2 + b sinα,z2 = sin α cosα− z2
1cos αsin α ,
z1 = z1z2cos αsin α ,
(19)
with parameterb, considered in the three-dimensionaldomain
S1α mod2π\α = 0, α = π×R2z1, z2 (20)
(such a system can also be reduced to an equiva-lent system on the tangent bundleT∗S2 of the two-dimensional sphereS2) and describing the spatial mo-tion of a rigid body in a resisting medium, we put incorrespondence the following nonautonomous systemwith algebraic right-hand side: (τ = sinα):
dz2
dτ=
τ − z21/τ
−z2 + bτ,
dz1
dτ=
z1z2/τ
−z2 + bτ. (21)
It is seen that system (19) is a zero mean vari-able dissipation system; in order to achieve completecorrespondence with the definition, it suffices to intro-duce a new phase variable
z∗1 = ln |z1|. (22)
If we calculate the divergence of the right-handside of the system (19) in Cartesian coordinatesα, z∗1 , z2, then we shall get that it is equal tob cosα.Herewith, if we consider (20), we shall have in thesense of principal value:
limε→0
∫ π−ε
εb cosα + lim
ε→0
∫ 2π−ε
π+εb cosα = 0. (23)
Moreover, the system (19) has two first integrals(i.e., a complete list) that are transcendental functionsand we expressed through a finite combination of el-ementary functions, which, as was mentioned above,
becomes possible after putting in correspondence toit a (nonautonomous in general) system of equationswith algebraic (polynomial) right-hand (21).
The above-presented systems (15), (16), and (19),along with the property that they belong to the classof systems (6) and are zero mean variable dissipationsystems, also have a complete list of transcendentalfirst integrals expressed through a finite combinationof elementary functions.
Therefore, to search for the first integrals of thesystem considered, it is better to reduce systems ofthe form (6) to systems (10) with polynomial right-hand sides, on whose form the possibility of integra-tion in elementary functions of the initial system de-pends. Therefore, we proceed as follows: we seek suf-ficient conditions for integrability in elementary func-tions of systems of equations with polynomial right-hand sides studying systems of the most general formin this process.
4 Systems on Tangent Bundle ofTwo-Dimensional Sphere
Let consider the following dynamic system
θ + bθ cos θ + sin θ cos θ − ψ2 sin θcos θ = 0,
ψ + bψ cos θ + θψ[
1+cos2 θsin θ cos θ
]= 0
(24)
on tangent bundleT∗S2 of two-dimensional sphereS2θ, ψ. This system describes the spherical pendu-lum, placed in the accumulating medium flow. Here-with, both conservative moment is present in the sys-tem
sin θ cos θ, (25)
and the force moment depending on the velocity aslinear one with the variable coefficient:
b
(θ
ψ
)cos θ. (26)
The coefficients remaining in the equations arethe coefficients of connectedness, i.e.,
Γθψψ = − sin θ
cos θ, Γψ
θψ =1 + cos2 θ
sin θ cos θ. (27)
The system (24) has an order 3 practically, sincethe variableψ is a cyclic, herewith, the derivativeψ ispresent in the system only.
Proposition 3 The equation
ψ = 0 (28)
define the family of integral planes for the system (24).
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Furthermore, the equation (28) reduces the sys-tem (24) to an equation describing the cylindric pen-dulum which placed in the accumulating mediumflow.
Proposition 4 The system (24) is equivalent to thefollowing system:
θ = −z2 + b sin θ,
z2 = sin θ cos θ − z21
cos θsin θ ,
z1 = z1z2cos θsin θ ,
ψ = z1cos θsin θ
(29)
on the tangent bundleT∗S2z1, z2, θ, ψ of two-dimensional sphereS2θ, ψ.
Moreover, the first three equations of the system(29) form the closed three order system and coincidewith the equations of the system (19) (if we denoteα = θ). The separation of fourth equation of the sys-tem (29) has also occurred by the reason of cyclicityof the variableψ.
On the construction of phase pattern of the system(24), expressed in Pic.??, see [9].
Example 3. Let us study a system of the form(19), which reduces to (21), and also the system
α = −z2 + b(z21 + z2
2) sin α++b sinα cos2 α,
z2 = sinα cosα + bz2(z21 + z2
2) cosα−−bz2 sin2 α cosα− z2
1cos αsin α ,
z1 = bz1(z21 + z2
2) cos α−−bz1 sin2 α cosα + z1z2
cos αsin α ,
(30)
which also arises in the three-dimensional dynamicsof a rigid body interacting with a medium and whichcorresponds to the following system with algebraicright-hand side:
dz2dτ = τ+bz2(z2
1+z22)−bz2τ2−z2
1/τ
−z2+bτ(z21+z2
2)+bτ(1−τ2),
dz1dτ = bz1(z2
1+z22)−bz1τ2+z1z2/τ
−z2+bτ(z21+z2
2)+bτ(1−τ2).
(31)
Therefore, as before, we consider a pair of sys-tems: the initial system (30) and the algebraic system(31) corresponding to it.
In a similar way, we pass to homogeneous coor-dinatesuk, k = 1, 2, by the formulas
zk = ukτ. (32)
Using the latter change, we reduce system (21) tothe form
τ du2dτ + u2 = τ−u2
1τ−u2τ+bτ ,
τ du1dτ + u1 = u1u2τ
−u2τ+bτ ,(33)
corresponding to the equation
du2
du1=
1− bu2 + u22 − u2
1
2u1u2 − bu1. (34)
This equation is integrated in elementary functionssince the identity
d
(1− βu2 + u2
2
u1
)+ du1 = 0, (35)
is integrated, and in the coordinates(τ, z1, z2), it hasa first integral of the form (compare with [9])
z21 + z2
2 − βz2τ + τ2
z1τ= const.
System (30) after its reduction corresponds to thesystem
τ du2dτ + u2 = τ+bu2τ3(u2
1+u22)−bu2τ3−u2
1τ
−u2τ+bτ3(u21+u2
2)+bτ(1−τ2),
τ du1dτ + u1 = bu1τ3(u2
1+u22)−bu1τ3+u1u2τ
−u2τ+bτ3(u21+u2
2)+bτ(1−τ2),
(36)
which also reduces to the form (34).
5 Certain GeneralizationsLet us pose the following question: What are the pos-sibilities of integrating in elementary functions thesystem
dzdx = ax+by+cz+c1z2/x+c2zy/x+c3y2/x
d1x+ey+fz ,dydx = gx+hy+iz+i1z2/x+i2zy/x+i3y2/x
d1x+ey+fz ,(37)
of a more general form, which includes the systems(21) and (31) considered above in three-dimensionalphase domains and which has a singularity of the form1/x?
Previously, a number of results concerning thisquestion were already obtained (see also [6]). Let uspresent these results and complement them by originalarguments.
As previously, introducing the substitutions
y = ux, z = vx, (38)
we obtain that system (37) reduces to the followingsystem
x dvdx + v = ax+bux+cvx+c1v2x+c2vux+c3u2x
d1x+eux+fvx , (39)
xdudx + u = gx+hux+ivx+i1v2x+i2vux+i3u2x
d1x+eux+fvx , (40)
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which is equivalent to
x dvdx =
= ax+bux+(c−d1)vx+(c1−f)v2x+(c2−e)vux+c3u2xd1x+eux+fvx ,
(41)xdu
dx == gx+(h−d1)ux+ivx+i1v2x+(i2−f)vux+(i3−e)u2x
d1x+eux+fvx ,
(42)we put in correspondence the following equation withalgebraic right-hand side:
dvdu = a+bu+cv+c1v2+c2vu+c3u2−v[d1+eu+fv]
g+hu+iv+i1v2+i2vu+i3u2−u[d1+eu+fv]. (43)
The integration of the latter equation reduces tothat of the following equation in total differentials:
[g+hu+iv+i1v2+i2vu+i3u
2−d1u−eu2−fuv]dv =
= [a + bu + cv + c1v2+
+c2vu + c3u2 − d1v − euv − fv2]du.
(44)
We have (in general) a 15-parameter family ofequations of the form (44). To integrate the latter iden-tity in elementary functions as a homogeneous equa-tion, it suffices to impose seven relations
g = 0, i = 0, i1 = 0,e = c2, h = c, i2 = 2c1 − f.
(45)
Introduce nine parametersβ1, . . . , β9 and con-sider them as independent parameters:
β1 = a, β2 = b, β3 = c, β4 = c1, β5 = c2,β6 = c3, β7 = d1, β8 = f, β9 = i3.
(46)
Therefore, under the group of conditions (45),and (46), Eq. (44) reduces to the form
dvdu = β1+β2u+(β3−β7)v+(β4−β8)v2+β6u2
(β3−β7)u+2(β4−β8)vu+(β9−β5)u2 , (47)
and the system (41), (42), respectively, to the form
x dvdx = β1+β2u+(β3−β7)v+(β4−β8)v2+β6u2
β7+β5u+β8v , (48)
xdudx = (β3−β7)u+2(β4−β8)vu+(β9−β5)u2
β7+β5u+β8v , (49)
after that, the equation (47) is integrated in finite com-bination of elementary functions.
Indeed, integrating the identity (44), we obtain therelation
d
[(β3 − β7)v
u
]+
+d
[(β4 − β8)v2
u
]+ d[(β9 − β5)v] + d
[β1
u
]−
−d[β2 ln |u|]− d[β6u] = 0, (50)
which for the beginning allows us to obtain the fol-lowing invariant relation:
(β3 − β7)vu
+(β4 − β8)v2
u+ (β9 − β5)v +
β1
u−
−β2 ln |u| − β6u = C1 = const, (51)
and in the coordinates(x, y, z), it allows us to obtainthe first integral in the form
A
yx− β2 ln
∣∣∣∣y
x
∣∣∣∣ = const, (52)
A = (β4 − β8)z2−−β6y
2 + (β3 − β7)zx + (β9 − β5)zy + β1x2.
Therefore, we can make a conclusion about theintegrability in elementary functions of the follow-ing, in general, nonconservative third-order systemdepending on nine parameters:
dzdx = β1x+β2y+β3z+β4z2/x+β5zy/x+β6y2/x
β7x+β5y+β8z ,dydx = β3y+(2β4−β8)zy/x+β9y2/x
β7x+β5y+β8z .(53)
Corollary 5 The third-order system
α = β7 sinα + β5z1 + β8z2,z2 = β1 sinα cosα + β2z1 cosα+
+β3z2 cosα++β4z
22
cos αsin α + β5z1z2
cos αsin α + β6z
21
cos αsin α ,
z1 = β3z1 cosα + (2β4 − β8)z1z2cos αsin α +
+β9z21
cos αsin α ,
(54)
on the set
S1α mod2π\α = 0, α = π×R2z1, z2, (55)
which depends on nine parameters, has a first inte-gral, which is in general transcendental and expressedthrough elementary functions:
B
z1 sinα− β2 ln
∣∣∣∣z1
sinα
∣∣∣∣ = const, (56)
B = (β4 − β8)z22 − β6z
21+
+(β3 − β7)z2 sinα + (β9 − β5)z2z1 + β1 sin2 α2.
In particular, forβ1 = 1, β2 = β3 = β4 = β5 =β9 = 0, β6 = β8 = −1, β7 = b system (54) reducesto system (19).
To find the additional first integral of the nonau-tonomous system (37), we use the found first integral
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(52), which is expressed through a finite combinationof elementary functions.
For the beginning let transform the relation (51)as follows:
(β4 − β8)v2+
+ [(β9 − β5)u + (β3 − β7)] v + f1(u) = 0, (57)
where
f1(u) = β1 − β6u2 − β2u ln |u| − C1u.
Herewith, the valuev formally can be found fromthe equality
v1,2(u) =1
2(β4 − β8)×
×
(β5 − β9)u + (β7 − β3)±√
f2(u)
, (58)
where
f2(u) = A1 + A2u + A3u2 + A4u ln |u|,
A1 = (β3 − β7)2 − 4β1(β4 − β8),
A2 = 2(β9 − β5)(β3 − β7) + 4C1(β4 − β8),
A3 = (β9 − β5)2 + 4β6(β4 − β8),
A4 = 4β2(β4 − β8).
Then the quadrature studied for the search of ad-ditional (in general) transcendental first integral (forexample, of the system (48), (49) or (41), (42), here-with, the equation (49) is used) has the following form
∫dx
x=
∫ [β7 + β5u + β8v1,2(u)]du
C=
=∫ [B1 + B2u + B3
√f2(u)]du
B4u√
f2(u), (59)
Bk = const, k= 1, . . . , 4,
C = (β3−β7)u+(β9−β5)u2 +2(β4−β8)uv1,2(u).
And the quadrature studied for the search of ad-ditional (in general) transcendental first integral (forexample, of the system (48), (49) or (41), (42), here-with, the equation (48) is used) has the following form
∫dx
x=
∫ [β7 + β5u(v) + β8v]dv
D, (60)
D = β1+β2u(v)+(β3−β7)v+(β4−β8)v2+β6u2(v).
herewith, the functionu(v) should be obtain as a re-sult of resolving of implicit equation (51) respectivelyto u (that, in general case, is not always evident).
The following lemma gives the necessary condi-tions of the expression of integrals in (60) through thefinite combination of elementary functions.
Lemma 6 The indefinite integral in (60) is expressedthrough the finite combination of elementary functionsfor A4 = 0, i.e., either
β2 = 0 (61)
orβ4 = β8. (62)
Theorem 7 The system (54) under assumption ofnecessary conditions of lemma 6 (the property (61)holds in given case), has the compete set of first inte-grals, which expressing through the finite combinationof elementary functions.
Therefore, the dynamical systems considered inthis work refer to zero mean variable dissipation sys-tems in the existing periodic coordinate. Moreover,such systems often have a complete list of first inte-grals expressed through elementary functions.
So, it was shown above the certain cases of com-plete integrability in spatial dynamics of a rigid bodymotion in a nonconservative field. Herewith, we dealtwith with three properties, which are independent forthe first glance:
i) the distinguished class of systems (6) with thesymmetries above;
ii) the fact that this class of systems consists ofsystems with zero mean variable dissipation (in thevariableα), which allows us to consider them as “al-most” conservative systems;
iii) in certain (although lower-dimensional) cases,these systems have the complete tuple of first inte-grals, which are transcendental in general (from theviewpoint of complex analysis).
The method for reducing the initial systems ofequations with right-hand sides containing polynomi-als in trigonometric functions to systems with polyno-mial right-hand sides allows us to search for the firstintegrals (or to prove their absence) for systems of amore general form, but not only for systems that havethe above symmetries (see also [9]).
6 ConclusionThe results of the presented work were appeared ow-ing to the study the applied problem of the rigid bodymotion in a resisting medium, where we have ob-tained complete lists of transcendental first integralsexpressed through a finite combination of elemen-tary functions. This circumstance allows the authorto carry out the analysis of all phase trajectories andshow those their properties which have the roughnessand are preserved for systems of a more general form.The complete integrability of such system is related
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to their symmetries of latent type. Therefore, it is ofinterest to study a sufficiently wide class of dynamicalsystems having analogous latent symmetries.
So, for example, the instability of the simplestbody motion, the rectilinear translational drag, is usedfor methodological purposes, precisely, for finding theunknown parameters of the medium action on a rigidbody under the quasi-stationarity conditions.
The experiment on the motion of homogeneouscircular cylinders in the water carried out in Instituteof Mechanics of M. V. Lomonosov State Universityjustified that in modelling the medium action on therigid body, it is also necessary to take into account anadditional parameter that brings a dissipation to thesystem.
In studying the class of body drags with finite an-gle of attack, the principal problem is finding thoseconditions under which there exist auto-oscillations ina finite neighborhood of the rectilinear translationaldrag. Therefore, there arises the necessity of a com-plete nonlinear study.
Generally speaking, the dynamics of a rigid bodyinteracting with a medium is just the field where therearise either nonzero mean variable dissipation systemsor systems in which the energy loss in the mean dur-ing a period can vanish. In the work, we have obtainedsuch a methodology owing to which it becomes possi-ble to finally and analytically study a number of planeand spatial model problems.
In qualitative describing the body interaction witha medium, because of using the experimental infor-mation about the properties of the streamline flowaround, there arises a definite dispersion in modellingthe force-model characteristics. This makes it naturalto introduce the definitions of relative roughness (rel-ative structural stability) and to prove such a rough-ness for the system studied. Moreover, many sys-tems considered are merely (absolutely) Andronov–Pontryagin rough.
Acknowledgements: This work was supported bythe Russian Foundation for Basic Research, projectno. 12–01–00020–a.
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