10 Back to planar nonlinear systems

Size: px
Start display at page:

Download "10 Back to planar nonlinear systems"

Transcription

1 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 ẋ = f(), U R 2, f : U R 2. (1) After this we discussed general properties of sstems of the form (1) and formulated the main goal: Given the sstem how can we obtain its phase portrait? This task can be full solved for linear sstems of the form ẋ = A, R 2, A = [a ij ] 2 2. (2) In this lecture I will show how the knowledge of the phase portraits of (2) can be used to obtain partial and essentiall local information about phase portraits of (1). The general idea is quite straightforward. Assume that sstem (1) has an equilibrium ˆ (i.e., f(ˆ) = 0) and epand f in its Talor series around this point: where f() = f(ˆ)( ˆ) + Df(ˆ)( ˆ) + O ( ˆ 2) = Df(ˆ)( ˆ) + O ( ˆ 2), f 1 f 1 Df(ˆ) = 1 f 2 2 f ( 1, 2 )=(ˆ 1,ˆ 2 ) is called the Jacobi (or Jacobian) matri of f evaluated at the point ˆ. There is a temptation to drop the terms O ( ˆ 2) and make the shift of the variables = ˆ. Then for we obtain a linear sstem ẏ = Df(ˆ) = A, A = Df(ˆ), (3) whose phase portrait we know. Sstem (3) is called the linearization of (1) around (or at) the equilibrium ˆ. The answer when we can use the linearization to figure out the behavior near ˆ is given b the following theorem. Theorem 1 (Grobman Hartman). Assume that the origin of the linearization of (1) at ˆ is hperbolic. Then sstem (1) is locall topologicall equivalent its linearization (3). Note that the theorem holds onl in some neighborhood of ˆ. To etend the terminolog from the linear sstem, I call the equilibrium ˆ hperbolic, if the Jacobi matri Df evaluated at this equilibrium has no eigenvalues with zero real part. In particular, denoting d, d +, d 0 the number of eigenvalues with negative, positive, and zero real parts, we have Theorem 2 (Lapunov, Poincaré). If ˆ hperbolic, then it is asmptoticall stable if d = d. d + > 0 for an ˆ (including non-hperbolic equilibria) then ˆ is unstable. If Math 484/684: Mathematical modeling of biological processes b Artem Novozhilov artem.novozhilov@ndsu.edu. Spring

2 If the equilibrium is non-hperbolic in the linear sstem, for the analsis of the orbit structure of the nonlinear equation we need to use some additional analsis. This is in particular true when the linearization is a center. For eample, for the Lotka Volterra model the nontrivial equilibrium is a center in the linearized sstem (check this), and how we argued, is also a center in the nonlinear sstem. However, according to the general theor, we could not conclude that the equilibrium is a center because the linearization is a center! Therefore, for hperbolic equilibria of (1) eactl the same classification of the phase portraits in some neighborhood of ˆ holds, as for linear sstems (see however homework problems). If the point is hperbolic, then we will see familiar nodes, spirals, and saddles. If we deal with saddles, even more can be said. Let me define first the stable and unstable manifolds of ˆ. B definition, the stable manifold of ˆ is defined as W (ˆ) := { 0 U : (t; 0 ) ˆ as t }, and the unstable manifold of ˆ is defined as W + (ˆ) := { 0 U : (t; 0 ) ˆ as t }. Theorem 3. Assume that ˆ is a hperbolic and denote T, T + the stable and unstable subspaces of its linearization. If T then W (ˆ) eists and is tangent to T at ˆ, and if T + then W + (ˆ) eists and is tangent to T + at ˆ. The last theorem means that the saddle structure of the equilibrium of the linearization is preserved in the nonlinear sstem, i.e., there are orbits that approach the equilibrium for t, and there are orbits that approach the equilibrium for t. To illustrate the last theorem consider a simple eample ẋ =, ẏ = + 2. This sstem has a unique equilibrium ˆ = (0, 0), and the linearization at this point is ẋ =. 0 1 (I use here and in the following, with a slight abuse of notations, the same variable to denote the linearization of the sstem). The linearization is a saddle point with eigenvalues λ 1 = 1 and λ 2 = 1 and eigenvectors v 1 = (1, 0) and v 2 = (0, 1) respectivel, therefore the stable subspace T is the 1 -ais and unstable subspace T + is the 2 -ais. For the full nonlinear sstem we also have the same orbit structure on 2 -ais (because if 1 = 0 then the sstem is linear), and the simple form of the equations allows us to find that the orbits on the plane (, ) are given b = C. Therefore for C = 0 we have orbits approaching the origin along the parabola = 3 3, which is the stable manifold for our equilibrium, and which is obviousl tangent to T (see the figure). 2

3 T + W + (ˆ) T W (ˆ) ˆ 1 Figure 1: Stable and unstable manifolds for a hperbolic equilibrium 10.2 Outside of equilibria The previous subsection tells us a lot of possible orbit behavior near equilibria. The natural question is of course how to get an idea about what happens outside of equilibria. Right at this point I would like to sa that there are no universal methods to analze the structure of the phase portrait of a non-linear ODE sstem, however, b now we have quite a rich arsenal of tools to etract at least partial knowledge on the orbit behavior. For the planar sstem ẋ = f(, ), ẏ = g(, ), (4) the essential fact is that a simple curve dissects R 2 into two connected regions. We can use this fact to identif the regions of R 2, where the solutions to (4) are monotone (i.e., the derivatives have definite signs in these regions). Such regions are found b plotting the null-clines f(, ) = 0, g(, ) = 0 and using the following almost obvious proposition Proposition 4. Let ϕ(t) = ( (t), (t) ) be a solution to (4). Consider an open bounded set V R 2. If (t) and (t) are strictl monotone in V then either ϕ(t) hits the boundar of V at some finite t, or ϕ(t) converges to an equilibrium (ˆ, ŷ) V. Eample 5. Consider the sstem ẋ =, ẏ = The Jacobi matri is Df =

4 ẋ > 0,ẏ < 0 ẋ < 0,ẏ < 0 ˆ ẋ > 0,ẏ > 0 ẋ < 0,ẏ > 0 l l ˆ Figure 2: The phase portrait of the sstem in eample (see tet) The sstem has two equilibria: ˆ 1 = (0, 1) and ˆ 2 = (0, 1). The first one is an asmptoticall stable node since Df(ˆ 1 ) =, 0 2 and the eigenvalues are 1 and 2. Therefore there eists a two-dimensional stable manifold W (ˆ 1 ). The second equilibrium is unstable because Df(ˆ 2 ) =, 0 2 with the stable manifold tangent to the line parallel to -ais and unstable manifold tangent (actuall, coincident with) to -ais. The null-clines of the sstem are given b l 1 = {(, ): = 0}, l 2 = {(, ): = 1}, therefore there are four regions of R 2 where the solutions are monotone (see the figure). Using the proposition above, it should be clear that the onl possible orbit structure is given in the figure. In general, it is not quite straightforward to guess a possible structure of the orbits, however, the null-clines are usuall quite useful for obtaining at least partial information. 4

5 10.3 Bifurcations of equilibria. Structural stabilit Recall that a bifurcation is the change of the topological tpe of the sstem. Equilibria can change their topological tpe under some parameter variation onl if the linearization of the sstem around these equilibria is non-hperbolic, or, in other words, a presence of a non-hperbolic equilibria is a necessar condition for a bifurcation. This implies that bifurcations of the equilibria of the nonlinear sstem ẋ = f(, α), U R 2, α R, are possible when either one of the eigenvalues of the Jacobi matri becomes zero, or when two eigenvalues cross the real ais. (I consider here onl the case when there is one parameter in the sstem. In general, of course, it is possible to have that both eigenvalues of the Jacobi matri becomes zero simultaneousl, but genericall for this it is necessar to have two parameters in the sstem.) Eample 6 (Saddle-node or fold bifurcation). Consider the planar sstem ẋ = α + 2, ẏ =. Here the equations are decoupled and it is quite straightforward to figure out the phase portraits depending on the sign of α. If α < 0 we have two equilibria, one is asmptoticall stable (node) and the other one is unstable (saddle), if α = 0 then the origin is a non-hperbolic equilibrium (unstable, this equilibrium is called the saddle-node, hence the name for the bifurcation), and finall if α > 0 then there are no equilibria in the sstem, see the figure. Here in this eample the bifurcation is essentiall one-dimensional, which we alread studied in the earlier lecture notes. It turns out that this situation is generic for the case when one of the eigenvalues of the Jacobi matri becomes zero. We can have for most of the cases either fold, or transcritical, or pitchfork bifurcation, which we alread studied. In the first case two equilibria of the opposite stabilit approach each other, collide and disappear; in the second case there is alwas an equilibrium for all parameter values that changes the stabilit properties as another equilibrium passes through it, and finall in the third case we should distinguish sub- and supercritical pitchfork bifurcation, this occurs in sstems with certain smmetries. It is possible to specif the eact mathematical conditions for these three bifurcation tpes, but in practice it is usuall much easier to determine the tpe of the bifurcation with one of the eigenvalues zero b analzing the tpes and number of equilibria for the parameter values close to the bifurcation point, we will see plent of eamples of such analsis. On the other hand, the case when two eigenvalues of the Jacobi matri cross the imaginar ais is essentiall two-dimensional. The corresponding bifurcation is called Hopf, or, more appropriatel, Poincaré Andronov Hopf bifurcation, and we will stud it in more details later. Before finishing this section, I would like to note that bifurcations of equilibria in two dimensional sstems are not the onl possible changes of the phase portrait that lead to topologicall non-equivalent pictures. Consider the following eample. Eample 7 (Heteroclinic bifurcation in a planar sstem). Consider the sstem depending on parameter α ẋ = 1 2 α, ẏ = + α(1 2 ). 5

6 α < 0 α = 0 α > 0 Figure 3: Saddle-node or fold bifurcation This sstem alwas has two saddle equilibria ˆ 1 = ( 1, 0) and ˆ 2 = (1, 0). At α = 0 the -ais is invariant and approaches both equilibria for t ±. Such trajectories are called heteroclinic. For α 0 this connection disappears. This is an eample of a global bifurcation, since to detect it we need to keep track of both equilibria. There is one more important notion that pertains to bifurcations in sstems of ODE. In the last eample it is clear that for α = 0 the sstem is such that a small sstem perturbation would lead to a qualitativel different phase portraits. To formalize this, consider two sstems defined in a closed and bounded region U (the region is chosen as closed and bounded, i.e, compact, not to deal with complications due to unboundness of the plane): and ẋ = f(), U, (5) ẋ = g(), U. (6) Now consider the C (1) distance between these two vector fields (5) and (6) in U as { } f g 1 = sup f() g(), Df() Dg(), U 6

7 α < 0 α = 0 α > 0 Figure 4: Heteroclinic bifurcation where the first norm inside the sup is the usual Euclidian norm, and the second one is the norm of a matri A. The C (0) distance is not enough for m purposes. Using this definition of a distance in the space of all vector fields we can define a neighborhood N (f) of the vector field f as a set ϵ-close to f with respect to the defined metric. Finall, we can state that Definition 8. A planar differential equation ẋ = f() (or, equivalentl, a vector field f) is called structurall stable if there is a neighborhood N (f) such that for an vector field g N (f) is topologicall equivalent in U to f. One of the indication of a structurall unstable sstem is the presence of a non-hperbolic equilibrium. For eample, the Lotka Volterra sstem is structurall unstable, which is sometimes used as an indication that this sstem of ODE cannot be considered as a reliable candidate for a mathematical model describing predator pre interaction. I will come back to the discussion of structural stabilit later in the course. 7

Chapter 8 Equilibria in Nonlinear Systems

Chapter 8 Equilibria in Nonlinear Systems Chapter 8 Equilibria in Nonlinear Sstems Recall linearization for Nonlinear dnamical sstems in R n : X 0 = F (X) : if X 0 is an equilibrium, i.e., F (X 0 ) = 0; then its linearization is U 0 = AU; A =

More information

7 Planar systems of linear ODE

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

More information

Local Phase Portrait of Nonlinear Systems Near Equilibria

Local Phase Portrait of Nonlinear Systems Near Equilibria Local Phase Portrait of Nonlinear Sstems Near Equilibria [1] Consider 1 = 6 1 1 3 1, = 3 1. ( ) (a) Find all equilibrium solutions of the sstem ( ). (b) For each equilibrium point, give the linear approimating

More information

7.7 LOTKA-VOLTERRA M ODELS

7.7 LOTKA-VOLTERRA M ODELS 77 LOTKA-VOLTERRA M ODELS sstems, such as the undamped pendulum, enabled us to draw the phase plane for this sstem and view it globall In that case we were able to understand the motions for all initial

More information

* τσ σκ. Supporting Text. A. Stability Analysis of System 2

* τσ σκ. Supporting Text. A. Stability Analysis of System 2 Supporting Tet A. Stabilit Analsis of Sstem In this Appendi, we stud the stabilit of the equilibria of sstem. If we redefine the sstem as, T when -, then there are at most three equilibria: E,, E κ -,,

More information

EE222: Solutions to Homework 2

EE222: Solutions to Homework 2 Spring 7 EE: Solutions to Homework 1. Draw the phase portrait of a reaction-diffusion sstem ẋ 1 = ( 1 )+ 1 (1 1 ) ẋ = ( 1 )+ (1 ). List the equilibria and their tpes. Does the sstem have limit ccles? (Hint:

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

Problem set 6 Math 207A, Fall 2011 Solutions. 1. A two-dimensional gradient system has the form

Problem set 6 Math 207A, Fall 2011 Solutions. 1. A two-dimensional gradient system has the form Problem set 6 Math 207A, Fall 2011 s 1 A two-dimensional gradient sstem has the form x t = W (x,, x t = W (x, where W (x, is a given function (a If W is a quadratic function W (x, = 1 2 ax2 + bx + 1 2

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

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

8 Autonomous first order ODE

8 Autonomous first order ODE 8 Autonomous first order ODE There are different ways to approach differential equations. Prior to this lecture we mostly dealt with analytical methods, i.e., with methods that require a formula as a final

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

Math Notes on sections 7.8,9.1, and 9.3. Derivation of a solution in the repeated roots case: 3 4 A = 1 1. x =e t : + e t w 2.

Math Notes on sections 7.8,9.1, and 9.3. Derivation of a solution in the repeated roots case: 3 4 A = 1 1. x =e t : + e t w 2. Math 7 Notes on sections 7.8,9., and 9.3. Derivation of a solution in the repeated roots case We consider the eample = A where 3 4 A = The onl eigenvalue is = ; and there is onl one linearl independent

More information

14 Periodic phenomena in nature and limit cycles

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.

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

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

520 Chapter 9. Nonlinear Differential Equations and Stability. dt =

520 Chapter 9. Nonlinear Differential Equations and Stability. dt = 5 Chapter 9. Nonlinear Differential Equations and Stabilit dt L dθ. g cos θ cos α Wh was the negative square root chosen in the last equation? (b) If T is the natural period of oscillation, derive the

More information

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

0 as an eigenvalue. degenerate

0 as an eigenvalue. degenerate Math 1 Topics since the third exam Chapter 9: Non-linear Sstems of equations x1: Tpical Phase Portraits The structure of the solutions to a linear, constant coefficient, sstem of differential equations

More information

EE222 - Spring 16 - Lecture 2 Notes 1

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

More information

ME 680- Spring Geometrical Analysis of 1-D Dynamical Systems

ME 680- Spring Geometrical Analysis of 1-D Dynamical Systems ME 680- Spring 2014 Geometrical Analysis of 1-D Dynamical Systems 1 Geometrical Analysis of 1-D Dynamical Systems Logistic equation: n = rn(1 n) velocity function Equilibria or fied points : initial conditions

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

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

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

More information

MATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation

MATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation MATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation 1 Bifurcations in Multiple Dimensions When we were considering one-dimensional systems, we saw that subtle changes in parameter

More information

Ordinary Differential Equations

Ordinary Differential Equations Instructor: José Antonio Agapito Ruiz UCSC Summer Session II, 00 Math Ordinar Differential Equations Final Solution This report does not contain full answers to all questions of the final. Instead, ou

More information

4 Second-Order Systems

4 Second-Order Systems 4 Second-Order Systems Second-order autonomous systems occupy an important place in the study of nonlinear systems because solution trajectories can be represented in the plane. This allows for easy visualization

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

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

Two Limit Cycles in a Two-Species Reaction

Two Limit Cycles in a Two-Species Reaction Two Limit Ccles in a Two-Species Reaction Brigita Ferčec 1 Ilona Nag Valer Romanovski 3 Gábor Szederkéni 4 and János Tóth 5 The Facult of Energ Technolog Krško 1 Center for Applied Mathematics and Theoretical

More information

1 The pendulum equation

1 The pendulum equation Math 270 Honors ODE I Fall, 2008 Class notes # 5 A longer than usual homework assignment is at the end. The pendulum equation We now come to a particularly important example, the equation for an oscillating

More information

Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes

Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.

More information

Math 2930 Worksheet Equilibria and Stability

Math 2930 Worksheet Equilibria and Stability Math 2930 Worksheet Equilibria and Stabilit Week 3 September 7, 2017 Question 1. (a) Let C be the temperature (in Fahrenheit) of a cup of coffee that is cooling off to room temperature. Which of the following

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

Midterm 2. MAT17C, Spring 2014, Walcott

Midterm 2. MAT17C, Spring 2014, Walcott Mierm 2 MAT7C, Spring 24, Walcott Note: There are SEVEN questions. Points for each question are shown in a table below and also in parentheses just after each question (and part). Be sure to budget our

More information

TRANSFER ORBITS GUIDED BY THE UNSTABLE/STABLE MANIFOLDS OF THE LAGRANGIAN POINTS

TRANSFER ORBITS GUIDED BY THE UNSTABLE/STABLE MANIFOLDS OF THE LAGRANGIAN POINTS TRANSFER ORBITS GUIDED BY THE UNSTABLE/STABLE MANIFOLDS OF THE LAGRANGIAN POINTS Annelisie Aiex Corrêa 1, Gerard Gómez 2, Teresinha J. Stuchi 3 1 DMC/INPE - São José dos Campos, Brazil 2 MAiA/UB - Barcelona,

More information

Dynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting

Dynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:05 55 Dynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting K. Saleh Department of Mathematics, King Fahd

More information

The Hopf Bifurcation Theorem: Abstract. Introduction. Transversality condition; the eigenvalues cross the imginary axis with non-zero speed

The Hopf Bifurcation Theorem: Abstract. Introduction. Transversality condition; the eigenvalues cross the imginary axis with non-zero speed Supercritical and Subcritical Hopf Bifurcations in Non Linear Maps Tarini Kumar Dutta, Department of Mathematics, Gauhati Universit Pramila Kumari Prajapati, Department of Mathematics, Gauhati Universit

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

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

(4p) See Theorem on pages in the course book. 3. Consider the following system of ODE: z 2. n j 1

(4p) See Theorem on pages in the course book. 3. Consider the following system of ODE: z 2. n j 1 MATEMATIK Datum -8- Tid eftermiddag GU, Chalmers Hjälpmedel inga A.Heintz Telefonvakt Aleei Heintz Tel. 76-786. Tenta i ODE och matematisk modellering, MMG, MVE6 Answer rst those questions that look simpler,

More information

DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS

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

More information

In these chapter 2A notes write vectors in boldface to reduce the ambiguity of the notation.

In these chapter 2A notes write vectors in boldface to reduce the ambiguity of the notation. 1 2 Linear Systems In these chapter 2A notes write vectors in boldface to reduce the ambiguity of the notation 21 Matrix ODEs Let and is a scalar A linear function satisfies Linear superposition ) Linear

More information

INTEGRABILITY AND GLOBAL DYNAMICS OF THE MAY LEONARD MODEL

INTEGRABILITY AND GLOBAL DYNAMICS OF THE MAY LEONARD MODEL This is a preprint of: Integrabilit and global dnamics of the Ma-Leonard model, Gamaliel Blé, Víctor Castellanos, Jaume Llibre, Ingrid Quilantán, Nonlinear Anal. Real World Appl., vol. 14, 280 293, 2013.

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

Appendix: A Computer-Generated Portrait Gallery

Appendix: A Computer-Generated Portrait Gallery Appendi: A Computer-Generated Portrait Galler There are a number of public-domain computer programs which produce phase portraits for 2 2 autonomous sstems. One has the option of displaing the trajectories

More information

Mathematical Foundations of Neuroscience - Lecture 7. Bifurcations II.

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

More information

Nonlinear Systems Examples Sheet: Solutions

Nonlinear Systems Examples Sheet: Solutions Nonlinear Sstems Eamples Sheet: Solutions Mark Cannon, Michaelmas Term 7 Equilibrium points. (a). Solving ẋ =sin 4 3 =for gives =as an equilibrium point. This is the onl equilibrium because there is onl

More information

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP Dnamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP E. Barrabés, J.M. Mondelo, M. Ollé December, 28 Abstract We consider the planar Restricted Three-Bod problem and the collinear

More information

6 General properties of an autonomous system of two first order ODE

6 General properties of an autonomous system of two first order ODE 6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x

More information

THE ROSENZWEIG-MACARTHUR PREDATOR-PREY MODEL

THE ROSENZWEIG-MACARTHUR PREDATOR-PREY MODEL THE ROSENZWEIG-MACARTHUR PREDATOR-PREY MODEL HAL L. SMITH* SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES ARIZONA STATE UNIVERSITY TEMPE, AZ, USA 8587 Abstract. This is intended as lecture notes for nd

More information

Nonlinear dynamics & chaos BECS

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

More information

Bifurcations of the Controlled Escape Equation

Bifurcations of the Controlled Escape Equation Bifurcations of the Controlled Escape Equation Tobias Gaer Institut für Mathematik, Universität Augsburg 86135 Augsburg, German gaer@math.uni-augsburg.de Abstract In this paper we present numerical methods

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

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

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 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

More information

MCE693/793: Analysis and Control of Nonlinear Systems

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

More information

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

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!

More information

UNFOLDING THE CUSP-CUSP BIFURCATION OF PLANAR ENDOMORPHISMS

UNFOLDING THE CUSP-CUSP BIFURCATION OF PLANAR ENDOMORPHISMS UNFOLDING THE CUSP-CUSP BIFURCATION OF PLANAR ENDOMORPHISMS BERND KRAUSKOPF, HINKE M OSINGA, AND BRUCE B PECKHAM Abstract In man applications of practical interest, for eample, in control theor, economics,

More information

Phase portraits in two dimensions

Phase portraits in two dimensions Phase portraits in two dimensions 8.3, Spring, 999 It [ is convenient to represent the solutions to an autonomous system x = f( x) (where x x = ) by means of a phase portrait. The x, y plane is called

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

Calculus and Differential Equations II

Calculus and Differential Equations II MATH 250 B Second order autonomous linear systems We are mostly interested with 2 2 first order autonomous systems of the form { x = a x + b y y = c x + d y where x and y are functions of t and a, b, c,

More information

4 Insect outbreak model

4 Insect outbreak model 4 Insect outbreak model In this lecture I will put to a good use all the mathematical machinery we discussed so far. Consider an insect population, which is subject to predation by birds. It is a very

More information

Math 312 Lecture Notes Linearization

Math 312 Lecture Notes Linearization Math 3 Lecture Notes Linearization Warren Weckesser Department of Mathematics Colgate University 3 March 005 These notes discuss linearization, in which a linear system is used to approximate the behavior

More information

Hello everyone, Best, Josh

Hello everyone, Best, Josh Hello everyone, As promised, the chart mentioned in class about what kind of critical points you get with different types of eigenvalues are included on the following pages (The pages are an ecerpt from

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

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

TWO DIMENSIONAL FLOWS. Lecture 5: Limit Cycles and Bifurcations

TWO DIMENSIONAL FLOWS. Lecture 5: Limit Cycles and Bifurcations TWO DIMENSIONAL FLOWS Lecture 5: Limit Cycles and Bifurcations 5. Limit cycles A limit cycle is an isolated closed trajectory [ isolated means that neighbouring trajectories are not closed] Fig. 5.1.1

More information

A PREY-PREDATOR MODEL WITH HOLLING II FUNCTIONAL RESPONSE AND THE CARRYING CAPACITY OF PREDATOR DEPENDING ON ITS PREY

A PREY-PREDATOR MODEL WITH HOLLING II FUNCTIONAL RESPONSE AND THE CARRYING CAPACITY OF PREDATOR DEPENDING ON ITS PREY Journal of Applied Analsis and Computation Volume 8, Number 5, October 218, 1464 1474 Website:http://jaac-online.com/ DOI:1.11948/218.1464 A PREY-PREDATOR MODEL WITH HOLLING II FUNCTIONAL RESPONSE AND

More information

MAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function

MAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function MAT 275: Introduction to Mathematical Analsis Dr. A. Rozenblum Graphs and Simplest Equations for Basic Trigonometric Functions We consider here three basic functions: sine, cosine and tangent. For them,

More information

NONLINEAR DYNAMICS AND CHAOS. Numerical integration. Stability analysis

NONLINEAR DYNAMICS AND CHAOS. Numerical integration. Stability analysis LECTURE 3: FLOWS NONLINEAR DYNAMICS AND CHAOS Patrick E McSharr Sstems Analsis, Modelling & Prediction Group www.eng.o.ac.uk/samp patrick@mcsharr.net Tel: +44 83 74 Numerical integration Stabilit analsis

More information

ARTICLE IN PRESS. Nonlinear Analysis: Real World Applications. Contents lists available at ScienceDirect

ARTICLE IN PRESS. Nonlinear Analysis: Real World Applications. Contents lists available at ScienceDirect Nonlinear Analsis: Real World Applications ( Contents lists available at ScienceDirect Nonlinear Analsis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa Qualitative analsis of

More information

Introduction to Bifurcation and Normal Form theories

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

More information

C. Non-linear Difference and Differential Equations: Linearization and Phase Diagram Technique

C. Non-linear Difference and Differential Equations: Linearization and Phase Diagram Technique C. Non-linear Difference and Differential Equations: Linearization and Phase Diaram Technique So far we have discussed methods of solvin linear difference and differential equations. Let us now discuss

More information

STABILITY AND NEIMARK-SACKER BIFURCATION OF A SEMI-DISCRETE POPULATION MODEL

STABILITY AND NEIMARK-SACKER BIFURCATION OF A SEMI-DISCRETE POPULATION MODEL Journal of Applied Analsis and Computation Website:http://jaac-online.com/ Volume 4, Number 4, November 14 pp. 419 45 STABILITY AND NEIMARK-SACKER BIFURCATION OF A SEMI-DISCRETE POPULATION MODEL Cheng

More information

LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector.

LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector. LESSON 5: EIGENVALUES AND EIGENVECTORS APRIL 2, 27 In this contet, a vector is a column matri E Note 2 v 2, v 4 5 6 () We might also write v as v Both notations refer to a vector (2) A vector can be man

More information

One-Dimensional Dynamics

One-Dimensional Dynamics One-Dimensional Dnamics We aim to characterize the dnamics completel describe the phase portrait) of the one-dimensional autonomous differential equation Before proceeding we worr about 1.1) having a solution.

More information

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv Math 1270 Honors ODE I Fall, 2008 Class notes # 1 We have learned how to study nonlinear systems x 0 = F (x; y) y 0 = G (x; y) (1) by linearizing around equilibrium points. If (x 0 ; y 0 ) is an equilibrium

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

Invariant Manifolds of Dynamical Systems and an application to Space Exploration

Invariant Manifolds of Dynamical Systems and an application to Space Exploration Invariant Manifolds of Dynamical Systems and an application to Space Exploration Mateo Wirth January 13, 2014 1 Abstract In this paper we go over the basics of stable and unstable manifolds associated

More information

Examples and counterexamples for Markus-Yamabe and LaSalle global asymptotic stability problems Anna Cima, Armengol Gasull and Francesc Mañosas

Examples and counterexamples for Markus-Yamabe and LaSalle global asymptotic stability problems Anna Cima, Armengol Gasull and Francesc Mañosas Proceedings of the International Workshop Future Directions in Difference Equations. June 13-17, 2011, Vigo, Spain. PAGES 89 96 Examples and counterexamples for Markus-Yamabe and LaSalle global asmptotic

More information

THE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A 2 2 SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS

THE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A 2 2 SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS THE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS Maria P. Skhosana and Stephan V. Joubert, Tshwane University

More information

CHAPTER 2: Partial Derivatives. 2.2 Increments and Differential

CHAPTER 2: Partial Derivatives. 2.2 Increments and Differential CHAPTER : Partial Derivatives.1 Definition of a Partial Derivative. Increments and Differential.3 Chain Rules.4 Local Etrema.5 Absolute Etrema 1 Chapter : Partial Derivatives.1 Definition of a Partial

More information

STABILITY. Phase portraits and local stability

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),

More information

x y plane is the plane in which the stresses act, yy xy xy Figure 3.5.1: non-zero stress components acting in the x y plane

x y plane is the plane in which the stresses act, yy xy xy Figure 3.5.1: non-zero stress components acting in the x y plane 3.5 Plane Stress This section is concerned with a special two-dimensional state of stress called plane stress. It is important for two reasons: () it arises in real components (particularl in thin components

More information

April 13, We now extend the structure of the horseshoe to more general kinds of invariant. x (v) λ n v.

April 13, We now extend the structure of the horseshoe to more general kinds of invariant. x (v) λ n v. April 3, 005 - Hyperbolic Sets We now extend the structure of the horseshoe to more general kinds of invariant sets. Let r, and let f D r (M) where M is a Riemannian manifold. A compact f invariant set

More information

+ i. cos(t) + 2 sin(t) + c 2.

+ i. cos(t) + 2 sin(t) + c 2. MATH HOMEWORK #7 PART A SOLUTIONS Problem 7.6.. Consider the system x = 5 x. a Express the general solution of the given system of equations in terms of realvalued functions. b Draw a direction field,

More information

QUALITATIVE ANALYSIS OF DIFFERENTIAL EQUATIONS

QUALITATIVE ANALYSIS OF DIFFERENTIAL EQUATIONS arxiv:1803.0591v1 [math.gm] QUALITATIVE ANALYSIS OF DIFFERENTIAL EQUATIONS Aleander Panfilov stable spiral det A 6 3 5 4 non stable spiral D=0 stable node center non stable node saddle 1 tr A QUALITATIVE

More information

Chapter 7. Nonlinear Systems. 7.1 Introduction

Chapter 7. Nonlinear Systems. 7.1 Introduction Nonlinear Systems Chapter 7 The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. - Jules Henri Poincaré (1854-1912)

More information

New approach to study the van der Pol equation for large damping

New approach to study the van der Pol equation for large damping Electronic Journal of Qualitative Theor of Differential Equations 2018, No. 8, 1 10; https://doi.org/10.1422/ejqtde.2018.1.8 www.math.u-szeged.hu/ejqtde/ New approach to stud the van der Pol equation for

More information

2 Discrete growth models, logistic map (Murray, Chapter 2)

2 Discrete growth models, logistic map (Murray, Chapter 2) 2 Discrete growth models, logistic map (Murray, Chapter 2) As argued in Lecture 1 the population of non-overlapping generations can be modelled as a discrete dynamical system. This is an example of an

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

Math 1280 Notes 4 Last section revised, 1/31, 9:30 pm.

Math 1280 Notes 4 Last section revised, 1/31, 9:30 pm. 1 competing species Math 1280 Notes 4 Last section revised, 1/31, 9:30 pm. This section and the next deal with the subject of population biology. You will already have seen examples of this. Most calculus

More information

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 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

More information

Continuous Threshold Policy Harvesting in Predator-Prey Models

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

More information

17.3. Parametric Curves. Introduction. Prerequisites. Learning Outcomes

17.3. Parametric Curves. Introduction. Prerequisites. Learning Outcomes Parametric Curves 7.3 Introduction In this Section we eamine et another wa of defining curves - the parametric description. We shall see that this is, in some was, far more useful than either the Cartesian

More information

6.3. Nonlinear Systems of Equations

6.3. Nonlinear Systems of Equations G. NAGY ODE November,.. Nonlinear Systems of Equations Section Objective(s): Part One: Two-Dimensional Nonlinear Systems. ritical Points and Linearization. The Hartman-Grobman Theorem. Part Two: ompeting

More information

Eigenvectors and Eigenvalues 1

Eigenvectors and Eigenvalues 1 Ma 2015 page 1 Eigenvectors and Eigenvalues 1 In this handout, we will eplore eigenvectors and eigenvalues. We will begin with an eploration, then provide some direct eplanation and worked eamples, and

More information

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.

More information

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 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

More information

2 Ordinary Differential Equations: Initial Value Problems

2 Ordinary Differential Equations: Initial Value Problems Ordinar Differential Equations: Initial Value Problems Read sections 9., (9. for information), 9.3, 9.3., 9.3. (up to p. 396), 9.3.6. Review questions 9.3, 9.4, 9.8, 9.9, 9.4 9.6.. Two Examples.. Foxes

More information