# BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs

Size: px
Start display at page:

## Transcription

1 BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs Yuri A. Kuznetsov August, 2010

2 Contents 1. Solutions and orbits. 2. Equilibria. 3. Periodic orbits and limit cycles. 4. Homoclinic orbits.

3 Literature 1. A.A. Andronov, E.A. Leontovich, I.I. Gordon, and A.G. Maier Qualitative Theory of Second-Order Dynamic Systems, Willey & Sons, London, L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev, and L.O. Chua Methods of Qualitative Theory in Nonlinear Dynamics, Part I, World Scientific, Singapore, F. Dumortier, J. Llibre, and J.C. Artés Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006

4 1. SOLUTIONS AND ORBITS Newton s Second Law: mẍ = F(x, ẋ) { ẋ = y, ẏ = m 1 F(x, y) General planar system: { ẋ = P(x, y), ẏ = Q(x, y) where X = ( x y ) or Ẋ = f(x), X R 2,, f(x) = ( P(x, y) Q(x, y) Theorem 1 If f is smooth than for any inital point there exists y ( ) 0 x(t) a unique locally defined solution t such that x(0) = x y(t) 0 and y(0) = y 0. ). ( x0 )

5 Definition 1 Let I be the maximal definition interval of a solution t X(t), t I. The oriented by the advance of time image X(I) R 2 is called the orbit. f(x) X Vector field: X f(x) f(x) 0 is tangent to the orbit through X orbits do not cross. Definition 2 Phase portrait of a planar system is the collection of all its orbits in R 2. We draw only key orbits, which determine the topology of the phase portrait.

6 Types of orbits: 1. Equilibria: X(t) X 0 so that f(x 0 ) = Periodic orbits (cycles): X(t) X 0, X(t + T) = X(t), t R The minimal T > 0 is called the period of the cycle. 3. Connecting orbits: lim t ± X(t) = X ± with f(x ± ) = 0. If X = X + the connecting orbit is called homoclinic If X X + the connecting orbit is called heteroclinic. 4. All other orbits

7 Theorem 2 (Poincaré-Bendixson) A bounded orbit of a smooth system Ẋ = f(x), X R 2, tends to one of the following sets in the phase plane: (i) an equilibrium point; (ii) a periodic orbit; (iii) a union of equilibria and their connecting orbits.

8 2. EQUILIBRIA: Null-isoclines f(x) = 0 { P(x, y) = 0, Q(x, y) = 0. P = ( ) Px P y and Q = ( ) Qx Q y are orthogonal to P = 0 and Q = 0, resp. Q(x,y) = 0 P Q Jacobian matrix of the equilibrium X 0 : A = f X (X 0 ) = ( Px P y Q x Q y ) x=x0,y=y 0 y 0 P(x, y) = 0 If det A 0 the null-isoclines intersect transversally at X 0. If det A = 0 the null-isoclines are tangent at X 0. x 0

9 Eigenvalues of the equilibrium X 0 are the eigenvalues of A, i.e. the solutions of where λ 2 σλ + = 0, σ = λ 1 + λ 2 = SpA = P x (x 0, y 0 ) + Q y (x 0, y 0 ), = λ 1 λ 2 = det A = P x (x 0, y 0 )Q y (x 0, y 0 ) P y (x 0, y 0 )Q x (x 0, y 0 ). λ 1,2 = σ 2 ± σ 2 4 Definition 3 An equilibrium X 0 is hyperbolic if R(λ) 0. Equilibrium X 0 with λ 1 = 0 (i.e. det A = 0) is called multiple. Equilibrium X 0 with λ 1 + λ 2 = 0 (i.e. Sp A = 0) is called neutral.

10 Phase portraits of planar linear systems Ẏ = AY (n u,n s ) Eigenvalues Phase portrait Stability (0, 2) node stable focus (1, 1) saddle unstable node (2, 0) unstable focus

11 Definition 4 Two systems are called topologically equivalent if their phase portraits are homeomorphic, i.e. there is a continuous invertible transformation that maps orbits of one system onto orbits of the other, preserving their orientation. Theorem 3 (Grobman-Hartman) Consider a smooth nonlinear system Ẋ = AX + F(X), F = O( X 2 ) O(2), and its linearization Ẏ = AY. If R(λ) 0 for all eigenvalues of A, then these systems are locally topologically equivalent near the origin. Warning: A stable/unstable node is locally topologically equivalent to a stable/unstable focus.

12 Trivial topological equivalences 1. Orbital equivalence: Ẋ = f(x) Ẏ = g(y )f(y ) where g : R 2 R is smooth positive function; Y = h(x) = X preserves orbits. 2. Smooth equivalence: Ẋ = f(x) Ẏ = h X (h 1 (Y ))f(h 1 (Y )), where h : R 2 R 2 is a smooth diffeomorphism; the substitution Y = h(x) transforms solutions onto solutions: Ẏ = h X (X)Ẋ = h X (X)f(X) where X = h 1 (Y ). 3. Smooth orbital equivalence:

13 Simplest critical cases λ 1 = 0, λ 2 0 By a linear diffeomorphism, Ẋ = f(x) can be transformed into { ẋ = ax 2 + bxy + cy 2 + O(3), ẏ = λ 2 y + O(2). If a 0 then Ẋ = f(x) is locally topologically equivalent near the origin to { ẋ = ax 2, Saddle-node (a > 0): ẏ = λ 2 y. λ 2 < 0 λ 2 > 0

14 λ 1,2 = ±iω, ω > 0 By a linear diffeomorphism, Ẋ = f(x) can be transformed into { ẋ = ωy + R(x, y), R = O(2), ẏ = ωx + S(x, y), S = O(2). Introduce z = x + iy C. Then this system becomes ż = iωz + g(z, z), where ( z + z g(z, z) = R 2, z z 2i Write its Taylor expansion in z, z: ) + is ( z + z 2, z z ). 2i g(z, z) = 1 2 g 20z 2 + g 11 z z g 02 z g 21z 2 z +... Definition 5 The first Lyapunov coefficient is l 1 = 1 2ω 2R(ig 20g 11 + ωg 21 ).

15 If l 1 0 then Ẋ = f(x) is locally topologically equivalent near the origin to { ρ = l1 ρ 3, ϕ = 1, where (ρ, ϕ) are polar coordinates: z = ρe iϕ. Weak focus: stable unstable l 1 < 0 l 1 > 0

16 3. PERIODIC ORBITS AND LIMIT CYCLES Poincaré map: ξ P(ξ) = µξ + O(2), ξ P(ξ) 0 where the multiplier µ = exp ( T 0 (div f)(x0 (t)) dt ) > 0 X 0 (t) Definition 6 A cycle of the planar system is hyperbolic if µ 1. The cycle is stable if µ < 1 and is unstable if µ > 1. ξ µ > 1 ξ = ξ µ < 1 ξ

17 Theorem 4 (Bendixson-Dulac) If (div f)(x) > 0 (< 0) in a disc D R 2, then Ẋ = f(x) has no periodic orbits in D. Proof: Suppose, there is a cycle C D and let Ω be a bounded domain with Ω = C. C Pdy Qdx = f, dx 0 C but C Pdy Qdx = (div f)dx > 0 Ω (or < 0), contradiction. f = ( P Q X dx ) f = ( C Q P )

18 Implications: 1. If div(gf) > 0 (< 0) in a disc D R 2 for a smooth positive function g : R 2 R, then Ẋ = f(x) has no periodic orbits in D. 2. If div(gf) > 0 (< 0) is an annulus A R 2 for a smooth positive function g : R 2 R, then Ẋ = f(x) has at most one periodic orbit in A. 3. If f(x) 0 and div(gf) < 0 in a trapping annulus A R 2 for a smooth positive function g : R 2 R, then Ẋ = f(x) has a unique stable periodic orbit in A.

19 Example: Consider { ẋ = y P(x, y), ẏ = ax + by + αx 2 + βy 2 Q(x, y). Define in R 2. Then g(x, y) = e 2βx > 0 x (gp) + y (gp) = x (e 2βx y) + y (e 2βx (ax + by + αx 2 + βy 2 )) = 2βe 2βx y + be 2βx + 2βe 2βx y = be 2βx 0 in R 2 if b 0. no periodic orbits.

20 Reversible systems Definition 7 A smooth system Ẋ = f(x) is called reversible if f(jx) = Jf(X) for a matrix J such that J 2 = E. The transformation X JX is called an involution. If there is an orbit segment without equilibria connecting two points in the fixed subspace {Y : JY = Y } of the involution, there is a periodic orbit of Ẋ = f(x). Periodic orbits occur in continuos familes. Example: 4 { ẋ = y, ẏ = x + xy x 3, J ( x y ) = ( x y ) y x

21 Example: A prey-predator model ξη ξ = ξ (1 + αξ)(1 + βη) f 1(ξ, η), ξη η = η + (1 + αξ)(1 + βη) f 2(ξ, η), where α, β > 0 and x, y 0. There is a family of closed orbits for α = β if 0 < α = β < 1 4. since the system is reversible with involution J : (ξ, η) (η, ξ). There are no closed orbits if α β, since the choice g(ξ, η) = ξ a η b (1 + αξ)(1 + βη), with appropriate a and b implies div(gf) = (α β)ξg.

22 Planar Hamiltonian systems: H : R 2 R (smooth) { ẋ = Hy (x, y), Ḣ = H ẏ = H x (x, y). x ẋ + H y ẏ 0 H(x(t), y(t)) = h Potential system: H(x, y) = y2 2 + U(x) (reversible: y y, t t). where T = ds dh h=h0 (1) (2) (3) y h H(x,y) = h (3) (3) (2) S(h) = area inside y2 + U(x) = h (1) 2 The Lotka-Volterra prey-predator model is orbitally equivalent to a Hamiltonian system. U(x) x

23 Dissipative perturbations of 2D Hamiltonian systems { ( ẋ = Hy (x, y) + εp(x, y), P(x, y) F(x, y) = ẏ = H x (x, y) + εq(x, y), Q(x, y) ). Let X 0 (t) correspond to the T 0 -periodic orbit C 0 at ε = 0 and let Ω 0 = domain bounded by C 0. X Ω C 0 Γ 0 Theorem 5 (Pontryagin-Melnikov) If Ω 0 div F(X) dx = 0 but T0 0 div F(X0 (t)) dt 0 then there exists an annulus contaning C 0 in which the system has a unique periodic orbit C ε for all sufficiently small ε, such that C ε C 0 as ε 0.

24 Example: Van der Pol equation ẍ + x = εẋ(1 x 2 ) { ẋ = y, ẏ = x + εy(1 x 2 ), For ε = 0, H(x, y) = 1 2 (x2 + y 2 ) and X 0 (t) = with T 0 = 2π. Then C 0 div F dxdy = and T0 ( rsin t rcos t F(x, y) = ), C 0 = {(x, y) : x 2 + y 2 = r 2, r > 0} ( P(x, y) Q(x, y) C 0 Pdy Qdx = ) 2π 0 2π = ( 0 y(1 x 2 ) ). r 2 cos 2 t(1 r 2 sin 2 t)dt = π 4 r2 (4 r 2 ) div 0 F(X0 (t)) dt = (1 0 4sin2 t)dt = 2π A cycle close to r = 2 exists for small ε 0.

25 4. HOMOCLINIC ORBITS Homoclinic orbits to saddles: small Γ 0 big Γ 0 X 0 X 0 Definition 8 The real number σ = λ 1 + λ 2 = (div f)(x 0 ) is called the saddle quantity of X 0. Γ 0 Γ 0 X 0 X 0 σ < 0 σ > 0

26 Singular map: { ẋ = λ1 x ẏ = λ 2 y y Q Regular map: ξ = (η) = η λ 1 λ2 η = Q(ξ) = Aξ + O(2), A > 0. 1 ξ η η 0 1 x Poincaré map: η η = Q( (η)) = Aη λ 1 λ The homoclinic orbit is stable if σ < 0 and is unstable if σ > 0.

27 If σ = λ 1 + λ 2 = 0, then if if (div f)(x0 (t)) dt < 0 the homoclinic orbit is stable; (div f)(x0 (t)) dt > 0 the homoclinic orbit is unstable. Homoclinic orbits to saddle-nodes: X 0 X 0 Γ 0 Γ 0 codim 1 codim 2

### 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

### 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

### NBA Lecture 1. Simplest bifurcations in n-dimensional ODEs. Yu.A. Kuznetsov (Utrecht University, NL) March 14, 2011

NBA Lecture 1 Simplest bifurcations in n-dimensional ODEs Yu.A. Kuznetsov (Utrecht University, NL) March 14, 2011 Contents 1. Solutions and orbits: equilibria cycles connecting orbits other invariant sets

### EE222 - Spring 16 - Lecture 2 Notes 1

EE222 - Spring 16 - Lecture 2 Notes 1 Murat Arcak January 21 2016 1 Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Essentially Nonlinear Phenomena Continued

### STABILITY. Phase portraits and local stability

MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),

### Introduction to Bifurcation and Normal Form theories

Introduction to Bifurcation and Normal Form theories Romain Veltz / Olivier Faugeras October 9th 2013 ENS - Master MVA / Paris 6 - Master Maths-Bio (2013-2014) Outline 1 Invariant sets Limit cycles Stable/Unstable

### 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

### 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

### CHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD

CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: August 22, 2018, at 08 30 12 30 Johanneberg Jan Meibohm,

### 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

### Lecture 5. Numerical continuation of connecting orbits of iterated maps and ODEs. Yu.A. Kuznetsov (Utrecht University, NL)

Lecture 5 Numerical continuation of connecting orbits of iterated maps and ODEs Yu.A. Kuznetsov (Utrecht University, NL) May 26, 2009 1 Contents 1. Point-to-point connections. 2. Continuation of homoclinic

### 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

### Stability Implications of Bendixson s Criterion

Wilfrid Laurier University Scholars Commons @ Laurier Mathematics Faculty Publications Mathematics 1998 Stability Implications of Bendixson s Criterion C. Connell McCluskey Wilfrid Laurier University,

### WIDELY SEPARATED FREQUENCIES IN COUPLED OSCILLATORS WITH ENERGY-PRESERVING QUADRATIC NONLINEARITY

WIDELY SEPARATED FREQUENCIES IN COUPLED OSCILLATORS WITH ENERGY-PRESERVING QUADRATIC NONLINEARITY J.M. TUWANKOTTA Abstract. In this paper we present an analysis of a system of coupled oscillators suggested

### 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

### 5.2.2 Planar Andronov-Hopf bifurcation

138 CHAPTER 5. LOCAL BIFURCATION THEORY 5.. Planar Andronov-Hopf bifurcation What happens if a planar system has an equilibrium x = x 0 at some parameter value α = α 0 with eigenvalues λ 1, = ±iω 0, ω

### 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

### 7 Planar systems of linear ODE

7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution

### 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

### 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

### Math 4200, Problem set 3

Math, Problem set 3 Solutions September, 13 Problem 1. ẍ = ω x. Solution. Following the general theory of conservative systems with one degree of freedom let us define the kinetic energy T and potential

### Practice Problems for Final Exam

Math 1280 Spring 2016 Practice Problems for Final Exam Part 2 (Sections 6.6, 6.7, 6.8, and chapter 7) S o l u t i o n s 1. Show that the given system has a nonlinear center at the origin. ẋ = 9y 5y 5,

### Problem Set Number 5, j/2.036j MIT (Fall 2014)

Problem Set Number 5, 18.385j/2.036j MIT (Fall 2014) Rodolfo R. Rosales (MIT, Math. Dept.,Cambridge, MA 02139) Due Fri., October 24, 2014. October 17, 2014 1 Large µ limit for Liénard system #03 Statement:

### 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

### 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

### Analysis of Dynamical Systems

2 YFX1520 Nonlinear phase portrait Mathematical Modelling and Nonlinear Dynamics Coursework Assignments 1 ϕ (t) 0-1 -2-6 -4-2 0 2 4 6 ϕ(t) Dmitri Kartofelev, PhD 2018 Variant 1 Part 1: Liénard type equation

### 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

### Nonlinear Control Lecture 2:Phase Plane Analysis

Nonlinear Control Lecture 2:Phase Plane Analysis Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 r. Farzaneh Abdollahi Nonlinear Control Lecture 2 1/53

### Analysis of the Takens-Bogdanov bifurcation on m parameterized vector fields

Analysis of the Takens-Bogdanov bifurcation on m parameterized vector fields Francisco Armando Carrillo Navarro, Fernando Verduzco G., Joaquín Delgado F. Programa de Doctorado en Ciencias (Matemáticas),

### 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

### 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

### Nonlinear Control Lecture 2:Phase Plane Analysis

Nonlinear Control Lecture 2:Phase Plane Analysis Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2009 Farzaneh Abdollahi Nonlinear Control Lecture 2 1/68

### Lectures on Dynamical Systems. Anatoly Neishtadt

Lectures on Dynamical Systems Anatoly Neishtadt Lectures for Mathematics Access Grid Instruction and Collaboration (MAGIC) consortium, Loughborough University, 2007 Part 3 LECTURE 14 NORMAL FORMS Resonances

### Now I switch to nonlinear systems. In this chapter the main object of study will be

Chapter 4 Stability 4.1 Autonomous systems Now I switch to nonlinear systems. In this chapter the main object of study will be ẋ = f(x), x(t) X R k, f : X R k, (4.1) where f is supposed to be locally Lipschitz

### 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

### LIMIT CYCLES FOR A CLASS OF ELEVENTH Z 12 -EQUIVARIANT SYSTEMS WITHOUT INFINITE CRITICAL POINTS

LIMIT CYCLES FOR A CLASS OF ELEVENTH Z 1 -EQUIVARIANT SYSTEMS WITHOUT INFINITE CRITICAL POINTS ADRIAN C. MURZA Communicated by Vasile Brînzănescu We analyze the complex dynamics of a family of Z 1-equivariant

### Perturbation Theory of Dynamical Systems

Perturbation Theory of Dynamical Systems arxiv:math/0111178v1 [math.ho] 15 Nov 2001 Nils Berglund Department of Mathematics ETH Zürich 8092 Zürich Switzerland Lecture Notes Summer Semester 2001 Version:

### Logistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations

Logistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations S. Y. Ha and J. Park Department of Mathematical Sciences Seoul National University Sep 23, 2013 Contents 1 Logistic Map 2 Euler and

Faculty of Electrical Engineering, Mathematics & Computer Science Homoclinic saddle to saddle-focus transitions in 4D systems Manu Kalia M.Sc. Thesis July 2017 Assessment committee: Prof. Dr. S. A. van

### 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

### Math 273 (51) - Final

Name: Id #: Math 273 (5) - Final Autumn Quarter 26 Thursday, December 8, 26-6: to 8: Instructions: Prob. Points Score possible 25 2 25 3 25 TOTAL 75 Read each problem carefully. Write legibly. Show all

### Chapter 4. Planar ODE. 4.1 Limit sets

Chapter 4 Planar ODE In this chapter we prove the famous Poincaré-Bendixson Theorem that classifies the asymptotic behaviour of bounded orbits in smooth planar ODEs. We shall see that in a generic planar

### DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS

WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Series Editor: Leon O. Chua Series A Vol. 66 DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS Jean-Marc Ginoux Université du Sud, France World Scientific

### Continuous Threshold Policy Harvesting in Predator-Prey Models

Continuous Threshold Policy Harvesting in Predator-Prey Models Jon Bohn and Kaitlin Speer Department of Mathematics, University of Wisconsin - Madison Department of Mathematics, Baylor University July

### Hilbert s 16. problem

Hilbert s 16. problem Aarhus University Aarhus 24 th March, 2017 Hilbert s Problem Does there exist (minimal) natural numbers H n, for n = 2, 3,..., such that any planar autonomous system of ODE s dx dt

### Math 232, Final Test, 20 March 2007

Math 232, Final Test, 20 March 2007 Name: Instructions. Do any five of the first six questions, and any five of the last six questions. Please do your best, and show all appropriate details in your solutions.

### MCE693/793: Analysis and Control of Nonlinear Systems

MCE693/793: Analysis and Control of Nonlinear Systems Systems of Differential Equations Phase Plane Analysis Hanz Richter Mechanical Engineering Department Cleveland State University Systems of Nonlinear

### 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

### 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:

### ẋ = f(x, y), ẏ = g(x, y), (x, y) D, can only have periodic solutions if (f,g) changes sign in D or if (f,g)=0in D.

4 Periodic Solutions We have shown that in the case of an autonomous equation the periodic solutions correspond with closed orbits in phase-space. Autonomous two-dimensional systems with phase-space R

### Shilnikov bifurcations in the Hopf-zero singularity

Shilnikov bifurcations in the Hopf-zero singularity Geometry and Dynamics in interaction Inma Baldomá, Oriol Castejón, Santiago Ibáñez, Tere M-Seara Observatoire de Paris, 15-17 December 2017, Paris Tere

### Mathematical Foundations of Neuroscience - Lecture 7. Bifurcations II.

Mathematical Foundations of Neuroscience - Lecture 7. Bifurcations II. Filip Piękniewski Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland Winter 2009/2010 Filip

### DYNAMICAL SYSTEMS WITH A CODIMENSION-ONE INVARIANT MANIFOLD: THE UNFOLDINGS AND ITS BIFURCATIONS

International Journal of Bifurcation and Chaos c World Scientific Publishing Company DYNAMICAL SYSTEMS WITH A CODIMENSION-ONE INVARIANT MANIFOLD: THE UNFOLDINGS AND ITS BIFURCATIONS KIE VAN IVANKY SAPUTRA

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

### DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS

DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS Morris W. Hirsch University of California, Berkeley Stephen Smale University of California, Berkeley Robert L. Devaney Boston University

### CHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD

CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: January 14, 2019, at 08 30 12 30 Johanneberg Kristian

### 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

### CLASSIFICATIONS OF THE FLOWS OF LINEAR ODE

CLASSIFICATIONS OF THE FLOWS OF LINEAR ODE PETER ROBICHEAUX Abstract. The goal of this paper is to examine characterizations of linear differential equations. We define the flow of an equation and examine

### PLANAR ANALYTIC NILPOTENT GERMS VIA ANALYTIC FIRST INTEGRALS

PLANAR ANALYTIC NILPOTENT GERMS VIA ANALYTIC FIRST INTEGRALS YINGFEI YI AND XIANG ZHANG Abstract. We generalize the results of [6] by giving necessary and sufficient conditions for the planar analytic

### A BENDIXSON DULAC THEOREM FOR SOME PIECEWISE SYSTEMS

A BENDIXSON DULAC THEOREM FOR SOME PIECEWISE SYSTEMS LEONARDO P. C. DA CRUZ AND JOAN TORREGROSA Abstract. The Bendixson Dulac Theorem provides a criterion to find upper bounds for the number of limit cycles

### 2D-Volterra-Lotka Modeling For 2 Species

Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya 2D-Volterra-Lotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose

### In essence, Dynamical Systems is a science which studies differential equations. A differential equation here is the equation

Lecture I In essence, Dynamical Systems is a science which studies differential equations. A differential equation here is the equation ẋ(t) = f(x(t), t) where f is a given function, and x(t) is an unknown

### CDS 101/110a: Lecture 2.1 Dynamic Behavior

CDS 11/11a: Lecture 2.1 Dynamic Behavior Richard M. Murray 6 October 28 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium

### On the fixed points set of differential systems reversibilities arxiv: v1 [math.ds] 6 Oct 2015

On the fixed points set of differential systems reversibilities arxiv:1510.01464v1 [math.ds] 6 Oct 2015 Marco Sabatini October 5, 2015 Abstract We extend a result proved in [7] for mirror symmetries of

### 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

### Stabilization and Passivity-Based Control

DISC Systems and Control Theory of Nonlinear Systems, 2010 1 Stabilization and Passivity-Based Control Lecture 8 Nonlinear Dynamical Control Systems, Chapter 10, plus handout from R. Sepulchre, Constructive

### Dynamical Systems: Ecological Modeling

Dynamical Systems: Ecological Modeling G Söderbacka Abstract Ecological modeling is becoming increasingly more important for modern engineers. The mathematical language of dynamical systems has been applied

### Connecting Orbits with Bifurcating End Points

Connecting Orbits with Bifurcating End Points Thorsten Hüls Fakultät für Mathematik Universität Bielefeld Postfach 100131, 33501 Bielefeld Germany huels@mathematik.uni-bielefeld.de June 24, 2004 Abstract

### UNIFORM SUBHARMONIC ORBITS FOR SITNIKOV PROBLEM

Manuscript submitted to Website: http://aimsciences.org AIMS Journals Volume 00, Number 0, Xxxx XXXX pp. 000 000 UNIFORM SUBHARMONIC ORBITS FOR SITNIKOV PROBLEM CLARK ROBINSON Abstract. We highlight the

### A Chaotic Phenomenon in the Power Swing Equation Umesh G. Vaidya R. N. Banavar y N. M. Singh March 22, 2000 Abstract Existence of chaotic dynamics in

A Chaotic Phenomenon in the Power Swing Equation Umesh G. Vaidya R. N. Banavar y N. M. Singh March, Abstract Existence of chaotic dynamics in the classical swing equations of a power system of three interconnected

### 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

### Qualitative Classification of Singular Points

QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 6, 87 167 (2005) ARTICLE NO. 94 Qualitative Classification of Singular Points Qibao Jiang Department of Mathematics and Mechanics, Southeast University, Nanjing

### CDS 101/110a: Lecture 2.1 Dynamic Behavior

CDS 11/11a: Lecture.1 Dynamic Behavior Richard M. Murray 6 October 8 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium

### Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games

Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Alberto Bressan ) and Khai T. Nguyen ) *) Department of Mathematics, Penn State University **) Department of Mathematics,

### Polynomial vector fields in R 3 having a quadric of revolution as an invariant algebraic surface

Polynomial vector fields in R 3 having a quadric of revolution as an invariant algebraic surface Luis Fernando Mello Universidade Federal de Itajubá UNIFEI E-mail: lfmelo@unifei.edu.br Texas Christian

### Outline. Learning Objectives. References. Lecture 2: Second-order Systems

Outline Lecture 2: Second-order Systems! Techniques based on linear systems analysis! Phase-plane analysis! Example: Neanderthal / Early man competition! Hartman-Grobman theorem -- validity of linearizations!

### Constructing a chaotic system with any number of equilibria

Nonlinear Dyn (2013) 71:429 436 DOI 10.1007/s11071-012-0669-7 ORIGINAL PAPER Constructing a chaotic system with any number of equilibria Xiong Wang Guanrong Chen Received: 9 June 2012 / Accepted: 29 October

### Lectures on Periodic Orbits

Lectures on Periodic Orbits 11 February 2009 Most of the contents of these notes can found in any typical text on dynamical systems, most notably Strogatz [1994], Perko [2001] and Verhulst [1996]. Complete

### 10 Back to planar nonlinear systems

10 Back to planar nonlinear sstems 10.1 Near the equilibria Recall that I started talking about the Lotka Volterra model as a motivation to stud sstems of two first order autonomous equations of the form

### Qualitative Analysis of Dynamical Systems and Models in Life Science

Ecole Polytechnique UPSay 2015-16 Master de Mécanique M2 Biomécanique Qualitative Analysis of Dynamical Systems and Models in Life Science Alexandre VIDAL Dernière modification : February 5, 2016 2 Contents

### 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.

### THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS

THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS GUILLAUME LAJOIE Contents 1. Introduction 2 2. The Hartman-Grobman Theorem 2 2.1. Preliminaries 2 2.2. The discrete-time Case 4 2.3. The

### MULTIPLE FOCUS AND HOPF BIFURCATIONS IN A PREDATOR-PREY SYSTEM WITH NONMONOTONIC FUNCTIONAL RESPONSE

SIAM J. APPL. MATH. Vol. 66, No. 3, pp. 802 89 c 2006 Society for Industrial and Applied Mathematics MULTIPLE FOCUS AND HOPF BIFURCATIONS IN A PREDATOR-PREY SYSTEM WITH NONMONOTONIC FUNCTIONAL RESPONSE

### An Undergraduate s Guide to the Hartman-Grobman and Poincaré-Bendixon Theorems

An Undergraduate s Guide to the Hartman-Grobman and Poincaré-Bendixon Theorems Scott Zimmerman MATH181HM: Dynamical Systems Spring 2008 1 Introduction The Hartman-Grobman and Poincaré-Bendixon Theorems

### E209A: Analysis and Control of Nonlinear Systems Problem Set 3 Solutions

E09A: Analysis and Control of Nonlinear Systems Problem Set 3 Solutions Michael Vitus Stanford University Winter 007 Problem : Planar phase portraits. Part a Figure : Problem a This phase portrait is correct.

### 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:

### CHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD

CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: January 08, 2018, at 08 30 12 30 Johanneberg Kristian

### Alligood, K. T., Sauer, T. D. & Yorke, J. A. [1997], Chaos: An Introduction to Dynamical Systems, Springer-Verlag, New York.

Bibliography Alligood, K. T., Sauer, T. D. & Yorke, J. A. [1997], Chaos: An Introduction to Dynamical Systems, Springer-Verlag, New York. Amann, H. [1990], Ordinary Differential Equations: An Introduction

### Multiple focus and Hopf bifurcations in a predator prey system with nonmonotonic functional response

Multiple focus and Hopf bifurcations in a predator prey system with nonmonotonic functional response Dongmei Xiao Department of Mathematics Shanghai Jiao Tong University, Shanghai 200030, China Email:

### ONE-PARAMETER BIFURCATIONS IN PLANAR FILIPPOV SYSTEMS

International Journal of Bifurcation and Chaos, Vol. 3, No. 8 (23) 257 288 c World Scientific Publishing Company ONE-PARAMETER BIFURCATIONS IN PLANAR FILIPPOV SYSTEMS YU. A. KUZNETSOV, S. RINALDI and A.

### On low speed travelling waves of the Kuramoto-Sivashinsky equation.

On low speed travelling waves of the Kuramoto-Sivashinsky equation. Jeroen S.W. Lamb Joint with Jürgen Knobloch (Ilmenau, Germany) Marco-Antonio Teixeira (Campinas, Brazil) Kevin Webster (Imperial College

### Poincaré Map, Floquet Theory, and Stability of Periodic Orbits

Poincaré Map, Floquet Theory, and Stability of Periodic Orbits CDS140A Lecturer: W.S. Koon Fall, 2006 1 Poincaré Maps Definition (Poincaré Map): Consider ẋ = f(x) with periodic solution x(t). Construct

### Complicated behavior of dynamical systems. Mathematical methods and computer experiments.

Complicated behavior of dynamical systems. Mathematical methods and computer experiments. Kuznetsov N.V. 1, Leonov G.A. 1, and Seledzhi S.M. 1 St.Petersburg State University Universitetsky pr. 28 198504

### An extended complete Chebyshev system of 3 Abelian integrals related to a non-algebraic Hamiltonian system

Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 6, No. 4, 2018, pp. 438-447 An extended complete Chebyshev system of 3 Abelian integrals related to a non-algebraic Hamiltonian

### 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

### Nonlinear dynamics & chaos BECS

Nonlinear dynamics & chaos BECS-114.7151 Phase portraits Focus: nonlinear systems in two dimensions General form of a vector field on the phase plane: Vector notation: Phase portraits Solution x(t) describes

### 14 Periodic phenomena in nature and limit cycles

14 Periodic phenomena in nature and limit cycles 14.1 Periodic phenomena in nature As it was discussed while I talked about the Lotka Volterra model, a great deal of natural phenomena show periodic behavior.

### 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