# The equation of motion of a dynamical system is given by a set of differential equations. That is (1)

Size: px
Start display at page:

Download "The equation of motion of a dynamical system is given by a set of differential equations. That is (1)"

Transcription

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 of external exctaton shown n the fgure, the angle θ and the angular velocty θ & unquely defne the state of the dynamcal system. Phase Space mg Fgure 1. Smple pendulum. Plots of the state varables aganst one another are referred to as the phase space representaton. Every pont n the phase space dentfes a unque state of the system. For the pendulum, a plot of θ & versus θ s the phase space representaton. Equaton of Moton The equaton of moton of a dynamcal system s gven by a set of dfferental equatons. That s x& = f(x, t) (1) where x s the state and t s tme. The dynamcal system s lnear f the governng equaton s lnear. For the pendulum shown n Fgure 1, the equaton of moton s gven as θ & = ω ω & = ζωoω ω o sn θ where ω = g o l () and the dynamcal system s nonlnear. For small ampltude oscllaton, the equaton of moton becomes θ & = ω ω & = ζωoω ωoθ sn θ θ, and (3) The dynamcal system s now lnear. In Equatons () and (3) ωo s the natural frequency and ζ s the dampng coeffcent. Autonomous and Nonautonomous Systems A system s sad to autonomous f tme does not appear explctly n the equaton of moton. The equaton of moton of nonautonomous systems, however, explctly 1

2 depends on tme. Thus, the equatons of moton gven by () and (3) for a pendulum n absence of external exctaton are for autonomous systems, whle a forced pendulum s a nonautonomous system. A nonautonomous system could be determnstc or stochastc. Orbt An orbt or trajectory s a curve n phase space, whch s obtaned by the soluton of the equaton of moton. Flow For a fxed tme, f(x,t), the rght-hand sde of the equaton of moton gven by (1), dentfes a vector feld n the phase space that s tangent to the trajectores. Fgure shows the phase space and the flow for a damped pendulum gven by Equaton (). The black arrows of the vector feld f = ( ω, ζω oω ωo sn θ) are tangental to all trajectores n the phase space. A two-dmensonal phase space provdes a qualtatve pcture of the behavor of the dynamcal systems ncludng ts orbt and ts flow. In Fgure the dashed red lnes are null clnes. These are the lnes where the tme dervatve of one component of the state varable s zero. For the damped pendulum t follows that θ & = ω = 0 ω = ωo sn θ / ζ (3) Therefore, one null clne s the angle θ axes and the other s a snusodal curve n the phase space. On the frst null clne the flow vector feld s vertcal (snce θ & = 0 ), whle on the second one, the vector feld s horzontal (snce ω& = 0 ). Fgure. Schematcs of the phase plane for the damped pendulum. (

3 Between the null clnes the drecton of the flow vector s determned by the sgns of θ & and ω. & At the ntersecton ponts of the null clnes, θ & =ω= & 0, and the flow vector feld s zero. These ntersecton ponts, whch correspond to the statonary solutons, are called fxed ponts or equlbrum ponts. Fxed ponts of a dynamcal system could be stable or unstable. Poncaré Secton and Poncaré Map Poncaré secton and Poncaré map are tools for vsualzaton of the flow n a phase space. Poncaré secton s a plane (or curved surface) n the phase space that s crossed by almost all orbts. The Poncaré map maps the ponts of the Poncaré secton onto tself. Consecutve ntersecton ponts of the orbt wth the Poncaré secton form the Poncaré map. A Poncaré map represents a contnuous dynamcal system by a dscrete one. Poncaré maps are nvertable maps because one gets from by followng the orbt u n u n + 1 Fgure 3. Schematcs of Poncaré secton and Poncaré map. backwards. For the case of the forced pendulum the Poncaré secton s defned by a certan phase of the tme-perodc exctaton. Non-Wanderng Set Non-wanderng set s a set of ponts n the phase space for whch all orbts startng from a pont of ths set come arbtrarly close and arbtrarly often to any pont of the set. Non-wanderng sets are of four types. These are Fxed (statonary) ponts. For the smple pendulum gven by Equaton (), ω = θ & = 0 and θ = 0 and θ = 180 are fxed ponts. Lmt cycles (perodc solutons). These solutons are common for the lnearzed oscllaton of a smple pendulum. Quas-perodc orbts. Perodc solutons wth at least two ncommensurable frequences. These solutons occur for an undamped pendulum under perodc exctatons. Chaotc orbts. Bound non-perodc solutons. These solutons occur for a (nonlnear) pendulum under certan external exctatons. The frst three types can also occur n lnear dynamcal systems. The fourth type appears only n nonlnear systems. The Poncaré map for lmt cycles become fxed ponts. A non-wanderng set can be ether stable or unstable. Changng a parameter of the system can change the stablty of a non-wanderng set. Ths s accompaned by a change of the number of non-wanderng sets due to a bfurcaton. 3

4 Stablty and Bfurcaton In nonlnear dynamcal system, the man questons are: What s the qualtatve behavor of the dynamcal system? How many non-wanderng sets (fxed ponts, lmt cycles, quas-perodc or chaotc orbts) occur? Whch of the non-wanderng sets are stable? How does the number of non-wanderng sets change wth changes n the parameters of the system? The appearance and dsappearance of non-wanderng sets s called a bfurcaton. Change of stablty and bfurcaton always concde. Stablty A non-wanderng set may be stable or unstable. The stablty could be n the sense of Lyapunov (weak) or Asymptotc (strong). Lyapunov (Margnal) Stablty A non-wanderng set s sad to be Lyapunov stable f every orbt startng n ts neghborhood remans n ts neghborhood. Asymptotc Stablty A non-wanderng set s sad to be asymptotcally stable f n addton to the Lyapunov stablty, every orbt n ts neghborhood approaches the non-wanderng set asymptotcally. Thus, a non-wanderng set s ether asymptotcally stable, margnally stable (Lyapunov stable), or unstable. Attractors and Basn of Attracton Asymptotcally stable non-wanderng sets are also called attractors. The basn of attracton s the set of all ntal states approachng the attractor n the long tme lmt. Lnear Stablty of Non-Wanderng Set To check the stablty of a non-wanderng set a lnear stablty analyss may be used. Let (t) be an orbt that satsfes Equaton (1). That s x o x & f(x t) (4) o = o, The soluton x o (t) s asymptotcally stable f any nfntesmal small perturbaton x(t) decays. Assume that x = xo ( t) + x s the perturbed soluton, and x satsfes the equaton of moton gven by (1). It then follows that 4

5 d( x) dt ( xo ) = x (5) The lnearzed equaton of moton for x(t) gven by (5) s justfed as long as the orbt s n the neghborhood of x o. For fxed ponts, the fundamental solutons for x(t) are x = e λst x s (6) d f where λ s and xs are the egenvalues and egenvector of the Jacoban, and s s the dmenson of phase space. The egenvalues λ s are the roots of the characterstc polynomal The fxed pont del λi =0 (6) x o s asymptotcally stable f the real parts of all egenvalues λ s are negatve. If at least one egenvalue has a postve real part the correspondng fundamental soluton would ncrease exponentally, and the fxed pont wll be unstable. Routh and Hurwtz theorem may be used to check the stablty wthout explctely calculatng the egenvalues. For the case of a two-dmensonal phase space the characterstc polynomal s quadratc. Routh and Hurwtz theorem mples that both egenvalues have negatve real part f and only f det s postve and the trace s negatve. The regon of stablty and nstablty are shown n Fgure 4. Nodes, Spral and Saddle Fgure 4 shows a classfcaton scheme of the fxed (statonary) ponts n twodmensonal phase spaces. The noton spral and node are nspred by the flow near the fxed pont. A par of conjugated complex egenvalues lead to a spral, whereas a node s causes by two real egenvalues of the same sgn. Real egenvalues of dfferent sgn lead to a saddle. That s, a saddle s a fxed pont where at least one egenvalue has a postve real part but also at least one egenvalue has a negatve real part. Near a saddle, an orbt s usually attracted at frst but repelled later on. There are ponts n the phase space that approach the fxed pont as t, and they form the stable manfold. Saddles and ther stable manfolds are usually the boundares of a basn of attracton. The pendulum shown n Fgure 1 has a saddle pont (correspondng to the upsde-down equlbrum poston) and a spral f ζ < 1 or a node f ζ > 1 as shown n Fgure. j 5

6 det j Fgure 4. The regon of stablty and nstablty and classfcaton scheme of the fxed (statonary) ponts n two-dmensonal phase spaces. 6

7 Bfurcaton The number of attractors n a nonlnear dynamcal system can change when a system parameter s vared. Ths change s called bfurcaton and t s accompaned by a change of the stablty of an attractor. In a bfurcaton pont, at least one egenvalue of the Jacoban wll attan a zero real part. There are three generc types of bfurcaton. Statonary Bfurcaton In a statonary bfurcaton, a sngle real egenvalue crosses the boundary of stablty. Hopf Bfurcaton Hopf bfurcaton occurs when a conjugated complex par crosses the boundary of stablty. In the tme-contnuous case, a lmt cycle bfurcates. It has an angular frequency that s gven by the magnary part of the crossng par. In ths secton the man forms one-dmensonal bfurcaton are llustrated. The phase space varable s u. The control parameter s µ. The bfurcaton pont s at µ = 0. The drecton of moton n the one-dmensonal phase space s shown by arrows. Stable (unstable) fxed ponts are drawn as red sold (dotted) lnes. Ptchfork Bfurcaton Ptchfork bfurcaton s possble n dynamcal systems wth an nverson or reflecton symmetry. That s, an equaton of moton that remans unchanged f one changes the sgn of all phase space varables. An example s a system governed by u& = ( µ u )u (7) As the control parameter µ vares the statonary solutons and ther stablty condtons changes. For µ < 0, there s one equlbrum soluton, u=0, whch s stable. For µ > 0, there are three equlbrum solutons u=0, and u = ± µ. Here, u=0 s unstable whle u = ± µ are stable. At µ = 0, a bfurcaton occurs whch s referred to as (supercrtcal) ptchfork bfurcaton. Smlarly for a system governed by 7

8 Fgure 5. Schematcs of a ptchfork bfurcaton. u& = ( µ + u )u (8) there are three equlbrum solutons u=0, and u = ± µ for µ < 0. In ths case u=0 s a stable soluton whle u = ± µ are unstable. For µ > 0, there s only one soluton, u=0, whch s stable. In ths case at µ = 0, a (subercrtcal) ptchfork bfurcaton occurs. Transcrtcal Bfurcaton Consder a dynamcal system gven by u& = ( µ u)u (9) For µ < 0, there are two equlbrum soluton, u=0, whch s stable, and u = µ, whch s unstable. For µ > 0, there are stll the same two equlbrum soluton. But u=0 s unstable, and u = µ s now stable. That s at µ = 0, the two statonary soluton exchanged ther stablty. In ths case t s sad that a transcrtcal bfurcaton occurred. 8

### CHAPTER 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

### Chapter 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

### Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

### Introduction. - 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

### 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

### CHAPTER II Recurrent Neural Networks

CHAPTER II Recurrent Neural Networks In ths chapter frst the dynamcs of the contnuous space recurrent neural networks wll be examned n a general framework. Then, the Hopfeld Network as a specal case of

### COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not

### Chapter 8. Potential Energy and Conservation of Energy

Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and non-conservatve forces Mechancal Energy Conservaton of Mechancal

### Difference Equations

Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

### CHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)

CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O

### Chapter 4 The Wave Equation

Chapter 4 The Wave Equaton Another classcal example of a hyperbolc PDE s a wave equaton. The wave equaton s a second-order lnear hyperbolc PDE that descrbes the propagaton of a varety of waves, such as

### NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

### 3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

### Physics 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

### Nice plotting of proteins II

Nce plottng of protens II Fnal remark regardng effcency: It s possble to wrte the Newton representaton n a way that can be computed effcently, usng smlar bracketng that we made for the frst representaton

### Delay 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

### = 1.23 m/s 2 [W] Required: t. Solution:!t = = 17 m/s [W]! m/s [W] (two extra digits carried) = 2.1 m/s [W]

Secton 1.3: Acceleraton Tutoral 1 Practce, page 24 1. Gven: 0 m/s; 15.0 m/s [S]; t 12.5 s Requred: Analyss: a av v t v f v t a v av f v t 15.0 m/s [S] 0 m/s 12.5 s 15.0 m/s [S] 12.5 s 1.20 m/s 2 [S] Statement:

### Modeling 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

### Lecture 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

### APPENDIX A Some Linear Algebra

APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

### Section 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

### DO 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.

### First Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.

Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act

### Physics 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

### 12. The Hamilton-Jacobi Equation Michael Fowler

1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and

### Math1110 (Spring 2009) Prelim 3 - Solutions

Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.

### C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1

C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned

### U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017

U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that

### Module 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

### Additional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty

Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,

### Kernel Methods and SVMs Extension

Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general

### Chapter 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

### SCHOOL OF FINANCE AND ECONOMICS

SCHOOL OF FINANCE AND ECONOMICS UTS:BUSINESS WORKING PAPER NO. 11 DECEMBER, 1991 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays Carl Charella ISSN: 1036-7373 http://www.busness.uts.edu.au/fnance/

### 1 Matrix representations of canonical matrices

1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:

### SIMULATION OF WAVE PROPAGATION IN AN HETEROGENEOUS ELASTIC ROD

SIMUATION OF WAVE POPAGATION IN AN HETEOGENEOUS EASTIC OD ogéro M Saldanha da Gama Unversdade do Estado do o de Janero ua Sào Francsco Xaver 54, sala 5 A 559-9, o de Janero, Brasl e-mal: rsgama@domancombr

### Notes 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

### A Quantum Gauss-Bonnet Theorem

A Quantum Gauss-Bonnet Theorem Tyler Fresen November 13, 2014 Curvature n the plane Let Γ be a smooth curve wth orentaton n R 2, parametrzed by arc length. The curvature k of Γ s ± Γ, where the sgn s postve

### Chapter Newton s Method

Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve

### Dynamical Systems and Information Theory

Dynamcal Systems and Informaton Theory Informaton Theory Lecture 4 Let s consder systems that evolve wth tme x F ( x, x, x,... That s, systems that can be descrbed as the evoluton of a set of state varables

Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton

### ELASTIC 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

### 6. Stochastic processes (2)

6. Stochastc processes () Lect6.ppt S-38.45 - Introducton to Teletraffc Theory Sprng 5 6. Stochastc processes () Contents Markov processes Brth-death processes 6. Stochastc processes () Markov process

### 6. Stochastic processes (2)

Contents Markov processes Brth-death processes Lect6.ppt S-38.45 - Introducton to Teletraffc Theory Sprng 5 Markov process Consder a contnuous-tme and dscrete-state stochastc process X(t) wth state space

### Linear Regression Analysis: Terminology and Notation

ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented

### Transfer 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

### Stability and Analytical Approximation of Limit. Cycles in Hopf Bifurcations of Four-dimensional. Economic Models

Appled Mathematcal Scences, Vol. 8, 4, no. 8, 967-99 HIKARI Ltd, www.m-hkar.com http://dx.do.org/.988/ams.4.445 Stablty and Analytcal Approxmaton of Lmt Cycles n Hopf Bfurcatons of Four-dmensonal Economc

### Second Order Analysis

Second Order Analyss In the prevous classes we looked at a method that determnes the load correspondng to a state of bfurcaton equlbrum of a perfect frame by egenvalye analyss The system was assumed to

### Solutions to exam in SF1811 Optimization, Jan 14, 2015

Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable

### The Feynman path integral

The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space

### coordinates. Then, the position vectors are described by

Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,

### 2 Finite difference basics

Numersche Methoden 1, WS 11/12 B.J.P. Kaus 2 Fnte dfference bascs Consder the one- The bascs of the fnte dfference method are best understood wth an example. dmensonal transent heat conducton equaton T

### Week 9 Chapter 10 Section 1-5

Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,

### Prof. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model

EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several

### Mathematical Preparations

1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the

### Affine transformations and convexity

Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/

### 11. Dynamics in Rotating Frames of Reference

Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons

### Digital Signal Processing

Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over

### Physics 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

### SCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors.

SCALARS AND ECTORS All phscal uanttes n engneerng mechancs are measured usng ether scalars or vectors. Scalar. A scalar s an postve or negatve phscal uantt that can be completel specfed b ts magntude.

### More 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

### Renormalised steepest descent converges to a two-point attractor

Renormalsed steepest descent converges to a two-pont attractor Luc Pronzato, Henry P. Wynn and Anatoly A. Zhglavsky Abstract The behavour of the standard steepest-descent algorthm for a quadratc functon

### Canonical transformations

Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,

### Snce 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

### Lecture 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

### Limit 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

### Frequency dependence of the permittivity

Frequency dependence of the permttvty February 7, 016 In materals, the delectrc constant and permeablty are actually frequency dependent. Ths does not affect our results for sngle frequency modes, but

### Physics 181. Particle Systems

Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system

### Convexity 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

### Boundary Layer to a System of Viscous Hyperbolic Conservation Laws

Acta Mathematcae Applcatae Snca, Englsh Seres Vol. 24, No. 3 (28) 523 528 DOI: 1.17/s1255-8-861-6 www.applmath.com.cn Acta Mathema ca Applcatae Snca, Englsh Seres The Edtoral Offce of AMAS & Sprnger-Verlag

### Lecture Notes on Linear Regression

Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume

### Gravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11)

Gravtatonal Acceleraton: A case of constant acceleraton (approx. hr.) (6/7/11) Introducton The gravtatonal force s one of the fundamental forces of nature. Under the nfluence of ths force all objects havng

### Physics 2A Chapter 3 HW Solutions

Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C

### Continuous 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

### Nonlinear 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

### Appendix 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

### Random Walks on Digraphs

Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected

### Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,

### Numerical 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

### MEEM 3700 Mechanical Vibrations

MEEM 700 Mechancal Vbratons Mohan D. Rao Chuck Van Karsen Mechancal Engneerng-Engneerng Mechancs Mchgan echnologcal Unversty Copyrght 00 Lecture & MEEM 700 Multple Degree of Freedom Systems (ext: S.S.

### Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2

Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to

### CS 468 Lecture 16: Isometry Invariance and Spectral Techniques

CS 468 Lecture 16: Isometry Invarance and Spectral Technques Justn Solomon Scrbe: Evan Gawlk Introducton. In geometry processng, t s often desrable to characterze the shape of an object n a manner that

### For all questions, answer choice E) NOTA" means none of the above answers is correct.

0 MA Natonal Conventon For all questons, answer choce " means none of the above answers s correct.. In calculus, one learns of functon representatons that are nfnte seres called power 3 4 5 seres. For

### Representation theory and quantum mechanics tutorial Representation theory and quantum conservation laws

Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac

### The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction

ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also

### Module 9. Lecture 6. Duality in Assignment Problems

Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept

### PHYSICS - CLUTCH CH 28: INDUCTION AND INDUCTANCE.

!! www.clutchprep.com CONCEPT: ELECTROMAGNETIC INDUCTION A col of wre wth a VOLTAGE across each end wll have a current n t - Wre doesn t HAVE to have voltage source, voltage can be INDUCED V Common ways

### Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)

I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes

### Limited Dependent Variables

Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages

### Flow 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,

### Convergence of random processes

DS-GA 12 Lecture notes 6 Fall 216 Convergence of random processes 1 Introducton In these notes we study convergence of dscrete random processes. Ths allows to characterze phenomena such as the law of large

### Indeterminate pin-jointed frames (trusses)

Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all

### Three views of mechanics

Three vews of mechancs John Hubbard, n L. Gross s course February 1, 211 1 Introducton A mechancal system s manfold wth a Remannan metrc K : T M R called knetc energy and a functon V : M R called potental

### INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING & TECHNOLOGY (IJEET)

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.

### Polynomial Regression Models

LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance

Chemstry 13 NT Be true to your work, your word, and your frend. Henry Davd Thoreau 1 Chem 13 NT Chemcal Equlbrum Module Usng the Equlbrum Constant Interpretng the Equlbrum Constant Predctng the Drecton

### This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as:

1 Problem set #1 1.1. A one-band model on a square lattce Fg. 1 Consder a square lattce wth only nearest-neghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors

### ρ some λ THE INVERSE POWER METHOD (or INVERSE ITERATION) , for , or (more usually) to

THE INVERSE POWER METHOD (or INVERSE ITERATION) -- applcaton of the Power method to A some fxed constant ρ (whch s called a shft), x λ ρ If the egenpars of A are { ( λ, x ) } ( ), or (more usually) to,

### ENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15

NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl 15 1. Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound