INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING & TECHNOLOGY (IJEET)
|
|
- Amberly Casey
- 5 years ago
- Views:
Transcription
1 INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING & TECHNOLOG (IJEET) Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME ISSN 976 6(Prnt) ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME: Journal Impact Factor ():.9 (Calculated by GISI) IJEET I A E M E A NEW CHAOTIC ATTRACTOR GENERATED FROM A -D AUTONOMOUS SSTEM WITH ONE EQUILIBRIUM AND ITS FRACTIONAL ORDER FORM Kshore Bng, Susy Thomas M.Tech Student, Electrcal Engneerng Department, Natonal Insttute of Technology, Calcut, Kerala, Inda Professor & Head, Electrcal Engneerng Department, Natonal Insttute of Technology, Calcut, Kerala, Inda ABSTRACT In ths paper, a novel three-dmensonal autonomous chaotc system s proposed. The proposed system contans four varatonal parameters, a cubc nonlnearty term (.e. product of all the three states) and exhbts a chaotc attractor n numercal smulatons. The basc dynamc propertes of the system are analyzed by means of equlbrum ponts, Egen values and Lyapunov exponents. Fnally, the commensurate and non-commensurate fractonal order form of the system whch exhbts chaotc attractor s also analyzed. Keywords: Chaos, Chaotc Systems, Chaotc Attractors, Commensurate Order System, Lyapunov Exponents, Non-Commensurate Order System.. INTRODUCTION Chaotc behavor of dynamc systems can be utlzed n a varety of dscplnes, such as algorthmc tradng, bology, computer scence, cvl engneerng, economcs, fnance, geology, mathematcs, mcrobology, meteorology, physcs, phlosophy, and robotcs and so on. In 98, G. Duffng ntroduced a duffng equaton whch can be extended to complex doman n order to study strange attractors and chaotc behavor of forced vbratons of ndustral machnery []. In 9, Van der Pol ntroduced a model known as VPO model to study oscllatons n vacuum tube crcuts. The Van der Pol oscllator (VPO) represents a nonlnear system wth an nterestng behavor that exhbts naturally n several applcatons, such as heartbeat, neurons, acoustc models etc. []. In 9, Alfred J. Lotka and Vto Volterra proposed predator-prey equatons to descrbe the dynamcs of bologcal systems n whch two speces nteract on each other, one s a predator and the other s ts prey []. In
2 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME 96, Lorenz found the chaotc attractor n a three-dmensonal autonomous system whle studyng atmospherc convecton []. In 976, Otto Rossler proposed Rossler s system wth strange attractor whch s useful n modelng equlbrum n chemcal reactons []. In 98, Newton and Lepnk obtaned the set of dfferental equatons from Euler rgd equatons whch are modfed wth the addton of a lnear feedback. Two strange attractors startng from dfferent ntal condtons and same parameter condtons were obtaned [6]. In 98, the chaotc phenomenon n macroeconomcs was found. The contnuous economcal system was descrbed and analyzed by Ma and Chen n [7]. In 988, the basc crcut unt of the Cellular Neural Network (CNN) was ntroduced by L.O.Chua whch contans lnear and non-lnear elements. Such types of CNN are able to show chaotc behavor [8]. In 999, Chen found a smple three-dmensonal autonomous system, whch s not topologcally equvalent to Lorenz s system and whch has a chaotc attractor as well [9]. In, Lu ntroduced a system whch s known as a brdge between the Lorenz system and Chen s system []. In, a new system was ntroduced whch contan two varatonal parameters and exhbts Lorenz lke attractor []. In the same year a new type of four wng chaotc attractor was generated from a smooth canoncal -D contnuous system []. Motvated by such prevous work, ths paper ntroduces another smple three-dmensonal autonomous system whch contans four varatonal parameters and one cubc nonlnearty term whch s a product of all the three states.e. dsplacement, velocty and acceleraton. Secton explans the basc defntons. In secton the new system s brefly ntroduced. In secton the dynamc behavors of the proposed system are dscussed. The fractonal order form of the system s dscussed n secton. Fnally, some concludng comments are gven n secton 6.. BASIC DEFINITIONS. CHAOS There s no unversally accepted defnton for chaos, but the followng characterstcs are nearly always dsplayed by the soluton of chaotc system.. Aperodc (non-perodc) behavor.. Bounded structure.. Senstvty to ntal condtons.. FRACTIONAL DERIVATIVE AND INTEGRAL The contnuous ntegral-dfferental operator s defned as d, > dt D f ( t), a t t ( dτ ), < a (). GRUNWALD-LETNIKOV FRACTIONAL DERIVATIVE The Grunwald-Letnkov fractonal order dervatve defnton of order s defned as t a h j D f ( t) lt ( ) f a t h h j j ( t jh) ()
3 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME For bnomal coeffcents calculaton we can use the relaton between Euler s Gamma Γ( + ) functon, defned as for j Γ( j + ) Γ( j + ) Γ Is Euler s Gamma functon and a, t are the bounds of operaton for f (t). ( ). RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE The Remann-Louvlle fractonal order dervatve defnton of order s defned as a D t a D t n d t f ( τ ) f ( t) n dτ () + Γ a n ( n ) dt ( t τ ) Γ( ) Is Euler s Gamma functon and t a, are the bounds of operaton for f (t).. STABILIT OF FRACTIONAL NONLINEAR SSTEMS Accordng to stablty theorem, the fractonal order system that m s the LCM of the denomnators we setγ m u ' s of q ' s, where. The fractonal order system s asymptotcally stable f π arg( λ ) > γ For all roots λ of the followng equaton mq mq mq ( dag( [ λ... λ n ] ) v q a D t q... q n and suppose + q, v u u, for,,... n and det λ ().6 CONDITION FOR MINIMUM COMMENSURATE ORDER Suppose that the unstable Egen values of scroll focus ponts are λ,, ± jβ,. The necessary condton to exhbt double scroll attractor of fractonal order system s the Egen values λ remanng n the unstable regon. The condton for commensurate order s, β q > a tan,, π () Ths condton can be used to determne the mnmum order for whch a nonlnear system can generate a chaos.. THE PROPOSED -D DNAMICAL SSTEM Consder the followng smple -D autonomous system: x& y y& z z& az by cx + dxyz (6)
4 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME Where [ x, y, z] T R the state vector, and a, b, c and d are postve constant parameters of the system (6). In the followng, some basc propertes of system (6) are analyzed.. EQUILIBRIA The equlbra of system (6) can be found by solvng the followng algebrac equatons: x& y y& z z& az by cx + dxyz (7) From the frst and second equatons of (7), y, z Substtutng ths nto the thrd equaton of (7), x Therefore O(,,) s the only equlbrum pont of the system (6).. STABILIT AND EISTANCE OF ATTRACTOR By lnearzng the system (6), one obtans the Jacoban J dyz c dxz b dxy a Therefore J O (,, ) c b a So, the Egen values of the lnearzed system are obtaned as follows: λ I J λ + aλ + bλ + c O Case : If a b c d the Egen values are, ± j, the crtcal case. Case : If a >, b c d the equlbrum O s stable, ensures that system (6) s not chaotc. Case : To ensure that system (6) s chaotc mplyng that the equlbrum O s saddle pont, the condton on the postve constant parameters of system should be consdered,.e. a <, b., c and d any value.. DNAMICAL BEHAVIOR OF THE PROPOSED SSTEM When a., b., c and d, the Egen values of the lnearzed system are.79,. ±. j. Therefore, based on the Egen values we know that equlbrum O s a saddle pont. In ths secton the fourth and ffth order Range-Kutta ntegraton algorthm was performed to solve the dfferental equatons. Settng the ntal condton to [...], the chaotc attractor s shown n fgure.
5 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME The Lyapunov spectrum of the system (6) versus tme s shown n fgure wth parameters a., b., c and d. When a., b, c and d, the chaotc attractor of the system (6) wth ntal condton [...] s shown n fgure. -D vew Projecton on - plane Projecton on - plane Projecton on - plane Fg : Chaotc attractor the system (6) wth parameters a., b., c and d, ntal condton [...]. Dynamcs of Lyapunov exponents Lyapunov exponents λ.98 λ -.8 λ -.9 tme Fg : Lyapunov spectrum of the system (6) wth parameters a., b., c and d, ntal condton [...]
6 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME -D vew Projecton on - plane Projecton on - plane Projecton on - plane Fg : Chaotc attractor the system (6) wth parameters a., b, c and d, ntal condton [...]. FRACTIONAL ORDER FORM OF THE PROPOSED SSTEM The fractonal order form of the system (6) s defned as follows q d x y q dt q d y z q dt q d z az by cx + dxyz q dt (8) Where q, q and q are the dervatve orders. For numercal smulaton of the fractonal order system (8), we have consdered the two cases: frst, commensurate order system and second, non-commensurate order system. Case : Commensurate order system From equaton () the commensurate order of the system s gven by q > a π. tan.96. 6
7 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME In fgure s depcted the chaotc attractor of the commensurate fractonal order (8) wth parameters a., b., c, d, dervatve orders q q q. 97 wth ntal condton [...] for smulaton tme T sm s and step tme h.. Case : Non-commensurate order system We consder non-commensurate order system wth parameters a., b., c, d, dervatve orders q.97, q.98 and q.99. Therefore γ m From equaton () the characterstc equaton of the lnearzed system s λ +.λ +.λ + π The unstable roots are λ,.89 ±.67j, because arg( λ, ).6 < γ In fgure s depcted the chaotc attractor of the non-commensurate fractonal order (8) wth parameters a., b., c, d, dervatve orders q.97, q.98, q. 99 wth ntal condton [...] for smulaton tme T sm s and step tme h.. In fgure 6 s depcted the chaotc attractor of the non-commensurate fractonal order (8) wth parameters a., b., c, d, dervatve orders q., q., q. wth ntal condton [...] for smulaton tme T sm s and step tme h.. -D vew Projecton on - plane Projecton on - plane Projecton on - plane Fg : Chaotc attractor of the commensurate fractonal order (8) wth parameters a., b., c, d, dervatve orders q q q. 97 wth ntal condton [...] 7
8 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME -D vew Projecton on - plane Projecton on - plane Projecton on - plane Fg : Chaotc attractor of the non-commensurate fractonal order (8) wth parameters a., b., c, d, dervatve orders q.97, q.98, q. 99 wth ntal condton [...] -D vew Projecton on - plane Projecton on - plane Projecton on - plane Fg 6: Chaotc attractor of the non-commensurate fractonal order (8) wth parameters a., b., c, d, dervatve orders q., q., q. wth ntal condton [...] 8
9 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME 6. CONCLUSION In ths paper, a new novel three-dmensonal chaotc system s proposed. The proposed system has only one equlbrum pont for any arbtrary set of parameters and also some dynamc propertes of the system have been nvestgated. The dynamcs of the proposed system wth commensurate and non-commensurate fractonal order forms was studed based on stablty theorems. On the other hand chaos control and synchronzaton of ths system are nterestng problems to be nvestgated and should also be consdered n the future work. REFERENCES [] Ivana Kovacc, Mchael J. Brennan, The Duffng equaton-nonlnear oscllators and ther behavor, A John Wley and sons, Ltd. Publcatons, [] B. Van der Pol, A theory of the ampltude of free and forced trode vbratons, Rado Revew, 9, 7-7, [] Alfred J. Lotka, Contrbuton to the Theory of Perodc Reactons, J. Phys.Chem.,, 9, 7-7. [] E.N. Lorenz, Determnstc non perodc flow, J. Atmos. Sc.,, 96,. [] O.E. Rossler, An equaton for contnuous chaos, Physcs Letters A, 7, 976, [6] Lepnk R. B. and Newton T. A., Double strange attractors n rgd body moton wth lnear feedback control, Physcs Letters, 86, 98, [7] Ma J. H. and Chen. S., Study for the bfurcaton topologcal structure and the global complcated character of a knd of nonlnear fnance system, Appled Mathematcs and Mechancs,,,. [8] L.O. Chua and L. ang, "Cellular Neural Networks: Theory," IEEE Trans. on Crcuts and Systems,, 988, 7-7. [9] G. Chen, T. Ueta. et another chaotc attractor. Int. J. Bfurcaton and Chaos, 9, 999, 666. [] Deng W. H. and L C. P., Chaos synchronzaton of the fractonal Lu system, Physca A,,, 6 7. [] Ihsan P, lmaz U, A chaotc attractor from General Lorenz system famly and ts electronc expermental mplementaton, Turk J Elec Eng & Comp Sc, 8,, 7-8. [] enghu Wang, Guoyuan Q, anxa Sun, Barend Jacobus van Wyk, A new type of fourwng chaotc attractors n -D quadratc autonomous systems, Nonlnear Dyn, 6,, 7. 9
Irregular vibrations in multi-mass discrete-continuous systems torsionally deformed
(2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationOdd/Even Scroll Generation with Inductorless Chua s and Wien Bridge Oscillator Circuits
Watcharn Jantanate, Peter A. Chayasena, Sarawut Sutorn Odd/Even Scroll Generaton wth Inductorless Chua s and Wen Brdge Oscllator Crcuts Watcharn Jantanate, Peter A. Chayasena, and Sarawut Sutorn * School
More informationIntroduction. - The Second Lyapunov Method. - The First Lyapunov Method
Stablty Analyss A. Khak Sedgh Control Systems Group Faculty of Electrcal and Computer Engneerng K. N. Toos Unversty of Technology February 2009 1 Introducton Stablty s the most promnent characterstc of
More informationScroll Generation with Inductorless Chua s Circuit and Wien Bridge Oscillator
Latest Trends on Crcuts, Systems and Sgnals Scroll Generaton wth Inductorless Chua s Crcut and Wen Brdge Oscllator Watcharn Jantanate, Peter A. Chayasena, and Sarawut Sutorn * Abstract An nductorless Chua
More informationWavelet chaotic neural networks and their application to continuous function optimization
Vol., No.3, 04-09 (009) do:0.436/ns.009.307 Natural Scence Wavelet chaotc neural networks and ther applcaton to contnuous functon optmzaton Ja-Ha Zhang, Yao-Qun Xu College of Electrcal and Automatc Engneerng,
More informationSolving Fractional Nonlinear Fredholm Integro-differential Equations via Hybrid of Rationalized Haar Functions
ISSN 746-7659 England UK Journal of Informaton and Computng Scence Vol. 9 No. 3 4 pp. 69-8 Solvng Fractonal Nonlnear Fredholm Integro-dfferental Equatons va Hybrd of Ratonalzed Haar Functons Yadollah Ordokhan
More informationApplication of B-Spline to Numerical Solution of a System of Singularly Perturbed Problems
Mathematca Aeterna, Vol. 1, 011, no. 06, 405 415 Applcaton of B-Splne to Numercal Soluton of a System of Sngularly Perturbed Problems Yogesh Gupta Department of Mathematcs Unted College of Engneerng &
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationCOEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN
Int. J. Chem. Sc.: (4), 04, 645654 ISSN 097768X www.sadgurupublcatons.com COEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN R. GOVINDARASU a, R. PARTHIBAN a and P. K. BHABA b* a Department
More informationAppendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
More informationNON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationModeling of Dynamic Systems
Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationNotes on Analytical Dynamics
Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame
More informationDelay equations with engineering applications Gábor Stépán Department of Applied Mechanics Budapest University of Technology and Economics
Delay equatons wth engneerng applcatons Gábor Stépán Department of Appled Mechancs Budapest Unversty of Technology and Economcs Contents Delay equatons arse n mechancal systems by the nformaton system
More informationNote 10. Modeling and Simulation of Dynamic Systems
Lecture Notes of ME 475: Introducton to Mechatroncs Note 0 Modelng and Smulaton of Dynamc Systems Department of Mechancal Engneerng, Unversty Of Saskatchewan, 57 Campus Drve, Saskatoon, SK S7N 5A9, Canada
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationAnti-Synchronization of two Different Chaotic Systems via Optimal Control with Fully Unknown Parameters *
ISSN 76-7659, England, UK Journal of Informaton and Computng Scence Vol. 5, No., 00, pp. 0-08 Ant-Synchronzaton of two Dfferent Chaotc Systems va Optmal Control wth Fully Unknown Parameters * Honglan Zhu
More information6.3.4 Modified Euler s method of integration
6.3.4 Modfed Euler s method of ntegraton Before dscussng the applcaton of Euler s method for solvng the swng equatons, let us frst revew the basc Euler s method of numercal ntegraton. Let the general from
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationDO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.
EE 539 Homeworks Sprng 08 Updated: Tuesday, Aprl 7, 08 DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED. For full credt, show all work. Some problems requre hand calculatons.
More information829. An adaptive method for inertia force identification in cantilever under moving mass
89. An adaptve method for nerta force dentfcaton n cantlever under movng mass Qang Chen 1, Mnzhuo Wang, Hao Yan 3, Haonan Ye 4, Guola Yang 5 1,, 3, 4 Department of Control and System Engneerng, Nanng Unversty,
More informationHongyi Miao, College of Science, Nanjing Forestry University, Nanjing ,China. (Received 20 June 2013, accepted 11 March 2014) I)ϕ (k)
ISSN 1749-3889 (prnt), 1749-3897 (onlne) Internatonal Journal of Nonlnear Scence Vol.17(2014) No.2,pp.188-192 Modfed Block Jacob-Davdson Method for Solvng Large Sparse Egenproblems Hongy Mao, College of
More informationDETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM
Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S111 DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM by Davood Domr GANJI
More information(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate
Internatonal Journal of Mathematcs and Systems Scence (018) Volume 1 do:10.494/jmss.v1.815 (Onlne Frst)A Lattce Boltzmann Scheme for Dffuson Equaton n Sphercal Coordnate Debabrata Datta 1 *, T K Pal 1
More informationProjective change between two Special (α, β)- Finsler Metrics
Internatonal Journal of Trend n Research and Development, Volume 2(6), ISSN 2394-9333 www.jtrd.com Projectve change between two Specal (, β)- Fnsler Metrcs Gayathr.K 1 and Narasmhamurthy.S.K 2 1 Assstant
More informationBinomial transforms of the modified k-fibonacci-like sequence
Internatonal Journal of Mathematcs and Computer Scence, 14(2019, no. 1, 47 59 M CS Bnomal transforms of the modfed k-fbonacc-lke sequence Youngwoo Kwon Department of mathematcs Korea Unversty Seoul, Republc
More informationPerforming Modulation Scheme of Chaos Shift Keying with Hyperchaotic Chen System
6 th Internatonal Advanced echnologes Symposum (IAS 11), 16-18 May 011, Elazığ, urkey Performng Modulaton Scheme of Chaos Shft Keyng wth Hyperchaotc Chen System H. Oğraş 1, M. ürk 1 Unversty of Batman,
More informationSnce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t
8.5: Many-body phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes
More informationHow Differential Equations Arise. Newton s Second Law of Motion
page 1 CHAPTER 1 Frst-Order Dfferental Equatons Among all of the mathematcal dscplnes the theory of dfferental equatons s the most mportant. It furnshes the explanaton of all those elementary manfestatons
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationHaar Wavelet Collocation Method for the Numerical Solution of Nonlinear Volterra-Fredholm-Hammerstein Integral Equations
Global Journal of Pure and Appled Mathematcs. ISS 0973-768 Volume 3, umber 2 (207), pp. 463-474 Research Inda Publcatons http://www.rpublcaton.com Haar Wavelet Collocaton Method for the umercal Soluton
More informationLecture 5.8 Flux Vector Splitting
Lecture 5.8 Flux Vector Splttng 1 Flux Vector Splttng The vector E n (5.7.) can be rewrtten as E = AU (5.8.1) (wth A as gven n (5.7.4) or (5.7.6) ) whenever, the equaton of state s of the separable form
More informationModule 1 : The equation of continuity. Lecture 1: Equation of Continuity
1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum
More informationarxiv: v1 [math.co] 12 Sep 2014
arxv:1409.3707v1 [math.co] 12 Sep 2014 On the bnomal sums of Horadam sequence Nazmye Ylmaz and Necat Taskara Department of Mathematcs, Scence Faculty, Selcuk Unversty, 42075, Campus, Konya, Turkey March
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationNONLINEAR NATURAL FREQUENCIES OF A TAPERED CANTILEVER BEAM
Advanced Steel Constructon Vol. 5, No., pp. 59-7 (9) 59 NONLINEAR NATURAL FREQUENCIES OF A TAPERED CANTILEVER BEAM M. Abdel-Jaber, A.A. Al-Qasa,* and M.S. Abdel-Jaber Department of Cvl Engneerng, Faculty
More informationSOME ASPECTS OF THE EXISTENCE OF COULOMB VIBRATIONS IN A COMPOSITE BAR
SISOM 006, Bucharest 7-9 May SOME ASPECTS OF THE EXISTECE OF COULOMB VIBRATIOS I A COMPOSITE BAR Ştefana DOESCU Techncal Unversty of Cvl Engneerng, Dept. of Mathematcs, emal: stefa05@rdsln.ro. In ths paper,
More informationTHE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS OF A TELESCOPIC HYDRAULIC CYLINDER SUBJECTED TO EULER S LOAD
Journal of Appled Mathematcs and Computatonal Mechancs 7, 6(3), 7- www.amcm.pcz.pl p-issn 99-9965 DOI:.75/jamcm.7.3. e-issn 353-588 THE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS
More informationCOMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationThe Two-scale Finite Element Errors Analysis for One Class of Thermoelastic Problem in Periodic Composites
7 Asa-Pacfc Engneerng Technology Conference (APETC 7) ISBN: 978--6595-443- The Two-scale Fnte Element Errors Analyss for One Class of Thermoelastc Problem n Perodc Compostes Xaoun Deng Mngxang Deng ABSTRACT
More informationChapter 22. Mathematical Chaos Stable and Unstable Manifolds Saddle Fixed Point
Chapter 22 Mathematcal Chaos The sets generated as the long tme attractors of physcal dynamcal systems descrbed by ordnary dfferental equatons or dscrete evoluton equatons n the case of maps such as the
More informationCalifornia State Science Fair
Calforna State Scence Far Mathematcal Modelng of Real World Systems Part 1 -- Explorng on Chaos on Your Computer Edward Ruth drruth@x.netcom.com 1) Introducton Mathematcs s the language that scentsts use
More informationLimit Cycle Generation for Multi-Modal and 2-Dimensional Piecewise Affine Control Systems
Lmt Cycle Generaton for Mult-Modal and 2-Dmensonal Pecewse Affne Control Systems atsuya Ka Department of Appled Electroncs Faculty of Industral Scence and echnology okyo Unversty of Scence 6-3- Njuku Katsushka-ku
More informationFlow Induced Vibration
Flow Induced Vbraton Project Progress Report Date: 16 th November, 2005 Submtted by Subhrajt Bhattacharya Roll no.: 02ME101 Done under the gudance of Prof. Anrvan Dasgupta Department of Mechancal Engneerng,
More informationTHE STURM-LIOUVILLE EIGENVALUE PROBLEM - A NUMERICAL SOLUTION USING THE CONTROL VOLUME METHOD
Journal of Appled Mathematcs and Computatonal Mechancs 06, 5(), 7-36 www.amcm.pcz.pl p-iss 99-9965 DOI: 0.75/jamcm.06..4 e-iss 353-0588 THE STURM-LIOUVILLE EIGEVALUE PROBLEM - A UMERICAL SOLUTIO USIG THE
More informationOn Tiling for Some Types of Manifolds. and their Folding
Appled Mathematcal Scences, Vol. 3, 009, no. 6, 75-84 On Tlng for Some Types of Manfolds and ther Foldng H. Rafat Mathematcs Department, Faculty of Scence Tanta Unversty, Tanta Egypt hshamrafat005@yahoo.com
More informationA MODIFIED METHOD FOR SOLVING SYSTEM OF NONLINEAR EQUATIONS
Journal of Mathematcs and Statstcs 9 (1): 4-8, 1 ISSN 1549-644 1 Scence Publcatons do:1.844/jmssp.1.4.8 Publshed Onlne 9 (1) 1 (http://www.thescpub.com/jmss.toc) A MODIFIED METHOD FOR SOLVING SYSTEM OF
More informationThe binomial transforms of the generalized (s, t )-Jacobsthal matrix sequence
Int. J. Adv. Appl. Math. and Mech. 6(3 (2019 14 20 (ISSN: 2347-2529 Journal homepage: www.jaamm.com IJAAMM Internatonal Journal of Advances n Appled Mathematcs and Mechancs The bnomal transforms of the
More information6.3.7 Example with Runga Kutta 4 th order method
6.3.7 Example wth Runga Kutta 4 th order method Agan, as an example, 3 machne, 9 bus system shown n Fg. 6.4 s agan consdered. Intally, the dampng of the generators are neglected (.e. d = 0 for = 1, 2,
More informationEnergy Storage Elements: Capacitors and Inductors
CHAPTER 6 Energy Storage Elements: Capactors and Inductors To ths pont n our study of electronc crcuts, tme has not been mportant. The analyss and desgns we hae performed so far hae been statc, and all
More informationMaejo International Journal of Science and Technology
Maejo Int. J. Sc. Technol. () - Full Paper Maejo Internatonal Journal of Scence and Technology ISSN - Avalable onlne at www.mjst.mju.ac.th Fourth-order method for sngularly perturbed sngular boundary value
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationPhysics 207: Lecture 20. Today s Agenda Homework for Monday
Physcs 207: Lecture 20 Today s Agenda Homework for Monday Recap: Systems of Partcles Center of mass Velocty and acceleraton of the center of mass Dynamcs of the center of mass Lnear Momentum Example problems
More informationDesign and Optimization of Fuzzy Controller for Inverse Pendulum System Using Genetic Algorithm
Desgn and Optmzaton of Fuzzy Controller for Inverse Pendulum System Usng Genetc Algorthm H. Mehraban A. Ashoor Unversty of Tehran Unversty of Tehran h.mehraban@ece.ut.ac.r a.ashoor@ece.ut.ac.r Abstract:
More informationNonlinear Oscillations in a three-dimensional Competition with Inhibition Responses in a Bio-Reactor
Internatonal Journal of Appled Scence Engneerng 7 5 : 9-5 Nonlnear Oscllatons n a three-dmensonal Competton wth Inhbton Responses n a Bo-Reactor Xaonng Xu a Xuncheng Huang ab* a Yangzhou Polytechnc Unversty
More informationmodeling of equilibrium and dynamic multi-component adsorption in a two-layered fixed bed for purification of hydrogen from methane reforming products
modelng of equlbrum and dynamc mult-component adsorpton n a two-layered fxed bed for purfcaton of hydrogen from methane reformng products Mohammad A. Ebrahm, Mahmood R. G. Arsalan, Shohreh Fatem * Laboratory
More informationThe Synchronous 8th-Order Differential Attack on 12 Rounds of the Block Cipher HyRAL
The Synchronous 8th-Order Dfferental Attack on 12 Rounds of the Block Cpher HyRAL Yasutaka Igarash, Sej Fukushma, and Tomohro Hachno Kagoshma Unversty, Kagoshma, Japan Emal: {garash, fukushma, hachno}@eee.kagoshma-u.ac.jp
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationA PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.
Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng -- ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng -- ABCM, Curtba, Brazl,- Dec. 5-8, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR
More informationPerfect Fluid Cosmological Model in the Frame Work Lyra s Manifold
Prespacetme Journal December 06 Volume 7 Issue 6 pp. 095-099 Pund, A. M. & Avachar, G.., Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold Perfect Flud Cosmologcal Model n the Frame Work Lyra
More informationComplex Network Dynamics:
Complex Network Dynamcs: Theory & Applcaton Yur Mastrenko E-mal: y.mastrenko@bomed.kev.ua Lecture 2 Network Scence Complex networks versus Dynamcal networks Oscllatory networks Ensembles of oscllators
More informationBuckling analysis of single-layered FG nanoplates on elastic substrate with uneven porosities and various boundary conditions
IOSR Journal of Mechancal and Cvl Engneerng (IOSR-JMCE) e-issn: 78-1684,p-ISSN: 30-334X, Volume 15, Issue 5 Ver. IV (Sep. - Oct. 018), PP 41-46 www.osrjournals.org Bucklng analyss of sngle-layered FG nanoplates
More informationContinuous Time Markov Chain
Contnuous Tme Markov Chan Hu Jn Department of Electroncs and Communcaton Engneerng Hanyang Unversty ERICA Campus Contents Contnuous tme Markov Chan (CTMC) Propertes of sojourn tme Relatons Transton probablty
More informationControl and Synchronization of Chaotic Fractional-Order Coullet System via Active Controller
Control and Synchronzaton of Chaotc Fractonal-Order Coullet System va Actve Controller M. Shahr T.*, A. Ranjbar N.*, R. Ghader*, S. H. Hossenna*, S. Moman** * Noushrvan Unversty of Technology, Faculty
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationHigh resolution entropy stable scheme for shallow water equations
Internatonal Symposum on Computers & Informatcs (ISCI 05) Hgh resoluton entropy stable scheme for shallow water equatons Xaohan Cheng,a, Yufeng Ne,b, Department of Appled Mathematcs, Northwestern Polytechncal
More informationThe Quadratic Trigonometric Bézier Curve with Single Shape Parameter
J. Basc. Appl. Sc. Res., (3541-546, 01 01, TextRoad Publcaton ISSN 090-4304 Journal of Basc and Appled Scentfc Research www.textroad.com The Quadratc Trgonometrc Bézer Curve wth Sngle Shape Parameter Uzma
More informationSTUDY ON TWO PHASE FLOW IN MICRO CHANNEL BASED ON EXPERI- MENTS AND NUMERICAL EXAMINATIONS
Blucher Mechancal Engneerng Proceedngs May 0, vol., num. www.proceedngs.blucher.com.br/evento/0wccm STUDY ON TWO PHASE FLOW IN MICRO CHANNEL BASED ON EXPERI- MENTS AND NUMERICAL EXAMINATIONS Takahko Kurahash,
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationVARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES
VARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES BÂRZĂ, Slvu Faculty of Mathematcs-Informatcs Spru Haret Unversty barza_slvu@yahoo.com Abstract Ths paper wants to contnue
More informationFundamental loop-current method using virtual voltage sources technique for special cases
Fundamental loop-current method usng vrtual voltage sources technque for specal cases George E. Chatzaraks, 1 Marna D. Tortorel 1 and Anastasos D. Tzolas 1 Electrcal and Electroncs Engneerng Departments,
More informationGlobal Sensitivity. Tuesday 20 th February, 2018
Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values
More informationTHE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationrisk and uncertainty assessment
Optmal forecastng of atmospherc qualty n ndustral regons: rsk and uncertanty assessment Vladmr Penenko Insttute of Computatonal Mathematcs and Mathematcal Geophyscs SD RAS Goal Development of theoretcal
More informationUncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
More informationHandout # 6 (MEEN 617) Numerical Integration to Find Time Response of SDOF mechanical system. and write EOM (1) as two first-order Eqs.
Handout # 6 (MEEN 67) Numercal Integraton to Fnd Tme Response of SDOF mechancal system State Space Method The EOM for a lnear system s M X + DX + K X = F() t () t = X = X X = X = V wth ntal condtons, at
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationTensor Smooth Length for SPH Modelling of High Speed Impact
Tensor Smooth Length for SPH Modellng of Hgh Speed Impact Roman Cherepanov and Alexander Gerasmov Insttute of Appled mathematcs and mechancs, Tomsk State Unversty 634050, Lenna av. 36, Tomsk, Russa RCherepanov82@gmal.com,Ger@npmm.tsu.ru
More informationAn identification algorithm of model kinetic parameters of the interfacial layer growth in fiber composites
IOP Conference Seres: Materals Scence and Engneerng PAPER OPE ACCESS An dentfcaton algorthm of model knetc parameters of the nterfacal layer growth n fber compostes o cte ths artcle: V Zubov et al 216
More informationCHEMICAL REACTIONS AND DIFFUSION
CHEMICAL REACTIONS AND DIFFUSION A.K.A. NETWORK THERMODYNAMICS BACKGROUND Classcal thermodynamcs descrbes equlbrum states. Non-equlbrum thermodynamcs descrbes steady states. Network thermodynamcs descrbes
More information1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 -Davd Klenfeld - Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys
More informationA Solution of the Harry-Dym Equation Using Lattice-Boltzmannn and a Solitary Wave Methods
Appled Mathematcal Scences, Vol. 11, 2017, no. 52, 2579-2586 HIKARI Ltd, www.m-hkar.com https://do.org/10.12988/ams.2017.79280 A Soluton of the Harry-Dym Equaton Usng Lattce-Boltzmannn and a Soltary Wave
More informationUncertainty and auto-correlation in. Measurement
Uncertanty and auto-correlaton n arxv:1707.03276v2 [physcs.data-an] 30 Dec 2017 Measurement Markus Schebl Federal Offce of Metrology and Surveyng (BEV), 1160 Venna, Austra E-mal: markus.schebl@bev.gv.at
More informationWeighted Fifth Degree Polynomial Spline
Pure and Appled Mathematcs Journal 5; 4(6): 69-74 Publshed onlne December, 5 (http://www.scencepublshnggroup.com/j/pamj) do:.648/j.pamj.546.8 ISSN: 36-979 (Prnt); ISSN: 36-98 (Onlne) Weghted Ffth Degree
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationElectrical Circuits 2.1 INTRODUCTION CHAPTER
CHAPTE Electrcal Crcuts. INTODUCTION In ths chapter, we brefly revew the three types of basc passve electrcal elements: resstor, nductor and capactor. esstance Elements: Ohm s Law: The voltage drop across
More information