# Lectures on Periodic Orbits

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 proofs have been omitted and wherever possible, references to the literature have been given instead. As these notes are being written during the term, they will be frequently updated. Week-by-week changes are detailed on the course web page. 1 Ruling Out Periodic Orbits Gradient Systems. A gradient system is a dynamical system of the form ẋ = V (x) (1.1) for a given function V (x) in R n. Theorem 1.1. Gradient systems cannot have periodic orbits. Proof. Suppose to the contrary that γ : t x(t) is a periodic orbit of the gradient system (1.1) with period T. Then V (x(t )) V (x(0)) = 0, but on the other hand V (x(t )) V (x(0)) = a contradiction. T 0 dv T T dt dt = V ẋ dt = ẋ 2 dt < 0, 0 0 Note that this result is valid for gradient systems in arbitrary dimensions, in contrast to the subsequent results, which are specific for the plane. Dulac s Criterion. Recall that a region R of the plane is said to be simply connected if every closed loop within R can be shrunk to a point without leaving R (intuitively, R contains no holes). Dulac s criterion gives sufficient conditions for the non-existence of periodic orbits of dynamical systems in simply connected regions of the plane. The downside of this method is that it depends on the choice of an appropriate multiplier, which might be hard to find. Theorem 1.2. Let R be a simply connected region in R 2 and consider a planar dynamical system in R given by ẋ = f(x, y) and ẏ = g(x, y), (1.2) 1

2 2 Index Theory 2 where f, g are C 1 functions in R. Suppose that there exists a C 1 function h(x, y) in R so that h(fe x + ge y ) has a definite sign in R. Then the dynamical system (1.2) cannot have any periodic orbits in R. Proof. Assume that (1.2) has a periodic orbit γ contained in R and let A be the area enclosed by γ. Since R is simply connected, A lies entirely in R. By Green s theorem, we have h(fe x + ge y ) dxdy = where n is the outward normal to γ. A γ h(fe x + ge y ) ndl, Now, note that the left-hand side of this expression is different from zero because of the sign-definiteness. However, the right-hand side is zero (since fe x + ge y is tangent to γ), a contradiction. A special case of this theorem, where the multiplier h = 1, is known as Bendixson s criterion. When asked to verify whether a vector field can have periodic orbits, the first thing to check is whether Bendixson s criterion is applicable: checking that the divergence of the vector field is sign definite can be a quick and straightforward way to rule out periodic orbits. When this fails, Dulac s criterion or other techniques can be used. Example 1. Show that the following dynamical system does not have any closed orbits in the region x, y > 0: ẋ = x(2 x y) and ẏ = y(4x x 2 3). By using the multiplier h = 1 xy, we find that the divergence is given by 1/y, which is sign-definite in the region considered. Lyapunov-like Functions. If a dynamical system ẋ = X(x) has a Lyapunov-like function V (x), i.e. V (x) is monotonically decreasing along trajectories, then it is easy to see that there cannot be any periodic orbits. Indeed, assume that there is a periodic orbit with period T, then T dv 0 = V (x(t )) V (x(0)) = dt dt, but the right-hand side is by assumption strictly smaller than zero, a contradiction. Like always, there is no explicit prescription for finding a Lyapunov function and in some examples, Lyapunov-like functions may be useful; see Strogatz [1994], example Index Theory Refer to 7.3 of Strogatz [1994]. We covered the intuitive definition of the index of a curve as the number of times the vector field rotates in counterclockwise fashion when traversing the curve once. The index satisfies a number of elementary topological properties:

3 3 The Poincaré-Bendixson Theorem 3 1. When C and C are two closed curves that can be continuously deformed into each other (without moving the curve over one of the fixed points of the vector field), then I C = I C. 2. If C does not enclose any fixed points, then I C = The index is unchanged when the direction of the vector field is reversed. 4. If C is a trajectory of the system rather than an arbitrary closed curve, then I C = +1. As a result, we define the index I x of a fixed point x by taking an arbitrary closed curve C around x such that C does not enclose any other fixed points, and we put I x = I C. By property 1 above, I x is independent of the choice of curve. Note that the index of a fixed point does not distinguish the stability types of a fixed point sources and sinks both have index +1. Lemma 2.1 (Thm in Strogatz [1994]). If a closed curve C encloses n fixed points x 1,..., x n, then I C = I x1 + + I xn. Some immediately useful consequences are Any closed orbit in the plane has to enclose fixed points whose total index sums to +1. In particular, if the vector field has no fixed points, there cannot be any periodic orbits. If the system has only one fixed point, it cannot be hyperbolic (since the index of a hyperbolic fixed point is minus one). These conclusions can be used to rule out the existence of closed orbits, or to provide qualitative information about fixed points and closed orbits. See examples and in Strogatz [1994]. 3 The Poincaré-Bendixson Theorem The results in the preceding sections are all concerned with the non-existence of periodic orbits, or (in the case of index theory) with the consequences of having a given periodic orbit. The Poincaré-Bendixson theorem on the other hand gives sufficient conditions for the existence of a periodic orbit. There are many versions of this theorem and the proof (which we omit from these lectures) is reasonably complicated. 3.1 Introductory Definitions Although the following concepts are strictly speaking not needed for the formulation of the Poincaré-Bendixson theorem, they are sufficiently fundamental for us to list them here anyway. Let ẋ = X(x) be a dynamical system and consider a trajectory γ : t x(t). A positive limit point is a point x for which there exists a sequence t 1, t 2,... + such that lim x(t n) = x. n

4 3.2 Statement of the Theorem 4 The concept of a negative limit point is defined similarly. The ω-limit set of a trajectory γ, denoted by ω(γ), is defined as the set of all positive limit points of that orbit. Similarly, the α-limit set is the set of all negative limit points. Example 2. Consider the system ẋ = x, ẏ = 2y. The origin is a positive limit point for all orbits and there are no other positive limit points. The ω-limit set is hence the origin for all orbits. The α-limit set is empty for all orbits except the origin. Example 3. Consider the harmonic oscillator equations ẋ = y, ẏ = x. Every point on an orbit is simultaneously a positive and a negative limit point for that orbit. The importance of the ω-limit set lies in the fact that trajectories in a bounded region of the plane will spiral inward to ω-limit set. Essentially, the Poincaré-Bendixson theorem tells us that ω(γ) will either contain a fixed point or it will be a closed orbit of the flow. The following lemma gives us a hint as to why this should be so (this is far from a complete proof, which is very complicated). Theorem 3.1 (thm. 4.2 in Verhulst [1996]). The sets α(γ) and ω(γ) are closed and invariant. If γ + is bounded, then ω(γ) is compact, connected and non-empty. Moreover d(x(t), ω(γ)) goes to zero as t +. Using the concept of the ω- and α-limit set, we can now define the concept of limit cycles rigorously. Definition 3.2 (def. 3.2 in Perko [2001]). A limit cycle γ of a dynamical system in the plane is a periodic orbit which is the α or ω-limit set of a trajectory γ other than γ. If a limit cycle γ is the ω-limit set of every other trajectory in a neighborhood of γ, γ is said to be an ω-limit cycle or stable limit cycle. Likewise, if γ is the α-limit set of neighboring trajectories, γ is said to be an α-limit cycle or unstable limit cycle. Limit cycles are hence a truly nonlinear feature. Strogatz [1994] defines limit cycles informally as closed orbits such that nearby trajectories are either attracted to or repelled by the limit cycle: by virtue of the theorem above, we see that this definition corresponds exactly to either an ω- or an α-limit cycle. Note that the condition that a limit cycle is the ω-limit set of a different trajectory is crucial. Otherwise, the trajectories of the simple oscillator (example 3) would be limit cycles, which does not capture the idea that a limit cycle is informally where trajectories end up. Example 4. The dynamical system given in polar coordinates by ṙ = r(1 r 2 ), θ = 1 has exactly one stable limit cycle, given by r = 1. All trajectories, except for the fixed point r = 0, are attracted to this limit cycle. See figure Statement of the Theorem Theorem 3.3 (Poincaré-Bendixson). Let R be a region of the plane which is closed and bounded. Consider a dynamical system ẋ = X(x) in R where the vector field X is at least C 1. Assume that R contains no fixed points of X. Assume furthermore that there exists a trajectory γ of X starting in R which stays in R for all future times. Then, either γ is a closed orbit or γ asymptotically approaches a closed orbit; in other words, there exists a limit cycle in R.

5 3.2 Statement of the Theorem 5 Figure 3.1: A system with a stable limit cycle. More formally, let γ + be the forward part of γ. Then if γ + is contained in a compact subset of the plane which contains no fixed points, ω(γ) will be the desired periodic orbit. This is how the Poincaré-Bendixson theorem is stated in [Perko, 2001, Sec. 3.7]. The hard part of the Poincaré-Bendixson theorem consists of finding a suitable trajectory γ. However, there are a number of special cases in which this becomes easier in practice: define a trapping region to be any region R of the plane which is positively invariant under the flow φ t of X: φ t (R) R for all t > 0. It follows that if R is a trapping region, then every trajectory of X starting in R stays in R for all future times. To check that R is a trapping region it is sufficient to verify that on the boundary of R, X is everywhere pointing inward. Practically speaking, this is how one proceeds to apply the Poincaré-Bendixson theorem: by constructing an (annular) region R in the plane so that on the boundary of R, the vector field points into R. Note that the region R will always be annular. In other words, R will contain at least one hole (topologically speaking, R is not simply connected). This can easily be seen through index theory. Denote the outer boundary of R by C: since the vector field is pointing inward on C, the index I C will be equal to +1. Hence C has to enclose a given number of fixed points with total index +1. These fixed points have to be excised from R in order for the Poincaré-Bendixson theorem to be applicable, leaving us with a non-simply connected domain. Remark 1. Note that there exists a time-reversed version of the Poincaré-Bendixson theorem, in the sense that if the backward trajectory γ, defined as γ = {(x(t), y(t)) : t 0} is contained in a compact subset R of the plane which contains no fixed points, then there exists a limit cycle in R. More specifically, the desired limit cycle will be α(γ). This follows from the usual Poincaré-Bendixson by reversing time in the original differential equation. The result is that γ turns into γ + and α(γ) into ω(γ).

6 4 Liénard Equations 6 If one can find a repelling region, 1 i.e. an annular region such that the vector field points outwards of this region, then this version of the Poincaré-Bendixson theorem may be applied. 4 Liénard Equations So far we have discussed existence results for limit cycles only. Deciding on the number of limit cycle and their relative location, however, is a wholly different matter. Indeed, this is exactly Hilbert s 16th problem, which is unresolved until this day. Neverthelesss, for certain classes of differential equations, sharper results than just existence can be obtained. One such class are the so-called Liénard equations, for which exactly one stable limit cycle exists. Liénard equations arose in the context of circuit theory and radiotransmitters, where having a stable regime of oscillations is very important. We refer to any differential equation of the form ẍ + f(x)ẋ + g(x) = 0 (4.1) as a Liénard equation. Here, f, g are unknown functions. For there to exist exactly one stable limit cycle, f and g have to satisfy a number of conditions. Theorem 4.1. Suppose the following conditions hold: 1. f, g are C 1 functions; 2. g(x) is an odd function of x; 3. g(x) > 0 for x > 0; 4. the primitive F (x) = x f(u)du satisfies 0 (a) F (x) has exactly one positive root at x = a; (b) F (x) < 0 for 0 < x < a; (c) F (x) is positive and non-decreasing for x > a; (d) F (x) + for x +. Then the Liénard equation (4.1) has exactly one stable limit cycle. In class, we proved this theorem for the van der Pol equation, where f(x) = µ(x 2 1) with µ > 0, and g(x) = x. Equations with similar characteristics arise throughout physics, chemistry and biology; one example is the Fitzhugh-Nagumo system describing the spiking of neurons. The existence proof for stable limit cycles of the van der Pol equation can be found in Hirsch et al. [2004], sec Note that if g(x) = x as in the case of the van der Pol equation, the system can be rewritten as a second-order equation. Kaushik raised an interesting question of whether a secondorder system can ever have more than one limit cycle. While the question of determining or even bounding the number of limit cycles of a given set of ODEs is highly intricate, it turns out that by taking g(x) = x and f(x) to be a polynomial of sufficiently high degree, multiple limit cycles may occur. An example is due to Zhang Perko [2001]: take 1 This terminology is not standard. F (x) = 8 25 x5 4 3 x x.

7 5 Quantitative Methods of Finding Periodic Orbits 7 The Liénard equation associated to this system has two limit cycles; see figure 4.1. Obviously this equation does not satify the hypotheses of theorem 4.1 as F (x) has multiple positive roots. Figure 4.1: Zhang s example of a system with two limit cycles. The inner limit cycle is unstable, which can inferred from the fact that the origin is a source. The outer limit cycle is stable. 5 Quantitative Methods of Finding Periodic Orbits Often the conclusions of the Poincaré-Bendixson, that there exists a periodic orbit in a given region, are somewhat lacking and we need an explicit expression for the periodic orbit. Series expansions provide one way of obtaining such expressions, but one should take care in interpreting them. In this section, we will look at periodic orbits of dynamical systems in the plane of the form ẍ + x + ɛh(x, ẋ) = 0, (5.1) where ɛ 0 is a small parameter. For ɛ = 0, this is just the harmonic oscillator. What happens to the periodic orbits of the harmonic oscillator when ɛ > 0? Naive Taylor Expansions. One way of approaching this question is by expanding the solution x ɛ (t) of (5.1) in powers of ɛ: x ɛ (t) = x 0 (t) + ɛx 1 (t) + ɛ 2 x 2 (t) +. This approach can and does run into difficulties, however: Strogatz considers the example where h(x, ẋ) = 2ẋ and initial conditions are given by x(0) = 0, ẋ(0) = 1. This example is explicitly solvable; see Strogatz [1994] for details. One special feature of the solution is that it goes to zero exponentially fast, since the added term represents a small damping term. From a perturbation point of view, x 0 (t) is easily seen to be equal to sin t, and the equation for x 1 (t) is given by ẍ 1 + x 1 = 2 cos t. This is an equation for a harmonic oscillator with

8 5 Quantitative Methods of Finding Periodic Orbits 8 resonant forcing: the right-hand side pumps energy into the system. The solution is given by x 1 (t) = t sin t: this is a disaster since x 1 (t) goes off to infinity, in contrast to the exact solution, which stays bounded. Terms in the perturbation expansion like x 1 (t), which grow without bound, are termed secular. What is the problem with all this? Mathematically speaking, x 0 (t) + ɛx 1 (t) represents, for fixed t, the first order Taylor expansion to x ɛ (t), and by calculating higher order contributions, we get progressively better approximations. From a dynamical systems point of view, we have no interest in fixing t of course. However, because of the secular nature of the second term, the expansion is then only accurate for times t < ɛ 1. See figure in Strogatz [1994]. This example sets the tone for the rest of this section: even by introducing just a small friction term, we already trigger qualitatively new phenomena. Figure 5.1: The true solution (damped) compared with its first-order Taylor expansion for ɛ = 0.1. Observe that both are in good agreement for t ɛ 1 = 10. Averaging. Another way of dealing with resonant terms is through the method of averaging, also referred to as Lagrange s method. A very thorough outline of this method can be obtained from Strogatz [1994], section 7.6. For periodic orbits, the first order approximation obtained with this method can be shown to stay within an O(ɛ) range of the true solution for all times. Poincaré-Lindstedt Theory. What is needed is a way to eliminate these secular terms. There are several related methods to do this: here we use the Poincaré-Lindstedt method. This method is only applicable to approximating periodic orbits and consists of proposing a perturbation expansion for both the periodic orbit x ɛ (t) and its frequency ω ɛ. 1. Introduce a new time scale τ = ωt so that the new period becomes 2π. 2. Substitute series expansions for x ɛ (τ) = x 0 (τ) + ɛx 1 (τ) + and ω ɛ = ω 0 + ɛω 1 +

9 6 Limitations to the Poincaré-Bendixson Theorem 9 into the equation. Note that ω 0 = 1 since the solution has period 2π when ɛ = 0. Substitute the same expansions into the initial conditions and find the resulting initial conditions for x i (t). 3. Collect terms of the same order in ɛ and solve the resulting equations for x i (t). Use the freedom in choosing the coefficients ω i to kill off any secular terms. In this way, we find an expansion to arbitrary high order to the desired periodic orbit of the system. Strogatz (exercise ) illustrates this on an example: the Duffing oscillator ẍ + x + ɛx 3 = 0 with initial conditions x(0) = a and ẋ(0) = The new equation becomes ω 2 x + x + ɛx 3 = 0, where the prime denotes derivation with respect to τ. 2. The zeroth- and first-order equations become x 0 + x 0 = 0 and x 1 + x 1 = 2ω 1 x 0 x 3 0 with initial conditions x 0 (0) = a and x 1 (0) = 0, as well as ẋ 0 (0) = ẋ 1 (0) = 0. So x 0 (τ) = a cos τ and the equation for x 1 becomes x 1 + x 1 = (2ω a2 )a cos τ 1 4 a3 cos 3τ. 3. To eliminate secular terms we must choose ω 1 = 3 8 a2. The resulting equation for x 1 can now be solved. In this way we can proceed order by order. As in the case of the averaging method, we will usually not be interested in these higher-order contributions. Just the fact that we ve eliminated secular terms in the equation for x 1 already yields important information: the first approximation x 0 is given by x 0 (t) = a cos ωt = a cos(1 + ɛ 3 8 a2 )t. In other words, we get a circular periodic orbit with a nearly-constant frequency. Notice that this periodic orbit is not a limit cycle. 6 Limitations to the Poincaré-Bendixson Theorem The Poincaré-Bendixson theorem essentially rules out chaos in the plane. This turns out to be a highly non-generic result, however, which does not seem to hold for other configuration spaces or other types of dynamical systems. In this section we explore a few of these possibilities. Two-dimensional Configuration Spaces. Dynamical systems on two-dimensional manifolds other than the plane may well violate the Poincaré-Bendixson theorem. Consider for instance the following vector field on the torus, which we identify with the unit square in the plane with opposite sides identified: ẋ = 1 and ẏ = π. (6.1) There is nothing special about the choice of π: any other irrational number would work just as well. Even though the torus is compact and the vector field (6.1) does not have any

10 REFERENCES 10 zeros, the orbits of (6.1) are not periodic: one can check that these orbits densely fill up the torus. This is referred to as quasi-periodic motion. Many systems have a torus (possibly in higher dimensions) as their configuration space. The so-called Control Moment Gyroscopes (CMGs), a control mechanism for satellite reorientation maneuvers, are defined on a two-dimensional torus. Integrable systems, a special subclass of mechanical system including the Euler equations for a free rigid body, can be written in a distinguished coordinate system as the motion of a constant vector field on an n-dimensional torus, where n is the number of degrees of freedom. Higher Dimensions. In dimension three or higher, orbits may approach a very complicated limit set known as a strange attractor, which is characterized by a non-integer dimension and the fact that the dynamics on it are sensitive to initial conditions. In other words, chaos occurs. A celebrated example of a strange attractor is the Lorentz attractor. Maps. There exists nothing even close to the Poincaré-Bendixson theorem for discretetime dynamical systems, i.e. maps. The celebrated Baker s map is a map from the unit square into itself with uncountably many chaotic orbits. References Hirsch, M. W., S. Smale, and R. L. Devaney [2004], Differential equations, dynamical systems, and an introduction to chaos, volume 60 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, second edition. Perko, L. [2001], Differential equations and dynamical systems, volume 7 of Texts in Applied Mathematics. Springer-Verlag, New York, third edition. Strogatz, S. [1994], Nonlinear Dynamics and Chaos. Addison-Wesley. Verhulst, F. [1996], Nonlinear differential equations and dynamical systems. Universitext. Springer-Verlag, Berlin, second edition.

### 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

### 3 Stability and Lyapunov Functions

CDS140a Nonlinear Systems: Local Theory 02/01/2011 3 Stability and Lyapunov Functions 3.1 Lyapunov Stability Denition: An equilibrium point x 0 of (1) is stable if for all ɛ > 0, there exists a δ > 0 such

### Lecture 5. Outline: Limit Cycles. Definition and examples How to rule out limit cycles. Poincare-Bendixson theorem Hopf bifurcations Poincare maps

Lecture 5 Outline: Limit Cycles Definition and examples How to rule out limit cycles Gradient systems Liapunov functions Dulacs criterion Poincare-Bendixson theorem Hopf bifurcations Poincare maps Limit

### 1 Lyapunov theory of stability

M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability

### 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

### 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

### WHAT IS A CHAOTIC ATTRACTOR?

WHAT IS A CHAOTIC ATTRACTOR? CLARK ROBINSON Abstract. Devaney gave a mathematical definition of the term chaos, which had earlier been introduced by Yorke. We discuss issues involved in choosing the properties

### 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

### Notes for Expansions/Series and Differential Equations

Notes for Expansions/Series and Differential Equations In the last discussion, we considered perturbation methods for constructing solutions/roots of algebraic equations. Three types of problems were illustrated

### 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

### 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

### MATH 415, WEEKS 14 & 15: 1 Recurrence Relations / Difference Equations

MATH 415, WEEKS 14 & 15: Recurrence Relations / Difference Equations 1 Recurrence Relations / Difference Equations In many applications, the systems are updated in discrete jumps rather than continuous

### An introduction to Birkhoff normal form

An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an

### Lyapunov Stability Theory

Lyapunov Stability Theory Peter Al Hokayem and Eduardo Gallestey March 16, 2015 1 Introduction In this lecture we consider the stability of equilibrium points of autonomous nonlinear systems, both in continuous

### A short introduction with a view toward examples. Short presentation for the students of: Dynamical Systems and Complexity Summer School Volos, 2017

A short introduction with a view toward examples Center of Research and Applications of Nonlinear (CRANS) Department of Mathematics University of Patras Greece sanastassiou@gmail.com Short presentation

### 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

### 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

### The Liapunov Method for Determining Stability (DRAFT)

44 The Liapunov Method for Determining Stability (DRAFT) 44.1 The Liapunov Method, Naively Developed In the last chapter, we discussed describing trajectories of a 2 2 autonomous system x = F(x) as level

### 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

### (x k ) sequence in F, lim x k = x x F. If F : R n R is a function, level sets and sublevel sets of F are any sets of the form (respectively);

STABILITY OF EQUILIBRIA AND LIAPUNOV FUNCTIONS. By topological properties in general we mean qualitative geometric properties (of subsets of R n or of functions in R n ), that is, those that don t depend

### Introduction to Dynamical Systems Basic Concepts of Dynamics

Introduction to Dynamical Systems Basic Concepts of Dynamics A dynamical system: Has a notion of state, which contains all the information upon which the dynamical system acts. A simple set of deterministic

### Chaotic motion. Phys 750 Lecture 9

Chaotic motion Phys 750 Lecture 9 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to

### Some Background Material

Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important

### We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero

Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.

### 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

### 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

### ENGI 9420 Lecture Notes 4 - Stability Analysis Page Stability Analysis for Non-linear Ordinary Differential Equations

ENGI 940 Lecture Notes 4 - Stability Analysis Page 4.01 4. Stability Analysis for Non-linear Ordinary Differential Equations A pair of simultaneous first order homogeneous linear ordinary differential

### 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

### Oscillations in Damped Driven Pendulum: A Chaotic System

International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 3, Issue 10, October 2015, PP 14-27 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org Oscillations

### Dynamical Systems and Ergodic Theory PhD Exam Spring Topics: Topological Dynamics Definitions and basic results about the following for maps and

Dynamical Systems and Ergodic Theory PhD Exam Spring 2012 Introduction: This is the first year for the Dynamical Systems and Ergodic Theory PhD exam. As such the material on the exam will conform fairly

### Part 2 Continuous functions and their properties

Part 2 Continuous functions and their properties 2.1 Definition Definition A function f is continuous at a R if, and only if, that is lim f (x) = f (a), x a ε > 0, δ > 0, x, x a < δ f (x) f (a) < ε. Notice

### Math 312 Lecture Notes Linear Two-dimensional Systems of Differential Equations

Math 2 Lecture Notes Linear Two-dimensional Systems of Differential Equations Warren Weckesser Department of Mathematics Colgate University February 2005 In these notes, we consider the linear system of

### Topic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis

Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of

### Nonlinear Systems Theory

Nonlinear Systems Theory Matthew M. Peet Arizona State University Lecture 2: Nonlinear Systems Theory Overview Our next goal is to extend LMI s and optimization to nonlinear systems analysis. Today we

### BIBO STABILITY AND ASYMPTOTIC STABILITY

BIBO STABILITY AND ASYMPTOTIC STABILITY FRANCESCO NORI Abstract. In this report with discuss the concepts of bounded-input boundedoutput stability (BIBO) and of Lyapunov stability. Examples are given to

### Perturbation theory for anharmonic oscillations

Perturbation theory for anharmonic oscillations Lecture notes by Sergei Winitzki June 12, 2006 Contents 1 Introduction 1 2 What is perturbation theory 1 21 A first example 1 22 Limits of applicability

### Definition 5.1. A vector field v on a manifold M is map M T M such that for all x M, v(x) T x M.

5 Vector fields Last updated: March 12, 2012. 5.1 Definition and general properties We first need to define what a vector field is. Definition 5.1. A vector field v on a manifold M is map M T M such that

### THEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)

4 Vector fields Last updated: November 26, 2009. (Under construction.) 4.1 Tangent vectors as derivations After we have introduced topological notions, we can come back to analysis on manifolds. Let M

### Physics 106b: Lecture 7 25 January, 2018

Physics 106b: Lecture 7 25 January, 2018 Hamiltonian Chaos: Introduction Integrable Systems We start with systems that do not exhibit chaos, but instead have simple periodic motion (like the SHO) with

### Multiple scale methods

Multiple scale methods G. Pedersen MEK 3100/4100, Spring 2006 March 13, 2006 1 Background Many physical problems involve more than one temporal or spatial scale. One important example is the boundary layer

### 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

### 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.

### 28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)

28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.

### Nonlinear Control Lecture 1: Introduction

Nonlinear Control Lecture 1: Introduction Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2011 Farzaneh Abdollahi Nonlinear Control Lecture 1 1/15 Motivation

### Numerical Algorithms as Dynamical Systems

A Study on Numerical Algorithms as Dynamical Systems Moody Chu North Carolina State University What This Study Is About? To recast many numerical algorithms as special dynamical systems, whence to derive

### Poincaré`s Map in a Van der Pol Equation

International Journal of Mathematical Analysis Vol. 8, 014, no. 59, 939-943 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.014.411338 Poincaré`s Map in a Van der Pol Equation Eduardo-Luis

### LINEAR CHAOS? Nathan S. Feldman

LINEAR CHAOS? Nathan S. Feldman In this article we hope to convience the reader that the dynamics of linear operators can be fantastically complex and that linear dynamics exhibits the same beauty and

### 4 Stability analysis of finite-difference methods for ODEs

MATH 337, by T. Lakoba, University of Vermont 36 4 Stability analysis of finite-difference methods for ODEs 4.1 Consistency, stability, and convergence of a numerical method; Main Theorem In this Lecture

### The Existence of Chaos in the Lorenz System

The Existence of Chaos in the Lorenz System Sheldon E. Newhouse Mathematics Department Michigan State University E. Lansing, MI 48864 joint with M. Berz, K. Makino, A. Wittig Physics, MSU Y. Zou, Math,

### CONSTRUCTION OF THE REAL NUMBERS.

CONSTRUCTION OF THE REAL NUMBERS. IAN KIMING 1. Motivation. It will not come as a big surprise to anyone when I say that we need the real numbers in mathematics. More to the point, we need to be able to

### Report E-Project Henriette Laabsch Toni Luhdo Steffen Mitzscherling Jens Paasche Thomas Pache

Potsdam, August 006 Report E-Project Henriette Laabsch 7685 Toni Luhdo 7589 Steffen Mitzscherling 7540 Jens Paasche 7575 Thomas Pache 754 Introduction From 7 th February till 3 rd March, we had our laboratory

### Damped harmonic motion

Damped harmonic motion March 3, 016 Harmonic motion is studied in the presence of a damping force proportional to the velocity. The complex method is introduced, and the different cases of under-damping,

### 7 Pendulum. Part II: More complicated situations

MATH 35, by T. Lakoba, University of Vermont 60 7 Pendulum. Part II: More complicated situations In this Lecture, we will pursue two main goals. First, we will take a glimpse at a method of Classical Mechanics

### PHY411 Lecture notes Part 4

PHY411 Lecture notes Part 4 Alice Quillen February 1, 2016 Contents 0.1 Introduction.................................... 2 1 Bifurcations of one-dimensional dynamical systems 2 1.1 Saddle-node bifurcation.............................

### The Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III)

Calculus Vector Calculus (III) Outline 1 The Divergence Theorem 2 Stokes Theorem 3 Applications of Vector Calculus The Divergence Theorem (I) Recall that at the end of section 12.5, we had rewritten Green

February 1, 2005 INTRODUCTION TO p-adic NUMBERS JASON PRESZLER 1. p-adic Expansions The study of p-adic numbers originated in the work of Kummer, but Hensel was the first to truly begin developing the

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### Orbital Motion in Schwarzschild Geometry

Physics 4 Lecture 29 Orbital Motion in Schwarzschild Geometry Lecture 29 Physics 4 Classical Mechanics II November 9th, 2007 We have seen, through the study of the weak field solutions of Einstein s equation

### Two-Body Problem. Central Potential. 1D Motion

Two-Body Problem. Central Potential. D Motion The simplest non-trivial dynamical problem is the problem of two particles. The equations of motion read. m r = F 2, () We already know that the center of

### z x = f x (x, y, a, b), z y = f y (x, y, a, b). F(x, y, z, z x, z y ) = 0. This is a PDE for the unknown function of two independent variables.

Chapter 2 First order PDE 2.1 How and Why First order PDE appear? 2.1.1 Physical origins Conservation laws form one of the two fundamental parts of any mathematical model of Continuum Mechanics. These

### Observability for deterministic systems and high-gain observers

Observability for deterministic systems and high-gain observers design. Part 1. March 29, 2011 Introduction and problem description Definition of observability Consequences of instantaneous observability

### 16. Local theory of regular singular points and applications

16. Local theory of regular singular points and applications 265 16. Local theory of regular singular points and applications In this section we consider linear systems defined by the germs of meromorphic

### 27. Topological classification of complex linear foliations

27. Topological classification of complex linear foliations 545 H. Find the expression of the corresponding element [Γ ε ] H 1 (L ε, Z) through [Γ 1 ε], [Γ 2 ε], [δ ε ]. Problem 26.24. Prove that for any

### MORE ON CONTINUOUS FUNCTIONS AND SETS

Chapter 6 MORE ON CONTINUOUS FUNCTIONS AND SETS This chapter can be considered enrichment material containing also several more advanced topics and may be skipped in its entirety. You can proceed directly

### Introduction to the Phase Plane

Introduction to the Phase Plane June, 06 The Phase Line A single first order differential equation of the form = f(y) () makes no mention of t in the function f. Such a differential equation is called

### Modelling biological oscillations

Modelling biological oscillations Shan He School for Computational Science University of Birmingham Module 06-23836: Computational Modelling with MATLAB Outline Outline of Topics Van der Pol equation Van

### Chapter 8. P-adic numbers. 8.1 Absolute values

Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.

### = 0. = q i., q i = E

Summary of the Above Newton s second law: d 2 r dt 2 = Φ( r) Complicated vector arithmetic & coordinate system dependence Lagrangian Formalism: L q i d dt ( L q i ) = 0 n second-order differential equations

### Lebesgue Measure. Dung Le 1

Lebesgue Measure Dung Le 1 1 Introduction How do we measure the size of a set in IR? Let s start with the simplest ones: intervals. Obviously, the natural candidate for a measure of an interval is its

### 1 The Observability Canonical Form

NONLINEAR OBSERVERS AND SEPARATION PRINCIPLE 1 The Observability Canonical Form In this Chapter we discuss the design of observers for nonlinear systems modelled by equations of the form ẋ = f(x, u) (1)

### Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) a n z n. n=0

Lecture 22 Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) Recall a few facts about power series: a n z n This series in z is centered at z 0. Here z can

### CALCULATION OF NONLINEAR VIBRATIONS OF PIECEWISE-LINEAR SYSTEMS USING THE SHOOTING METHOD

Vietnam Journal of Mechanics, VAST, Vol. 34, No. 3 (2012), pp. 157 167 CALCULATION OF NONLINEAR VIBRATIONS OF PIECEWISE-LINEAR SYSTEMS USING THE SHOOTING METHOD Nguyen Van Khang, Hoang Manh Cuong, Nguyen

### Series Solutions. 8.1 Taylor Polynomials

8 Series Solutions 8.1 Taylor Polynomials Polynomial functions, as we have seen, are well behaved. They are continuous everywhere, and have continuous derivatives of all orders everywhere. It also turns

### Numerical Sequences and Series

Numerical Sequences and Series Written by Men-Gen Tsai email: b89902089@ntu.edu.tw. Prove that the convergence of {s n } implies convergence of { s n }. Is the converse true? Solution: Since {s n } is

### 1 Random walks and data

Inference, Models and Simulation for Complex Systems CSCI 7-1 Lecture 7 15 September 11 Prof. Aaron Clauset 1 Random walks and data Supposeyou have some time-series data x 1,x,x 3,...,x T and you want

### Lecture2 The implicit function theorem. Bifurcations.

Lecture2 The implicit function theorem. Bifurcations. 1 Review:Newton s method. The existence theorem - the assumptions. Basins of attraction. 2 The implicit function theorem. 3 Bifurcations of iterations.

### Topic 6: Projected Dynamical Systems

Topic 6: Projected Dynamical Systems John F. Smith Memorial Professor and Director Virtual Center for Supernetworks Isenberg School of Management University of Massachusetts Amherst, Massachusetts 01003

### 1.3.1 Definition and Basic Properties of Convolution

1.3 Convolution 15 1.3 Convolution Since L 1 (R) is a Banach space, we know that it has many useful properties. In particular the operations of addition and scalar multiplication are continuous. However,

### 3 The language of proof

3 The language of proof After working through this section, you should be able to: (a) understand what is asserted by various types of mathematical statements, in particular implications and equivalences;

### Infinite series, improper integrals, and Taylor series

Chapter 2 Infinite series, improper integrals, and Taylor series 2. Introduction to series In studying calculus, we have explored a variety of functions. Among the most basic are polynomials, i.e. functions

### 6.2 Brief review of fundamental concepts about chaotic systems

6.2 Brief review of fundamental concepts about chaotic systems Lorenz (1963) introduced a 3-variable model that is a prototypical example of chaos theory. These equations were derived as a simplification

### Removing the Noise from Chaos Plus Noise

Removing the Noise from Chaos Plus Noise Steven P. Lalley Department of Statistics University of Chicago November 5, 2 Abstract The problem of extracting a signal x n generated by a dynamical system from

### Extremum Seeking with Drift

Extremum Seeking with Drift Jan aximilian ontenbruck, Hans-Bernd Dürr, Christian Ebenbauer, Frank Allgöwer Institute for Systems Theory and Automatic Control, University of Stuttgart Pfaffenwaldring 9,

### To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.

Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function

### 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

### 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

### 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

### BIFURCATIONS OF PERIODIC ORBITS IN THREE-WELL DUFFING SYSTEM WITH A PHASE SHIFT

J Syst Sci Complex (11 4: 519 531 BIFURCATIONS OF PERIODIC ORBITS IN THREE-WELL DUFFING SYSTEM WITH A PHASE SHIFT Jicai HUANG Han ZHANG DOI: 1.17/s1144-1-89-3 Received: 9 May 8 / Revised: 5 December 9

### Solutions of Math 53 Midterm Exam I

Solutions of Math 53 Midterm Exam I Problem 1: (1) [8 points] Draw a direction field for the given differential equation y 0 = t + y. (2) [8 points] Based on the direction field, determine the behavior

### Topic 4 Notes Jeremy Orloff

Topic 4 Notes Jeremy Orloff 4 auchy s integral formula 4. Introduction auchy s theorem is a big theorem which we will use almost daily from here on out. Right away it will reveal a number of interesting

### Large Deviations for Small-Noise Stochastic Differential Equations

Chapter 22 Large Deviations for Small-Noise Stochastic Differential Equations This lecture is at once the end of our main consideration of diffusions and stochastic calculus, and a first taste of large

### Cyclicity of common slow fast cycles

Available online at www.sciencedirect.com Indagationes Mathematicae 22 (2011) 165 206 www.elsevier.com/locate/indag Cyclicity of common slow fast cycles P. De Maesschalck a, F. Dumortier a,, R. Roussarie

### 11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes

Parametric Differentiation 11.6 Introduction Sometimes the equation of a curve is not be given in Cartesian form y f(x) but in parametric form: x h(t), y g(t). In this Section we see how to calculate the

### Period Three, Chaos and Fractals

Imperial, May 2012 The Sarkovskii Theorem Let us motivate the so-called Sarkovskii ordering on N 3 5 7 9 2 3 2 5 8 4 2 1. Let f : [0, 1] [0, 1] be continuous and consider successive iterates x n+1 = f

### Why are Discrete Maps Sufficient?

Why are Discrete Maps Sufficient? Why do dynamical systems specialists study maps of the form x n+ 1 = f ( xn), (time is discrete) when much of the world around us evolves continuously, and is thus well

### Large Deviations for Small-Noise Stochastic Differential Equations

Chapter 21 Large Deviations for Small-Noise Stochastic Differential Equations This lecture is at once the end of our main consideration of diffusions and stochastic calculus, and a first taste of large

### The fundamental theorem of calculus for definite integration helped us to compute If has an anti-derivative,

Module 16 : Line Integrals, Conservative fields Green's Theorem and applications Lecture 47 : Fundamental Theorems of Calculus for Line integrals [Section 47.1] Objectives In this section you will learn

### means is a subset of. So we say A B for sets A and B if x A we have x B holds. BY CONTRAST, a S means that a is a member of S.

1 Notation For those unfamiliar, we have := means equal by definition, N := {0, 1,... } or {1, 2,... } depending on context. (i.e. N is the set or collection of counting numbers.) In addition, means for