CENTER MANIFOLD AND NORMAL FORM THEORIES

Size: px
Start display at page:

Download "CENTER MANIFOLD AND NORMAL FORM THEORIES"

Transcription

1 3 rd Sperlonga Summer School on Mechanics and Engineering Sciences 3-7 September 013 SPERLONGA CENTER MANIFOLD AND NORMAL FORM THEORIES ANGELO LUONGO 1

2 THE CENTER MANIFOLD METHOD Existence of an invariant manifold Linear systems The state-space N X of the linear system x () t Jx() t is direct sum of three invariant sub-spaces, i.e. X Xc Xs X, u where: Xc is the center subspace, of dimension N c, spanned by the (generalized) eigenvectors associated with non-hyperbolic eigenvalues of J; X s is the stable subspace, of dimension N s, spanned by the (generalized) eigenvectors associated with hyperbolic eigenvalues of J having negative real part; X u is the unstable subspace, of dimension N u, spanned by the (generalized) eigenvectors associated with non-hyperbolic eigenvalues of J having positive real part.

3 Nonlinear systems We consider the nonlinear system (in local form): x () t F( x(), t μ) admitting the critical equilibrium ( x 0, μc 0 ). We assume that J: F x ( 00, ) posses N c >0 critical eigenvalues, N s >0 stable eigenvalues and N u =0 unstable eigenvalues. The Center Manifold Theorem states that the asymptotic dynamics of the system around the equilibrium point x 0, at the critical value of the parameters μ μ c, takes place on a (critical) manifold M X c, which has the following properties: M c has dimension N c ; M c is tangent to the critical subspace Xc at x 0 ; M c is attractive, i.e. all the orbits tend to it when t. 3

4 The center manifold is therefore an N c -dimensional surface in the N=N c +N s -dimensional state-space. Example: To analyze the asymptotic dynamics, it needs: (a) to find the center manifold M c ; (b) to obtain the reduced N c dimensional equations governing the motion on M (bifurcation equations). c 4

5 Dependence of the CM on parameters Since we are interested not only in the dynamics at μ μ c, but also at the dynamics at μ close to μ c, we can use the trick to consider μ as additional critical variables, by considering the extended dynamical system: x () t F( x(), t μ()) t μ () t 0 Therefore, the critical subspace becomes X c : X c P with P : { μ} the parameter space. Hence: M c has dimension N c +M; M c is tangent to the critical subspace X c ; M c is attractive, i.e. all the orbits tend to it when t. 5

6 Reduction process Equations of motion, expanded: By expanding x () t F( x() t, μ) (and ignoring the dummy equations μ 0 ) for small x() t and μ close to μ c, we have: x () t Jx() t f( x(), t μ), f O( x() t, μ ) Linear transformation of the variables: After letting x(): t ( x (), t x ()) t, with x () t X critical variables and x s c s c () t X stable variables, a proper linear transformation uncouples the linear s part of the equations (t omitted): x c Jc 0 xc fc( xc, xs, μ) s s s( c, s, ) x 0 Js x f x x μ c 6

7 Center manifold: active and passive coordinates Cartesian equations for the CM: Tangency requirements: x h( x, μ ) s c h( 0, 0) 0, h ( 0, 0) 0, h ( 0, 0) 0. x The equation expresses the stable variables x s as (unknown) functions of the critical variables x c ; therefore x c are also said active coordinates, and x s passive coordinates. Time derivative of x s : The passive character of x s also holds for time-derivatives. By using the chain rule and the upper partition of the equations of motion: d xs h( xc, μ) hx ( xc, μ) xc d t h ( x, μ)( Jx f( x, hx ( ), μ)) x c c c c c c μ 7

8 Equation for the center manifold: The lower-partition of the equation of motion supplies the equation for determining M : c h ( x, μ)( Jx f( x, hx ( ), μ)) Jhx ( ) f( x, hx ( ), μ ) x c c c c c c s c s c c This equation can be solved, e.g., by using power series expansions, (starting from degree- polynomial, due to the required tangency): hx (, μ) α z α z, z: { x, μ} c 3 3 c Bifurcation equations: The upper-partition governs the dynamics on M c : x J x f ( x, h( x ), μ) c c c c c c 8

9 Example 1: a static bifurcation Nonlinear, -dimensional system: x x xy cx y 0 1y bx 3 0 Critical point, critical and stable coordinates: Equation for the Center Manifold: 0, x { x}, x { y} c c s y hx (, ) h( x, )[ xxhx (, ) cx] hx (, ) bx x 3 Power series expansion: y h( x, ) { x x } { x }

10 Zeroing separately the different powers: x : b0 0 x: 0 11 : 0 By solving for α s coefficients, at the lowest order, we have: 0 3 ybx x O( (, ) ) Bifurcation equation: x x( bc) x 3 If b+c<0, a supercritical (stable) pitchfork occurs; if b+c>0, a sub-critical (unstable) pitchfork occurs; if b+c=0, higher-order terms must be evaluated. 10

11 Example : a dynamical bifurcation 3-dimensional nonlinear system: x 1 0 x xz y 1 0 y yz z 0 0 1z x kxy y Equation for CM: z x y xy x y O(3) z-equation (passive coordinate), transformed: ( ) xy ( x y ) x y ( x y xy x y) x kxy y O(3)

12 Zeroing separately the different powers in the z-equation: x 31 1 : k /5 y : k /5 xy : ( 1) 3k 0 3 k /5 x : y : Bifurcation equations: x x yx k x k xy k y y y x y k x k xy k y [(1 /5) /5 (1 /5) ] [(1 / 5) / 5 (1 / 5) ] 1

13 THE NORMAL FORM THEORY Scope: To use a smooth nonlinear coordinate transformation, in order to put the bifurcation equation in the simplest form. Algorithm: Equations of motion: x Jx f( x) Transformed Equations: y Jy g( y) Near-identity transformation: x y h( y ) where x (possibly) includes the dummy variables µ. 13

14 o Transformed equation in the h(y) unknown: By differentiating the nearly-identity transformation: x [ I h y ( y)] y the equation of motion is transformed into: [ I h ( y)][ Jyg( y)] J[ yh( y)] f( yh( y)) y or: h ( y) Jy Jh( y) f ( y h( y)) [ Ih ( y)] g( y ) y y which is a differential equation for h(y) for any given g(y). A series solution is often necessary. 14

15 o By letting: fy ( ) f( y) f3( y) gy ( ) g( y) g3( y) hy ( ) h ( y) h ( y) 3 with the homogeneous polynomial of k-degree: Mk Mk Mk f ( y) p ( y), g ( y) p ( y), h ( y) p ( y) k km km k km km k km km m1 m1 m1 15

16 o Zeroing the independent monomials: where (if J is diagonal): L o Resonance: L γ α β k k k k diag[ ], : ( m ), m k k i i j j i j j1 j1 N j 0,0,, i, i,, 0 Since 1 i for some i (i.e. L k is singular); hence, β k 0 must be taken for compatibility, and resonant terms survive in the normal form!!! N 16

17 Example 1: System independent of parameters o System at a Hopf bifurcation: 3 3 x i 0 x 31x 3x x 33xx 34x, x, 3 3 x 0 i x 31x 3x x33xx 34x o Normal form: o Near-identity transformation: y iy y y y yy y x y y y y yy y

18 o Equation for the coefficients: i i i o Solution: 0, o Normal form: y iy3 y y The term y y is resonant, since 1 1, i.e. ( i) ( i) i 18

19 o Amplitude equation: Changing the variables according to: it yt () At ()e, At () the normal form is transformed into: [ At () iat ()]e iat ()e A() t At ()e it it (1) it 3 or: A t () 3 A () t A() t This is called Amplitude Modulation Equation (AME). Since A 3 O( A ), the AME describes a slow modulation. Therefore, the change of variable filters the fast dynamics. 19

20 Example : Non-diagonalizable system o Double-zero bifurcation equations: x 1 0 1x1 1x1 xx 1 3x x 0 0 x 4x1 5xx 1 6x o Normal Form: y 1 0 1y1 1y1 y1 y 3 y y 0 0 y 4 y1 5 y1 y 6 y o Near-Identity transformation: x1 y1 1y1 y1 y 3 y x y 4 y1 5 y1 y 6 y 0

21 o By zeroing the coefficients of the three monomials in the two transformed equations: Since Rank[ L ] 4 : K ( L ) span{ u, u }, u : (0,1,0,0,0,1), u (0,0,1,0,0,0) T 1 1 T K ( L ) span{ v, v }, v (,0,0,0,1,0), v (0,0,0,1,0,0) T T T 1 1 1

22 o Compatibility condition: ( ) ( ) 0, It is possible to take and 1 0 or 5 0. o By taking 5 0, / the NF reads (Takens normal form): 1 y 1 0 1y1 ( 1 5) y1 y 0 0 y 4 y 1 o By taking 1 0, the NF reads (Bogdanov normal form): y 1 0 1y 0 1 y 0 0 y 4 y1 ( 5 1) y1 y The Normal Form is not unique!!

Problem set 7 Math 207A, Fall 2011 Solutions

Problem set 7 Math 207A, Fall 2011 Solutions Problem set 7 Math 207A, Fall 2011 s 1. Classify the equilibrium (x, y) = (0, 0) of the system x t = x, y t = y + x 2. Is the equilibrium hyperbolic? Find an equation for the trajectories in (x, y)- phase

More information

Parameter-dependent normal forms for bifurcation equations of dynamical systems

Parameter-dependent normal forms for bifurcation equations of dynamical systems Parameter-dependent normal forms for bifurcation equations of dynamical systems Angelo uongo Dipartento di Ingegneria delle Strutture delle Acque e del Terreno Università dell Aquila Monteluco Roio 67040

More information

Lotka Volterra Predator-Prey Model with a Predating Scavenger

Lotka Volterra Predator-Prey Model with a Predating Scavenger Lotka Volterra Predator-Prey Model with a Predating Scavenger Monica Pescitelli Georgia College December 13, 2013 Abstract The classic Lotka Volterra equations are used to model the population dynamics

More information

B5.6 Nonlinear Systems

B5.6 Nonlinear Systems B5.6 Nonlinear Systems 4. Bifurcations Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Local bifurcations for vector fields 1.1 The problem 1.2 The extended centre

More information

Half of Final Exam Name: Practice Problems October 28, 2014

Half of Final Exam Name: Practice Problems October 28, 2014 Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half

More information

Entrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.

Entrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4. Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the

More information

Bifurcation of Fixed Points

Bifurcation of Fixed Points Bifurcation of Fixed Points CDS140B Lecturer: Wang Sang Koon Winter, 2005 1 Introduction ẏ = g(y, λ). where y R n, λ R p. Suppose it has a fixed point at (y 0, λ 0 ), i.e., g(y 0, λ 0 ) = 0. Two Questions:

More information

2.10 Saddles, Nodes, Foci and Centers

2.10 Saddles, Nodes, Foci and Centers 2.10 Saddles, Nodes, Foci and Centers In Section 1.5, a linear system (1 where x R 2 was said to have a saddle, node, focus or center at the origin if its phase portrait was linearly equivalent to one

More information

Elements of Applied Bifurcation Theory

Elements of Applied Bifurcation Theory Yuri A. Kuznetsov Elements of Applied Bifurcation Theory Third Edition With 251 Illustrations Springer Introduction to Dynamical Systems 1 1.1 Definition of a dynamical system 1 1.1.1 State space 1 1.1.2

More information

Dynamical systems tutorial. Gregor Schöner, INI, RUB

Dynamical systems tutorial. Gregor Schöner, INI, RUB Dynamical systems tutorial Gregor Schöner, INI, RUB Dynamical systems: Tutorial the word dynamics time-varying measures range of a quantity forces causing/accounting for movement => dynamical systems dynamical

More information

Economics 204 Fall 2013 Problem Set 5 Suggested Solutions

Economics 204 Fall 2013 Problem Set 5 Suggested Solutions Economics 204 Fall 2013 Problem Set 5 Suggested Solutions 1. Let A and B be n n matrices such that A 2 = A and B 2 = B. Suppose that A and B have the same rank. Prove that A and B are similar. Solution.

More information

CHAPTER 6 HOPF-BIFURCATION IN A TWO DIMENSIONAL NONLINEAR DIFFERENTIAL EQUATION

CHAPTER 6 HOPF-BIFURCATION IN A TWO DIMENSIONAL NONLINEAR DIFFERENTIAL EQUATION CHAPTER 6 HOPF-BIFURCATION IN A TWO DIMENSIONAL NONLINEAR DIFFERENTIAL EQUATION [Discussion on this chapter is based on our paper entitled Hopf-Bifurcation Ina Two Dimensional Nonlinear Differential Equation,

More information

Linear Algebra Practice Problems

Linear Algebra Practice Problems Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,

More information

APPPHYS217 Tuesday 25 May 2010

APPPHYS217 Tuesday 25 May 2010 APPPHYS7 Tuesday 5 May Our aim today is to take a brief tour of some topics in nonlinear dynamics. Some good references include: [Perko] Lawrence Perko Differential Equations and Dynamical Systems (Springer-Verlag

More information

PROJECTS on. Multiple Bifurcations of Sample Dynamical Systems

PROJECTS on. Multiple Bifurcations of Sample Dynamical Systems Course on Bifurcation Theory, a.y. 2010/11 PROJECTS on Multiple Bifurcations of Sample Dynamical Systems Students of the Bifurcation Theory course can carry out an individual homework, to be discussed

More information

B5.6 Nonlinear Systems

B5.6 Nonlinear Systems B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A

More information

1. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal

1. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal . Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal 3 9 matrix D such that A = P DP, for A =. 3 4 3 (a) P = 4, D =. 3 (b) P = 4, D =. (c) P = 4 8 4, D =. 3 (d) P

More information

Nonlinear Autonomous Dynamical systems of two dimensions. Part A

Nonlinear Autonomous Dynamical systems of two dimensions. Part A Nonlinear Autonomous Dynamical systems of two dimensions Part A Nonlinear Autonomous Dynamical systems of two dimensions x f ( x, y), x(0) x vector field y g( xy, ), y(0) y F ( f, g) 0 0 f, g are continuous

More information

8.1 Bifurcations of Equilibria

8.1 Bifurcations of Equilibria 1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integro-differential equations

More information

1. (i) Determine how many periodic orbits and equilibria can be born at the bifurcations of the zero equilibrium of the following system:

1. (i) Determine how many periodic orbits and equilibria can be born at the bifurcations of the zero equilibrium of the following system: 1. (i) Determine how many periodic orbits and equilibria can be born at the bifurcations of the zero equilibrium of the following system: ẋ = y x 2, ẏ = z + xy, ż = y z + x 2 xy + y 2 + z 2 x 4. (ii) Determine

More information

Chapter #4 EEE8086-EEE8115. Robust and Adaptive Control Systems

Chapter #4 EEE8086-EEE8115. Robust and Adaptive Control Systems Chapter #4 Robust and Adaptive Control Systems Nonlinear Dynamics.... Linear Combination.... Equilibrium points... 3 3. Linearisation... 5 4. Limit cycles... 3 5. Bifurcations... 4 6. Stability... 6 7.

More information

UNIVERSIDADE DE SÃO PAULO

UNIVERSIDADE DE SÃO PAULO UNIVERSIDADE DE SÃO PAULO Instituto de Ciências Matemáticas e de Computação ISSN 010-577 GLOBAL DYNAMICAL ASPECTS OF A GENERALIZED SPROTT E DIFFERENTIAL SYSTEM REGILENE OLIVEIRA CLAUDIA VALLS N o 41 NOTAS

More information

154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below.

154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below. 54 Chapter 9 Hints, Answers, and Solutions 9. The Phase Plane 9.. 4. The particular trajectories are highlighted in the phase portraits below... 3. 4. 9..5. Shown below is one possibility with x(t) and

More information

1 Linear Algebra Problems

1 Linear Algebra Problems Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and

More information

Homework For each of the following matrices, find the minimal polynomial and determine whether the matrix is diagonalizable.

Homework For each of the following matrices, find the minimal polynomial and determine whether the matrix is diagonalizable. Math 5327 Fall 2018 Homework 7 1. For each of the following matrices, find the minimal polynomial and determine whether the matrix is diagonalizable. 3 1 0 (a) A = 1 2 0 1 1 0 x 3 1 0 Solution: 1 x 2 0

More information

BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs

BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs Yuri A. Kuznetsov August, 2010 Contents 1. Solutions and orbits. 2. Equilibria. 3. Periodic orbits and limit cycles. 4. Homoclinic orbits.

More information

THE RING OF POLYNOMIALS. Special Products and Factoring

THE RING OF POLYNOMIALS. Special Products and Factoring THE RING OF POLYNOMIALS Special Products and Factoring Special Products and Factoring Upon completion, you should be able to Find special products Factor a polynomial completely Special Products - rules

More information

A matching pursuit technique for computing the simplest normal forms of vector fields

A matching pursuit technique for computing the simplest normal forms of vector fields Journal of Symbolic Computation 35 (2003) 59 65 www.elsevier.com/locate/jsc A matching pursuit technique for computing the simplest normal forms of vector fields Pei Yu, Yuan Yuan Department of Applied

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL Y G y G Y 87 y Y 8 Y - $ X ; ; y y q 8 y $8 $ $ $ G 8 q < 8 6 4 y 8 7 4 8 8 < < y 6 $ q - - y G y G - Y y y 8 y y y Y Y 7-7- G - y y y ) y - y y y y - - y - y 87 7-7- G G < G y G y y 6 X y G y y y 87 G

More information

Chapter 4: First-order differential equations. Similarity and Transport Phenomena in Fluid Dynamics Christophe Ancey

Chapter 4: First-order differential equations. Similarity and Transport Phenomena in Fluid Dynamics Christophe Ancey Chapter 4: First-order differential equations Similarity and Transport Phenomena in Fluid Dynamics Christophe Ancey Chapter 4: First-order differential equations Phase portrait Singular point Separatrix

More information

Elements of Applied Bifurcation Theory

Elements of Applied Bifurcation Theory Yuri A. Kuznetsov Elements of Applied Bifurcation Theory Second Edition With 251 Illustrations Springer Preface to the Second Edition Preface to the First Edition vii ix 1 Introduction to Dynamical Systems

More information

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator

More information

Physics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics

Physics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics Applications of nonlinear ODE systems: Physics: spring-mass system, planet motion, pendulum Chemistry: mixing problems, chemical reactions Biology: ecology problem, neural conduction, epidemics Economy:

More information

APPM 3310 Problem Set 4 Solutions

APPM 3310 Problem Set 4 Solutions APPM 33 Problem Set 4 Solutions. Problem.. Note: Since these are nonstandard definitions of addition and scalar multiplication, be sure to show that they satisfy all of the vector space axioms. Solution:

More information

Linear ODEs. Existence of solutions to linear IVPs. Resolvent matrix. Autonomous linear systems

Linear ODEs. Existence of solutions to linear IVPs. Resolvent matrix. Autonomous linear systems Linear ODEs p. 1 Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems Linear ODEs Definition (Linear ODE) A linear ODE is a differential equation taking the form

More information

7 Two-dimensional bifurcations

7 Two-dimensional bifurcations 7 Two-dimensional bifurcations As in one-dimensional systems: fixed points may be created, destroyed, or change stability as parameters are varied (change of topological equivalence ). In addition closed

More information

Examples include: (a) the Lorenz system for climate and weather modeling (b) the Hodgkin-Huxley system for neuron modeling

Examples include: (a) the Lorenz system for climate and weather modeling (b) the Hodgkin-Huxley system for neuron modeling 1 Introduction Many natural processes can be viewed as dynamical systems, where the system is represented by a set of state variables and its evolution governed by a set of differential equations. Examples

More information

Nonlinear Stability and Bifurcation of Multi-D.O.F. Chatter System in Grinding Process

Nonlinear Stability and Bifurcation of Multi-D.O.F. Chatter System in Grinding Process Key Engineering Materials Vols. -5 (6) pp. -5 online at http://www.scientific.net (6) Trans Tech Publications Switzerland Online available since 6//5 Nonlinear Stability and Bifurcation of Multi-D.O.F.

More information

DIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix

DIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix DIAGONALIZATION Definition We say that a matrix A of size n n is diagonalizable if there is a basis of R n consisting of eigenvectors of A ie if there are n linearly independent vectors v v n such that

More information

ALGEBRAIC GEOMETRY HOMEWORK 3

ALGEBRAIC GEOMETRY HOMEWORK 3 ALGEBRAIC GEOMETRY HOMEWORK 3 (1) Consider the curve Y 2 = X 2 (X + 1). (a) Sketch the curve. (b) Determine the singular point P on C. (c) For all lines through P, determine the intersection multiplicity

More information

0, otherwise. Find each of the following limits, or explain that the limit does not exist.

0, otherwise. Find each of the following limits, or explain that the limit does not exist. Midterm Solutions 1, y x 4 1. Let f(x, y) = 1, y 0 0, otherwise. Find each of the following limits, or explain that the limit does not exist. (a) (b) (c) lim f(x, y) (x,y) (0,1) lim f(x, y) (x,y) (2,3)

More information

review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17

review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17 1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient

More information

Generalized Eigenvectors and Jordan Form

Generalized Eigenvectors and Jordan Form Generalized Eigenvectors and Jordan Form We have seen that an n n matrix A is diagonalizable precisely when the dimensions of its eigenspaces sum to n. So if A is not diagonalizable, there is at least

More information

Dynamical Systems. Bernard Deconinck Department of Applied Mathematics University of Washington Campus Box Seattle, WA, 98195, USA

Dynamical Systems. Bernard Deconinck Department of Applied Mathematics University of Washington Campus Box Seattle, WA, 98195, USA Dynamical Systems Bernard Deconinck Department of Applied Mathematics University of Washington Campus Box 352420 Seattle, WA, 98195, USA June 4, 2009 i Prolegomenon These are the lecture notes for Amath

More information

Remark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Remark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero. Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called

More information

UNIVERSITY of LIMERICK

UNIVERSITY of LIMERICK UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering Department of Mathematics & Statistics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MS08 SEMESTER: Spring 0 MODULE TITLE: Dynamical

More information

Homework 6 Solutions. Solution. Note {e t, te t, t 2 e t, e 2t } is linearly independent. If β = {e t, te t, t 2 e t, e 2t }, then

Homework 6 Solutions. Solution. Note {e t, te t, t 2 e t, e 2t } is linearly independent. If β = {e t, te t, t 2 e t, e 2t }, then Homework 6 Solutions 1 Let V be the real vector space spanned by the functions e t, te t, t 2 e t, e 2t Find a Jordan canonical basis and a Jordan canonical form of T on V dened by T (f) = f Solution Note

More information

BIFURCATIONS AND STRANGE ATTRACTORS IN A CLIMATE RELATED SYSTEM

BIFURCATIONS AND STRANGE ATTRACTORS IN A CLIMATE RELATED SYSTEM dx dt DIFFERENTIAL EQUATIONS AND CONTROL PROCESSES N 1, 25 Electronic Journal, reg. N P23275 at 7.3.97 http://www.neva.ru/journal e-mail: diff@osipenko.stu.neva.ru Ordinary differential equations BIFURCATIONS

More information

Hopf bifurcations analysis of a three-dimensional nonlinear system

Hopf bifurcations analysis of a three-dimensional nonlinear system BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 358), 28, Pages 57 66 ISSN 124 7696 Hopf bifurcations analysis of a three-dimensional nonlinear system Mircea Craioveanu, Gheorghe

More information

Stability and bifurcation in a two species predator-prey model with quintic interactions

Stability and bifurcation in a two species predator-prey model with quintic interactions Chaotic Modeling and Simulation (CMSIM) 4: 631 635, 2013 Stability and bifurcation in a two species predator-prey model with quintic interactions I. Kusbeyzi Aybar 1 and I. acinliyan 2 1 Department of

More information

OWELL WEEKLY JOURNAL

OWELL WEEKLY JOURNAL Y \»< - } Y Y Y & #»»» q ] q»»»>) & - - - } ) x ( - { Y» & ( x - (» & )< - Y X - & Q Q» 3 - x Q Y 6 \Y > Y Y X 3 3-9 33 x - - / - -»- --

More information

Stable and Unstable Manifolds for Planar Dynamical Systems

Stable and Unstable Manifolds for Planar Dynamical Systems Stable and Unstable Manifolds for Planar Dynamical Systems H.L. Smith Department of Mathematics Arizona State University Tempe, AZ 85287 184 October 7, 211 Key words and phrases: Saddle point, stable and

More information

Definition (T -invariant subspace) Example. Example

Definition (T -invariant subspace) Example. Example Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin

More information

Summary of topics relevant for the final. p. 1

Summary of topics relevant for the final. p. 1 Summary of topics relevant for the final p. 1 Outline Scalar difference equations General theory of ODEs Linear ODEs Linear maps Analysis near fixed points (linearization) Bifurcations How to analyze a

More information

The Higgins-Selkov oscillator

The Higgins-Selkov oscillator The Higgins-Selkov oscillator May 14, 2014 Here I analyse the long-time behaviour of the Higgins-Selkov oscillator. The system is ẋ = k 0 k 1 xy 2, (1 ẏ = k 1 xy 2 k 2 y. (2 The unknowns x and y, being

More information

MANY BILLS OF CONCERN TO PUBLIC

MANY BILLS OF CONCERN TO PUBLIC - 6 8 9-6 8 9 6 9 XXX 4 > -? - 8 9 x 4 z ) - -! x - x - - X - - - - - x 00 - - - - - x z - - - x x - x - - - - - ) x - - - - - - 0 > - 000-90 - - 4 0 x 00 - -? z 8 & x - - 8? > 9 - - - - 64 49 9 x - -

More information

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left

More information

ADVANCED TOPICS IN ALGEBRAIC GEOMETRY

ADVANCED TOPICS IN ALGEBRAIC GEOMETRY ADVANCED TOPICS IN ALGEBRAIC GEOMETRY DAVID WHITE Outline of talk: My goal is to introduce a few more advanced topics in algebraic geometry but not to go into too much detail. This will be a survey of

More information

Part II. Dynamical Systems. Year

Part II. Dynamical Systems. Year Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 34 Paper 1, Section II 30A Consider the dynamical system where β > 1 is a constant. ẋ = x + x 3 + βxy 2, ẏ = y + βx 2

More information

8.385 MIT (Rosales) Hopf Bifurcations. 2 Contents Hopf bifurcation for second order scalar equations. 3. Reduction of general phase plane case to seco

8.385 MIT (Rosales) Hopf Bifurcations. 2 Contents Hopf bifurcation for second order scalar equations. 3. Reduction of general phase plane case to seco 8.385 MIT Hopf Bifurcations. Rodolfo R. Rosales Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts MA 239 September 25, 999 Abstract In two dimensions a Hopf bifurcation

More information

Stability of Dynamical systems

Stability of Dynamical systems Stability of Dynamical systems Stability Isolated equilibria Classification of Isolated Equilibria Attractor and Repeller Almost linear systems Jacobian Matrix Stability Consider an autonomous system u

More information

FFTs in Graphics and Vision. Homogenous Polynomials and Irreducible Representations

FFTs in Graphics and Vision. Homogenous Polynomials and Irreducible Representations FFTs in Graphics and Vision Homogenous Polynomials and Irreducible Representations 1 Outline The 2π Term in Assignment 1 Homogenous Polynomials Representations of Functions on the Unit-Circle Sub-Representations

More information

Diagonalization. P. Danziger. u B = A 1. B u S.

Diagonalization. P. Danziger. u B = A 1. B u S. 7., 8., 8.2 Diagonalization P. Danziger Change of Basis Given a basis of R n, B {v,..., v n }, we have seen that the matrix whose columns consist of these vectors can be thought of as a change of basis

More information

Introduction to Applied Nonlinear Dynamical Systems and Chaos

Introduction to Applied Nonlinear Dynamical Systems and Chaos Stephen Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos Second Edition With 250 Figures 4jj Springer I Series Preface v L I Preface to the Second Edition vii Introduction 1 1 Equilibrium

More information

spring, math 204 (mitchell) list of theorems 1 Linear Systems Linear Transformations Matrix Algebra

spring, math 204 (mitchell) list of theorems 1 Linear Systems Linear Transformations Matrix Algebra spring, 2016. math 204 (mitchell) list of theorems 1 Linear Systems THEOREM 1.0.1 (Theorem 1.1). Uniqueness of Reduced Row-Echelon Form THEOREM 1.0.2 (Theorem 1.2). Existence and Uniqueness Theorem THEOREM

More information

MATH 369 Linear Algebra

MATH 369 Linear Algebra Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine

More information

Local properties of plane algebraic curves

Local properties of plane algebraic curves Chapter 7 Local properties of plane algebraic curves Throughout this chapter let K be an algebraically closed field of characteristic zero, and as usual let A (K) be embedded into P (K) by identifying

More information

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 222 3. M Test # July, 23 Solutions. For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For

More information

Lecture 3 : Bifurcation Analysis

Lecture 3 : Bifurcation Analysis Lecture 3 : Bifurcation Analysis D. Sumpter & S.C. Nicolis October - December 2008 D. Sumpter & S.C. Nicolis General settings 4 basic bifurcations (as long as there is only one unstable mode!) steady state

More information

Permutations and Polynomials Sarah Kitchen February 7, 2006

Permutations and Polynomials Sarah Kitchen February 7, 2006 Permutations and Polynomials Sarah Kitchen February 7, 2006 Suppose you are given the equations x + y + z = a and 1 x + 1 y + 1 z = 1 a, and are asked to prove that one of x,y, and z is equal to a. We

More information

Exercise The solution of

Exercise The solution of Exercise 8.51 The solution of dx dy = sin φ cos θdr + r cos φ cos θdφ r sin φ sin θdθ sin φ sin θdr + r cos φ sin θdφ r sin φ cos θdθ is dr dφ = dz cos φdr r sin φdφ dθ sin φ cos θdx + sin φ sin θdy +

More information

HW3 - Due 02/06. Each answer must be mathematically justified. Don t forget your name. 1 2, A = 2 2

HW3 - Due 02/06. Each answer must be mathematically justified. Don t forget your name. 1 2, A = 2 2 HW3 - Due 02/06 Each answer must be mathematically justified Don t forget your name Problem 1 Find a 2 2 matrix B such that B 3 = A, where A = 2 2 If A was diagonal, it would be easy: we would just take

More information

Mathematical Modeling I

Mathematical Modeling I Mathematical Modeling I Dr. Zachariah Sinkala Department of Mathematical Sciences Middle Tennessee State University Murfreesboro Tennessee 37132, USA November 5, 2011 1d systems To understand more complex

More information

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»

More information

PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.

PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them. Prof A Suciu MTH U37 LINEAR ALGEBRA Spring 2005 PRACTICE FINAL EXAM Are the following vectors independent or dependent? If they are independent, say why If they are dependent, exhibit a linear dependence

More information

Dr. Allen Back. Oct. 6, 2014

Dr. Allen Back. Oct. 6, 2014 Dr. Allen Back Oct. 6, 2014 Distribution Min=48 Max=100 Q1=74 Median=83 Q3=91.5 mean = 81.48 std. dev. = 12.5 Distribution Distribution: (The letters will not be directly used.) Scores Freq. 100 2 98-99

More information

1. < 0: the eigenvalues are real and have opposite signs; the fixed point is a saddle point

1. < 0: the eigenvalues are real and have opposite signs; the fixed point is a saddle point Solving a Linear System τ = trace(a) = a + d = λ 1 + λ 2 λ 1,2 = τ± = det(a) = ad bc = λ 1 λ 2 Classification of Fixed Points τ 2 4 1. < 0: the eigenvalues are real and have opposite signs; the fixed point

More information

Computational Methods. Eigenvalues and Singular Values

Computational Methods. Eigenvalues and Singular Values Computational Methods Eigenvalues and Singular Values Manfred Huber 2010 1 Eigenvalues and Singular Values Eigenvalues and singular values describe important aspects of transformations and of data relations

More information

FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS III: Autonomous Planar Systems David Levermore Department of Mathematics University of Maryland

FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS III: Autonomous Planar Systems David Levermore Department of Mathematics University of Maryland FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS III: Autonomous Planar Systems David Levermore Department of Mathematics University of Maryland 4 May 2012 Because the presentation of this material

More information

Problem List MATH 5173 Spring, 2014

Problem List MATH 5173 Spring, 2014 Problem List MATH 5173 Spring, 2014 The notation p/n means the problem with number n on page p of Perko. 1. 5/3 [Due Wednesday, January 15] 2. 6/5 and describe the relationship of the phase portraits [Due

More information

MA 527 first midterm review problems Hopefully final version as of October 2nd

MA 527 first midterm review problems Hopefully final version as of October 2nd MA 57 first midterm review problems Hopefully final version as of October nd The first midterm will be on Wednesday, October 4th, from 8 to 9 pm, in MTHW 0. It will cover all the material from the classes

More information

Linearization and Invariant Manifolds

Linearization and Invariant Manifolds CHAPTER 4 Linearization and Invariant Manifolds Coordinate transformations are often used to transform a dynamical system that is analytically intractable with respect to its native coordinates, into a

More information

Dynamics and Bifurcations in Predator-Prey Models with Refuge, Dispersal and Threshold Harvesting

Dynamics and Bifurcations in Predator-Prey Models with Refuge, Dispersal and Threshold Harvesting Dynamics and Bifurcations in Predator-Prey Models with Refuge, Dispersal and Threshold Harvesting August 2012 Overview Last Model ẋ = αx(1 x) a(1 m)xy 1+c(1 m)x H(x) ẏ = dy + b(1 m)xy 1+c(1 m)x (1) where

More information

2 Lecture 2: Amplitude equations and Hopf bifurcations

2 Lecture 2: Amplitude equations and Hopf bifurcations Lecture : Amplitude equations and Hopf bifurcations This lecture completes the brief discussion of steady-state bifurcations by discussing vector fields that describe the dynamics near a bifurcation. From

More information

4 Film Extension of the Dynamics: Slowness as Stability

4 Film Extension of the Dynamics: Slowness as Stability 4 Film Extension of the Dynamics: Slowness as Stability 4.1 Equation for the Film Motion One of the difficulties in the problem of reducing the description is caused by the fact that there exists no commonly

More information

Chapter 7. Extremal Problems. 7.1 Extrema and Local Extrema

Chapter 7. Extremal Problems. 7.1 Extrema and Local Extrema Chapter 7 Extremal Problems No matter in theoretical context or in applications many problems can be formulated as problems of finding the maximum or minimum of a function. Whenever this is the case, advanced

More information

Long-wave Instability in Anisotropic Double-Diffusion

Long-wave Instability in Anisotropic Double-Diffusion Long-wave Instability in Anisotropic Double-Diffusion Jean-Luc Thiffeault Institute for Fusion Studies and Department of Physics University of Texas at Austin and Neil J. Balmforth Department of Theoretical

More information

Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero. Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial

More information

= F ( x; µ) (1) where x is a 2-dimensional vector, µ is a parameter, and F :

= F ( x; µ) (1) where x is a 2-dimensional vector, µ is a parameter, and F : 1 Bifurcations Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 A bifurcation is a qualitative change

More information

Plan What is a normal formal of a mathematical object? Similarity and Congruence Transformations Jordan normal form and simultaneous diagonalisation R

Plan What is a normal formal of a mathematical object? Similarity and Congruence Transformations Jordan normal form and simultaneous diagonalisation R Pencils of skew-symmetric matrices and Jordan-Kronecker invariants Alexey Bolsinov, Loughborough University September 25-27, 27 Journée Astronomie et Systèmes Dynamiques MCCE / Observatoire de Paris Plan

More information

BIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs

BIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs BIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs Yuri A. Kuznetsov August, 2010 Contents 1. Solutions and orbits: equilibria cycles connecting orbits compact invariant manifolds strange

More information

Chapter 6 Inner product spaces

Chapter 6 Inner product spaces Chapter 6 Inner product spaces 6.1 Inner products and norms Definition 1 Let V be a vector space over F. An inner product on V is a function, : V V F such that the following conditions hold. x+z,y = x,y

More information

homogeneous 71 hyperplane 10 hyperplane 34 hyperplane 69 identity map 171 identity map 186 identity map 206 identity matrix 110 identity matrix 45

homogeneous 71 hyperplane 10 hyperplane 34 hyperplane 69 identity map 171 identity map 186 identity map 206 identity matrix 110 identity matrix 45 address 12 adjoint matrix 118 alternating 112 alternating 203 angle 159 angle 33 angle 60 area 120 associative 180 augmented matrix 11 axes 5 Axiom of Choice 153 basis 178 basis 210 basis 74 basis test

More information

Andronov Hopf and Bautin bifurcation in a tritrophic food chain model with Holling functional response types IV and II

Andronov Hopf and Bautin bifurcation in a tritrophic food chain model with Holling functional response types IV and II Electronic Journal of Qualitative Theory of Differential Equations 018 No 78 1 7; https://doiorg/10143/ejqtde018178 wwwmathu-szegedhu/ejqtde/ Andronov Hopf and Bautin bifurcation in a tritrophic food chain

More information

LECTURE 5, FRIDAY

LECTURE 5, FRIDAY LECTURE 5, FRIDAY 20.02.04 FRANZ LEMMERMEYER Before we start with the arithmetic of elliptic curves, let us talk a little bit about multiplicities, tangents, and singular points. 1. Tangents How do we

More information

One Dimensional Dynamical Systems

One Dimensional Dynamical Systems 16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with one-dimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar

More information

UNIVERSIDADE DE SÃO PAULO

UNIVERSIDADE DE SÃO PAULO UNIVERSIDADE DE SÃO PAULO Instituto de Ciências Matemáticas e de Computação ISSN 0103-2577 GLOBAL DYNAMICAL ASPECTS OF A GENERALIZED CHEN-WANG DIFFERENTIAL SYSTEM REGILENE D. S. OLIVEIRA CLAUDIA VALLS

More information

MA 265 FINAL EXAM Fall 2012

MA 265 FINAL EXAM Fall 2012 MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators

More information

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #2 Solutions

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #2 Solutions YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 3. M Test # Solutions. (8 pts) For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For this

More information

Unit 3 Factors & Products

Unit 3 Factors & Products 1 Unit 3 Factors & Products General Outcome: Develop algebraic reasoning and number sense. Specific Outcomes: 3.1 Demonstrate an understanding of factors of whole number by determining the: o prime factors

More information