ACTA
WASAENSIA
No 55
MATHEMATICS No 7
MATTI LAAKSONEN
A Nice Portrait of Restless Systems — Uniqueness of the Limit Cycle
of Some Li´enard Equations
UNIVERSITAS WASAENSIS
VAASA 1997
Editorial Board:
Kauko Mikkonen, Petri Julkunen, Juha-Pekka Kallunki, Teija Laitinen, Jorma Larimo, Vuokko Palonen, Juhani Stenfors, Pirkko Vartiainen
Editor:
Antero Niemikorpi Assistant editor:
Tarja Salo Address:
Vaasan Yliopisto University of Vaasa P.O. Box 700 (Wolffintie 34)
65101 VAASA Finland
ISBN 951-683-????-?
c Vaasan yliopisto
Contents
ABSTRACT 5
1. INTRODUCTION 7
1.1. Features of the restless bounded system 7
1.2. The objectives and the structure of the thesis 13
2. THE FLOW OF THE DYNAMICAL SYSTEM 15
2.1. Flow, orbits and limit sets 15
2.2. Stability and robustness 19
2.3. Limit Cycles in IR2 20
2.4. Linear systems 26
2.5. The critical points of a nonlinear system 31
3. LIMIT CYCLES OF THE LI ´ENARD EQUATION 33
3.1. The Li´enard equation 33
3.2. The critical point of the model 35
3.3. The existence of the limit cycle 37
3.4. The uniqueness of the limit cycle 45
3.5. The periodic solutions 49
4. COMPARATIVE ANALYSIS 54
4.1. Intersecting orbits 54
4.2. The majorant principle 55
4.3. The contraction of the system 60
4.4. The ∩∪-Function 63
4.5. The uniqueness of the limit cycle revisited 66
5. APPLICATIONS 67 5.1. Two schemes of the application leading to the simple nice model 67
5.2. Business cycle models 69
5.3. Samuelson’s original discrete model 73
5.4. A continuous form of Samuelson’s model 75
5.5. Nonlinear modification of the model 77
6. CONCLUSIONS 81
ACKNOWLEDGEMENTS 82
REFERENCES 82
Abstract
Laaksonen, Matti (1997). A Nice Portrait of Restless Systems — Uniqueness of the Limit Cycle of Some Li´enard Equations. Acta WasaensiaNo 57, 83p.
In this thesis we consider a simple family of Li´enard equations ¨x + f(x)x˙ + x = 0, generated by smooth functions f. Some theorems for the existence and the uniqueness of the limit cycles of the systems are presented. If f and g are two generating functions with unique limit cycles and x 0xf(z)dz < x 0xg(z)dz, ∀x = 0, then according to the ‘bounding theorem’ proved in the paper, the limit cycle of the system generated byf is bounded by the limit cycle of the system generated by g. This gives us a method to estimate the amplitude of the oscillations also for the systems for which we do not know the generating function exactly.
Also a new class of functions called N-functions (or ∩∪-function) are introduced. If the generating function is a ∩∪-function with three zeros, then the system has one and only one stable periodic solution. This is a new result and very useful because of the lack of the symmetry assumption.
As an application we consider the nonlinear business cycle model pro- posed by T¨onu Puu (1989).
Matti Laaksonen, University of Vaasa, P.O. Box 700 (Wolffintie 34), FIN-65101 VAASA, Finland.
Key words
limit cycle, Li´enard equation, non-linear dynamics, business cycle theory.
.
1. Introduction
1.1 Features of the restless bounded system
Dynamical systems are mathematical models for the corresponding real systems. We observe the real system by making measurements. After some observing we identify essential measurable features of the system called variables. The values of variables, x1, x2, x3, . . ., vary in time or at least they can vary. The values of the variable form the vector we call the state of the system. Other measures called parameters are constants during the time interval of observations. At most they fluctuate slowly and thus they can be taken as constants. It is not trivial to distinguish parameters and variables.
Often one greatly simplifies the model by taking a variable as a parameter with a constant value.
From the observations we try to find permanent relations between the values of the variables and the speeds of change of the values of the variables. Relation on variables restrict the set of possible states. Relations on the values of the variables and the values of the time derivatives of the variables describe the change of the state and are called the laws of the motion.
In the modeling of the dynamical system we usually construct a first or a second order difference or differential equation. If the laws of the motion are known (as in the case of a moving body in physics), then the modeling can be a trivial task. Even in physics we must most of the time make simplifying assumptions. We reject friction, or
some less relevant effects like the one caused by the motion of the earth in the solar system. The main reason for making a simple model is to get a solvable model.
In other fields, for instance in economy, the laws of the motion are not obvious.
We demand that the relations between the variables in the model are reasonable and the model has predictive power at least in the short run. We also expect to be able to estimate the parameters of the model. We also expect the model to have the same features in its behavior as the real system.
The basic features of the system includes equilibrium states and periodically re- peated forms of the behavior. The stability of the behavior and the limit behavior which the system is tending to are also important features. If the equilibrium states are unstable, then the system is restless in the sense that the limit behavior is not an equilibrium.
As an example let us consider a system for which we have identified two basic variablesx1, x2. At the time t =ti the observed values are xj(ti),j = 1,2. The double x(ti) = (x1(ti), x2(ti)) ∈ IR2 is the state of the system and the set of possible states, IR2, is thestate space. We make a visual representation of the system by drawing the observed states x(ti)∈IR2 of the observations,i= 0,1,2, . . . , n.
The system is periodic if there exists a period T for which x(i+T) = x(i). The variable xj repeats the same form of behavior after the period T (See Figure 1(a)). In the state space the pointx(t) of the state comes back to the starting point afterT steps (see Figure 1(b)). Often the behavior of the real system is not strictly periodic but we still use periodic mathematical models to describe the system because periodic models are good enough and easy to handle and maybe the real system without external shocks would also tend to periodic forms.
After a large shock, some systems return to periodic behavior. In Figure 2 is drawn an orbit after a shock. After some time, if there will be no more shocks, the orbit will be near the solid circle drawn in Figure 2(b). We call the circle the limit set of the motion.
The orbit in Figure 2(b) starts at pointx(0). With a different starting point we get a different orbit, but normally they both have a common limit set. In Figure 3 we have
a)
t x t1( )
b)
x 0( )x 1( ) x 2( )
( ) = ( , ) x t x t1( ) x t2( ) ( ) x t1 ( )
x t2
Figure 1. Two ways to express the periodic behavior: a) x1(t) as a function of time, b) the orbit in the state space IR2.
a)
t x t1( )
b)
x 2( )
( ) = ( , ) x t x t1( ) x t2( ) x t2( )
( ) x t1 x 0( ) x 1( )
Figure 2. Periodic behavior after a shock: a)x1(t) as a function of time, b) the orbit in the state space IR2.
a)
x t1( )
t
b) x t1( )
x t2( )
Figure 3. Several solutions of the same system with different starting states. a) Functions of time, b) the phase portrait.
several orbits with different starting points. Figure 3(a) is quite useless while Figure 3(b) well describes theflow of the dynamics. It is called the phase portrait. The phase portrait in Figure 3(b) shows the attracting nature of the limit set. All orbits starting near the limit set tend to the limit set. We say the limit set in Figure 3(b) is stable because the behavior after a small shock will return to the same limit set.
If the system meets small disturbances repeatedly then the orbit never settles down on the limit set. The system is stochastic and we can not make exact predictions of the future state of the system. On the other hand, if the limit set of the underlying flow is stable, then the expected value of the state will stay near the limit set. So the limit behavior is near the expected behavior. In Figure 4 we have a simulated orbit in the phase space of a stochastic system.
A critical point of the flow is a point where the motion of the flow stops. If the critical point is stable, all orbits with a starting point near the critical point will tend to the stable critical point. In a stochastic system the orbit will stay near the stable critical point. In this sense the stable critical point represents the equilibrium state (deterministic or stochastic) where the system will rest its future after the transient period at the beginning.
The system (especially the stochastic one) will not rest at an unstable critical point.
If all critical points are unstable then the system is called restless. The orbits in Figures 1—4 are restless and the orbits in Figures 5 and 6 are restful. So the restlessness is a property of the underlying flow and we must distinguish restlessness and stochastic noisiness.
When we make mathematical models for real systems, the basic features of the underlyingflow are.
• Critical points and their stability.
• Limit sets.
• Periodic orbits and their stability.
• The number of stable periodic orbits.
a)
x t1( )
t
b) x t1( )
x t2( )
Figure 4. A stochastic orbit on restlessflow.
a)
t x t1( )
b)
( ) x t2
( ) x t1
Figure 5. Silent (deterministic) orbit tending to the stable critical point.
a)
t x t1( )
b)
( ) x t2
x t1( )
Figure 6. Noisy (stochastic) orbit tending to the neighborhood of the stable critical point.
In the laws of the motion of the flow there are constant parameters, which have their corresponding measures in the real world (e.g. the rate of interest in finance, the density in physics, etc.) If the value of the parameter changes, then theflow deforms.
If the small change in the parameter does not cause a drastic change in the qualitative list of the features of the flow, then the value of the parameter is robust. If all the parameters have robust values, then the flow is robust. A robust flow is not sensitive to the values of the parameters.
A robust flow which has one and only one stable periodic orbit is not sensitive to the disturbances on the variables and on the parameters. Thus we call it nice.
From empirical observations of the real system, we can identify only the present stable structures. The mathematical model used to describe the real system must have the same stable features as the real system we observe to have. The rest of the features of the flow (mostly unstable structures and robustness) are a matter of taste and reasoning. One policy for a restless system is to keep the flow nice and simple.
Let y ∈ IR. The linear autonomous n:th order differential equation with constant coefficients is
y(n)+cn−1y(n−1)+. . .+c2y +c1y +c0y = 0, (1.1)
whereck,k = 1,2, . . . , n−1 are real constants. Let us denoteyj =y(j). Theny˙j =yj+1, for j = 0,1, . . . , n−2. Then the equation (1.1) is equivalent to
˙ y0
˙ y1
˙ y2
...
˙
yn−2y˙n−1
=
0 1 0 0 · · · 0 0 0 1 0 · · · 0 0 0 0 1 · · · 0 ... I ... 0 0 0 0 · · · 1 c0 c1 c2 c3 · · · cn−1
y0
y1
y2
... yn−2
yn−1
(1.2)
Lety= (y0, y1, . . . , yn−1) ∈IRn. The matrix equation
˙
y=Ay, (1.3)
where A is a n×n-matrix, is a first order linear differential equation for n variables.
In the next chapter we show for two dimensional case that there is no stable periodic
orbit of the system (1.3). The result is easily generalized to a higher dimension. So in any dimension a nice restless autonomousflow must be nonlinear.
In this thesis we restrict ourselves to two dimensional restless time invariant systems for which we construct the continuous flow by an ordinary differential equation. The analytic tools we are using work only in the two dimensional case. The research of higher dimensional nonlinear modeling is an apocalyptic adventure which maybe will never be accomplished.
When we are searching for simple and nice two-dimensional models the natural candidates are
¨
x+x˙ +f(x) = 0 (1.4)
¨
x+f(x) +˙ x = 0 (1.5)
¨
x+x˙ +f(x)x˙ = 0 (1.6)
¨
x+f(x)x˙+x = 0 (1.7)
where f(x) is a continuous nonlinear real function.
In section 2.5 we will show that (1.4) and (1.6) are not nice and simple , while the flows (1.5) and (1.7) are simple, and they are nice on conditions we analyze in chapters 3 and 4.
Equation (1.7) is the classical Li´enard equation and (1.5) is essentially equivalent to (1.7). They are the natural choice for a nice and simple model of the restless bounded two-dimensional system.
1.2 The objectives and the structure of the thesis
In this thesis we consider the two dimensional continuous flow defined by an ordinary differential equation called the Li´enard equation.
¨
x+xf˙ (x) +x= 0 (1.8)
which is equivalent (topologically conjugate) to
¨
x+F(x) +˙ x= 0 (1.9)
where F(x) = 0xf(t)dt is the generating function of the system.
First we introduce the majorant principle by which we can estimate reliable bounds for the periodic orbit of the unknown system if we know reliable bounds for the gener- ating function of the system. The method of the majorant principle uses simulation to determine the bounds for the orbit. Thus it is reasonable to use models with one and only one stable periodic orbit (i.e. nice models).
Secondly we will show in theorem 4.5.1a sufficient condition for F on which the flow has one and only one stable periodic orbit. The condition is not necessary but it is general and for many applications weak and natural. As an application we consider the business cycle model proposed by T¨onu Puu.
The thesis is organized as follows. In Chapter 2 we introduce the basic concepts of continuous flows on IR2. We prove the well known Poincar´e—Bendixon theorem which we will use in subsequent chapters. We also show that nice and simple models must be nonlinear. From the natural families of models we pick up the nice ones which turn out to be Li´enard equations.
In Chapter 3 we consider theflow of the Li´enard equation. We prove some theorems concerning the sufficient conditions for the existence and uniqueness of the limit cycle.
We also introduce a lower bound for the time period (T ≥2π).
In Chapter 4 we compare two flows to each other. We prove the majorant principle which is thefirst main contribution of this thesis. We also introduce the concept of the N-function, and show that if the generating function is an N-function with three zeros then the system has a unique limit cycle. This is a new result and the second main contribution of this thesis.
In Chapter 5 we use the existence and uniqueness theorems to analyze Puu’s busi- ness cycle model. In Chapter 6 we draw some conclusions on what has been brought to light.
2. The flow of the dynamical system
In this chapter we define several concepts in the n-dimensional framework. In
the first two sections we work out basic definitions. All the concepts will appear in
subsequent parts of this paper and we collect the needed definitions here. In the third section we define the limit cycle, which is the key concept of this thesis.
2.1 Flow, orbits and limit sets
Normally we can describe the state of the dynamical system by afinite set of continuous variables x1, x2, . . . , xn. We call the vector x = (x1, x2, . . . , xn) a state of the system.
The possible states form a phase spaceX as a subset of the Euclidean vector space IRn. Often there are some constraints on the values of the variables so that X is a proper subset of IRn. If there is an equation relating some variables to each other, then X can be a sub manifold of IRn. Thus in the following definitions we take X as a topological space.
Letϕ:X×IR→X be a continuous mapping. For eacht∈IR, we define a mapping ϕt:X →X,ϕt(x) =ϕ(x, t). If these mappings satisfy the following conditions
ϕs◦ϕt = ϕs+t
(2.1)
ϕ0 = 1X
(2.2)
then we say that (X,ϕ) isa continuousflow onX. By definition, the mapϕt:X →X is continuous and from (2.1) and (2.2) we see that ϕ−t1 =ϕ−t. So for each t ∈ IR the map ϕt :X →X is a homeomorphism.
For each x ∈ X, we define a mapping ϕx : IR →X,ϕx(t) = ϕ(x, t). The mapping ϕx is called a motion (or a solution) through x. The image of the motion γ(x) = {ϕx(t)|t ∈ IR} is called the orbit (or the trajectory) throughx. An orbit is nonempty, and two orbits are either identical or mutually disjoint. The family of orbits fills up the phase space X and is called the phase portrait.
The positive orbit starting from x is γ+(x) ={ϕx(t)|t ≥0} and the negative orbit is (mutatis mutandis) γ−(x) = {ϕx(t)|t ≤ 0}. A subset A of X is called invariant if ϕt(A) ⊂ A for all t ∈ IR. A subset A of X is positively (negatively) invariant if ϕt(A)⊂Afor allt≥0 (t≤0). A positively invariant setA is such that every positive orbit γ+(x) starting from a point x∈ Ais a subset of A.
A point x∈X is acritical point(or asingular point, or afixed point) ifγ(x) ={x}. If the point x is not critical, then we call it regular. A point x ∈ X is periodic if there exists a number T = 0 called period such that ϕ(x, t+T) = ϕ(x, t),∀t. Let x1 ∈ γ(x). Then x1 = ϕ(x, t1) for some t ∈ IR. If x is periodic with period T, then ϕ(x1, t+T) =ϕ(x, t+t1+T) =ϕ(x, t+t1) =ϕ(x1, t),∀t, and so all the points ofγ(x) are periodic with period T. If x is periodic with period T, then the motion ϕx and the orbit γ(x) are also said to be periodic with period T. An orbit through a regular periodic point is called a closed orbit. A periodic orbit can not contain any critical point. On plane IR2 we have the following theorem relating closed orbits and critical points.
Theorem 2.1.1 Let γ be a closed orbit in a continuous flow (IR2,ϕ) on the plane.
Then there exists a critical point in the bounded region bordered by γ.
Proof. LetA be the simply connected open region bounded byγ (which exists by the Jordan separation Theorem). Then F0 =A∪γ is a compact set. If the period of γ is T then let t1 < T. The map ϕt1|F0 : F0 → F0 is an automorphism and according to the fixed point theorem there exists a point p1 for which ϕt1(p1) = p1. So p1 is either
a critical or periodic point. p1 ∈/ γ because its period is less than the period of γ. If p1 is critical then let pn=p1,∀n >1, otherwise let A1 be the simply connected region bounded by γ(p1) and F1 = A1 ∪γ(p1). Let t2 = t1/2. The map ϕt2|F1 : F1 → F1
is an automorphism and it has a fixed point p2 ∈F1, where ϕt2(p2) = p2. Continuing the process we get a sequence {p1, p2, . . .} of periodic points. In a compact set F this sequence has an accumulation point p.
Next we show the point p to be critical. Let us make a counter assumption that there exists a t >0 such that ϕt(p) =p. Let U andV be disjoint open neighborhoods of p and ϕt(p), p ∈ U and ϕt(p) ∈ V. The map ϕ : X ×IR → X,(p, t) → ϕt(p) is continuous. Thus there exists an open neighborhood W of p and an open interval I = (τ1,τ2), t∈I, such that W ⊂U andϕ(W ×I)⊂V. Particularly the sets W and ϕ(W ×I) are disjoint. From the sequence of periodic points (p1, p2, . . .) we can choose a point pk such that pk ∈W and its period tk <τ2−τ1. So ntk ∈ I for some integer n. Now pk=ϕntk(pk)∈V, which is a contradiction becauseW andV are disjoint. 2
Let (X,ϕ) be a continuous flow and x ∈ X. A point y ∈ X is called an ω-limit point of x if there exists a sequence {tn} of real numbers such that
tn→ ∞, as n→ ∞, ϕ(x, tn)→y, as n→ ∞. (2.3)
The set ofω-limit points ofxis denoted by ω(x) and is called theω-limit set of x. Let γ(x) be an orbit through the point x and let z = ϕx(τ) ∈ γ(x) be any point of the orbit. Then ϕx(x, tn) → y if and only if ϕx(z, tn−τ) → y. So the ω-limit sets ω(x) and ω(z) are equal, and we can define the ω-limit set of the orbit, denoted by ω(γ), as the ω-limit set of any point of the orbit.
A point y∈X is called anα-limit point ofx if there exists a sequence{tn} of real numbers such that
tn→ −∞, as n→ ∞, ϕ(x, tn)→y, as n→ ∞. (2.4)
The set ofα-limit points ofxis denoted by α(x) and is called theα-limit set ofx. The α-limit set of the orbit, α(γ), is defined in a natural way.
The orbit does not usually reach the ω-limit set. Still it is a good approximation
to the far future of the orbit. In applications where the real system contains random noise (not perhaps included in the model), the ω-limit set is a good approximation even for the near future. Next we consider some properties of limit-sets. We will use these results later on.
Theorem 2.1.2 Let γ be an orbit of the flow (X,ϕ). The sets ω(γ) and α(γ) are closed and invariant.
Proof. Let s ∈ ω(γ) be a accumulation point of ω(γ) and let {Uk} be a decreasing sequence of open neighborhoods of s such that kUk = s. Then for every Uk there exists an ω-limit point pk ∈ Uk. Let p be a point of the orbit γ. Uk is an open neighborhood ofpkand so there exists a sequence of numbers tk1, tk2, . . .→ ∞such that ϕtk
j(p))∈Uk,∀j. We can now construct a new sequence of numbersτk =tknk such that τk+1 > τk,∀k. Then τ1,τ2, . . . → ∞ and ϕτk(p) → s, when k → ∞. But this means thats is anω-limit point. So theω-limit set contains its own accumulation points and is thus closed. By similar arguments we see that α(γ) is closed.
Next we consider the invariance. Let p∈ω(γ) be an arbitrary point of the ω-limit set. So there exist an p0 ∈ γ and an increasing sequence t1, t2, . . . → ∞ such that pk = ϕtk(p0) →p, when k → ∞. Let q = ϕt(p)∈ γ+(p) be an arbitrary point of the positive orbit starting fromp. The mapϕtis continuous. Thus ϕt+tk(p0) =ϕt(pk)→q and q ∈ω(γ) So the positive orbit starting from an ω-limit point stays in the ω-limit set. The invariance of α(γ) is shown by similar arguments. 2
Theorem 2.1.3 If the positive orbit γ+ ⊂ IR2 is bounded, then the ω-limit set is compact, connected and not empty.
Proof. [Verhulst pages 42 - 43] Because ω(γ) ⊂ γ+ ⊂ IR2 is bounded and closed, it is compact. The sequence ϕ1(p),ϕ2(p), . . . has an accumulation point in the bounded compact set and so ω(γ) = ∅. Let p1 and p2 be two ω-limit points and let U and V be two disjoint open sets such that p1 ∈ U and p2 ∈ V. The ω-limit set is connected if we can now show that ω(γ)−(U ∪V) = ∅. Let t1, t2, . . . → ∞ and τ1,τ2, . . .→ ∞
++ ++
++++
.
UV p2
p1
.
++ ++++++
q
.
++
++++
++++ ++ t
τ
ξ
Figure 7. The ω-limit set can not be divided into two components, one in U and the other in V.
be sequences such that ϕtn → p1 as tn → ∞, and ϕτn → p2 as τn → ∞ (see Figure (7)). We can make such sub-sequences that tn <τn< tn+1. Because IR2 is connected, there exists for every n a numberξn such that tn<ξn<τn and ϕξn(p)∈/ U ∪V. The sequence of points {ϕξn(p)} has an accumulation point q such that q /∈ U ∪V. But
q∈ω(γ) and thusω(γ)−(U ∪V) =∅. 2
2.2 Stability and robustness
A nonempty setA⊂X is called stable if every neighborhood ofAcontains a positively invariant neighborhood ofA. A nonempty setA is called asymptotically stable if A is stable and there exists a neighborhood V of A such that ω(x) ⊂A for any x ∈V. If A is stable and ω(x)⊂A for any x∈X, thenA is globally asymptotically stable.
LetA be a nonempty closed invariant set. A is called an attractor if it has an open neighborhood U satisfying the following conditions:
(i) U is positively invariant,
(ii) for each open neighborhood V of A, there is a T > 0 such that ϕt(U)⊂V for allt ≥T.
From (ii) we see that an attractor is asymptotically stable. The basin of attraction of A is the union of all open neighborhoods of A satisfying (i) and (ii).
Let (X,ϕ) be a continuous flow. Are the properties of the behavior of the system
preserved if we replace the mapping ϕ by another mapping ψ near to ϕ? There are several alternatives to define the ”nearness” in the set of possible mappings. The
”nearness” of two behaviors is a more vague concept. In both cases we choose the easiest way to advance.
Let (X,ϕ) and (Y,ψ) be two continuous flows. We say that a homeomorphism h:X →Y is a conjugacy between flows (X,ϕ) and (Y,ψ) if the diagram
X →h Y ϕt ↓ ↓ψt
X →
h Y (2.5)
commutes for all t ∈ IR. This means that ψt◦h = h◦ϕt or h−1 ◦ψt◦h = ϕt. We say that two flows are conjugate if there exists a conjugacy between them. The phase portraits for the conjugateflows are topologically equal. Thus the conjugateflows have topologically similar behavior.
Let (X,ϕ(p))) be a continuousflow such that in the model there exists a parameter p. We say that the value of the parameter p is robust if there exists an open neigh- borhood V of p such that (X,ϕ(p))) and (X,ϕ(q))) are conjugate for all q ∈ V. If the parameter has a robust value, then we say that the model is robust. For a robust model a small change in the value of the parameter does not cause a large change in the behavior of the model.
In the next section we will consider autonomous two-dimensional linear systems. We try tofind out if there is any way to build a linear robust model with an asymptotically stable periodic orbit. To be thorough we will go through all the linear structures.
2.3 Limit Cycles in IR
2In the two-dimensional phase space IR2 the closed orbit splits the phase space into two parts, the interior and the exterior (the Jordan separation theorem). Because orbits cannot intersect, the orbits with the initial point in the interior of a closed orbit stay in the bounded region. Then any infinite sequence of points on the positive orbit must
have a limit point. This kind of argument from the plane topology leads to some results named after Poincar´e and Bendixson.
Definition 2.3.1 If the ω-limit set is a closed orbit, then we call it a limit cycle.
Definition 2.3.2 The smooth curve τ will be called a transversal of the continuous flow (IR2,ϕ) if τ contains ordinary points only and if τ is nowhere a tangent to any orbit of the flow (IR2,ϕ).
By a parameterization h : [0,1] → τ the transversal gets direction. Let x ∈ τ be a point of the transversal and let θ(x) be the angle between the direction of the flow and the direction of transversal at x. θ(x) is positive when the flow penetrates the transversal from the right to the left (according to the parameterization) and the angle is negative when the flow penetrates the transversal from the left to the right. The angle θ(x) is well defined, continuous, nonzero and has same sign at all points ofτ. Lemma 2.3.1 A transversal of the smooth flow in IR2 can intersect a closed orbit at most at one point.
Proof. Letγ be a closed orbit of the smooth flow (IR2,ϕ) and letτ be a transversal of theflow. Letx1 andx2 be two points of intersection of the transversalτ and the closed orbit γ. Because both τ and γ are smooth and τ is not tangent to the flow, we can assume there exists no point of intersection betweenx1 and x2. If the transversal atx1
goes from outside to inside of γ, then at x2 it goes from inside to outside of γ. Then θ(x1) and θ(x2) have different sign, which is a contradiction sinceτ is a transversal. 2
Letτ be a closed transversal andτ0 its interior. Let V ⊂τ0 be the set of the points x for which there exists tx such that ϕx(tx)∈ τ0 and ϕx(t) ∈/ τ0 for all 0< t < tx. (If x1 =x and x2 =ϕx(tx), then x2 is the subsequent point of intersection of γ+(x1) and τ0 after the initial pointx1. If the orbit and the transversal intersect several times we get the corresponding sequencesx1, x2, x3, . . .andt1, t2, t3, . . ., wheretn=Σnk=1txk with xk+1 = ϕx1(tk) ∈ τ0. (See Figure 8.) Let h : [0,1] → τ be a homeomorphism. If the
τ x1
x2 x3
h-1( )x3 I h-1( )x1 h-1( )x2
h :I τ
ϕ
Figure 8. Successive intersections of the orbit and the transversal.
τ x
x ( ) =
ψ ϕ
x( ) t
xFigure 9. The Poincar´e map.
sequence of real numbers h−1(x1), h−1(x2), h−1(x3), . . .is monotonic then the sequence x1, x2, x3, . . .of points of the transversal is monotonic.
The flow defines the map ψ :V →τ0, x→ϕx(tx) which we call the Poincar´e-map.
(See Figure 9.)
Lemma 2.3.2 Let (IR2,ϕ) be a continuous flow and τ a transversal to the flow. Let V be the domain of the Poincar´e-map ψ : V → τ. If x ∈ V then the sequence x,ψ(x),ψ2(x), . . . is monotonic or ψ(x) =x (i.e. γ(x) is a periodic orbit).
Proof. Let h : [0,1] → τ be a homeomorphism. Let us denote x1 = x, x2 = ψ(x), x3 =ψ2(x) andw1 =h−1(x1),w2 =h−1(x2) and w3 =h−1(x3).
First we assumew1 < w2. Ifw1> w2we use homeomorphism ˜h(w) =h(1−w). The homeomorphism h maps the interval [w1, w2] onto the arc [x1, x2]τ of the transversal fromx1 to x2. We must now show thatw3 > w2.
As the transversalτ is nowhere tangent to theflow, all orbits intersectingτ intersect it in the same direction. The arc of the orbit fromx1tox2 and the arc of the transversal fromx2 tox1 form a closed curveΓwith the bounded regionGi inside and unbounded region Go outside. (Γ is not an orbit of the flow.) On the arc [x1, x2]τ the direction of intersection is either from inside to outside or from outside to inside. The positive orbitγ+(x2) can not intersect Γbecause it can not intersect itself (arc [x1, x2]γ) and it can not go back in the opposite direction through [x2, x1]τ. Thus x3 is not betweenx1
and x2.
If x3 = x2 then x1 = ψ−1(x2) = ψ−1(x3) = x2. This is a contradiction to the assumption w1 < w2. If x3 = x1, then x1 is a periodic point and γ(x1) is a periodic orbit. According to the lemma (2.3.1) the transversal τ intersects γ(x1) only at one point. So x2 =x1, which is again a contradiction.
Considering the inverse flow, we see in a similar way that x1 is not between x2 and x3. We did assume w1 < w2 and we did exclude the relations w3 = w1, w3 = w2, w1 < w3 < w2 and w3 < w1 < w2. So the only possible relation left is w1 < w2 < w3. The whole sequence is monotonic by induction.
If w1=w2, then ψ(x) =x2 =x1 =x. 2
Theorem 2.3.3 The ω-limit set of an orbit γ(x) with an initial point x can intersect the interior τ0 of a transversal τ at one point only. If there exists such a point x0, there exists a sequence tk→ ∞ such that ϕx(tk)→x0 monotonically inτ0, or we have x0 ∈γ(x) with γ(x) a cycle corresponding with a periodic solution.
Proof. Suppose that ω(γ(x)) intersects τ0 at x0. If x0 ∈/ γ(x) then there exists a sequence {tn}, tn → ∞ as n → ∞, such that ϕx(tn)→ x0 as n→ ∞. Let {Uk}, k = 1,2, . . ., be a decreasing sequence of open neighborhoods of x0 such that Uk+1 ⊂ Uk
and kUk = x0. Because the transversal τ is nowhere a tangent to the smooth flow
(IR2,ϕ) and x0 is not a critical point, we can choose a subsequence {tk = tnk} and corresponding points of the orbit xk = ϕx(tk) such that the connected component of Uk∩γ(xk) containing xk intersects the transversal at one point which we denote as xk =ϕx(tk). We can make the subsequence so that tk+1 > tk,∀k. By construction we have xk ∈ Uk and so xk → x0 on τ0 and tk → ∞ as k → ∞. By lemma (2.3.2) the sequence {xk} is monotonic.
If x0 ∈ γ(x) we can make the previous construction with x1 = x0. The sequence {xk} is monotonic and tends to x0 =x1. This is possible only if xk =x0,∀k. So x0 is a periodic point and γ is a periodic orbit.
As a last step we show thatx0 is the only point of intersection of the transversalτ0
and the ω-limit set ω(γ(x)). Let y0 be another point of intersection. Then by similar construction we get a sequence {xk} on τ0 such that the odd points x1, x3, . . .tend to x0 and the even points x2, x4, . . . tend to y0. According to lemma (2.3.2) the sequence {xk}is monotonic, which is possible only if y0 =x0. 2
Theorem 2.3.4 If ω(γ) contains a periodic orbit γ0, then we have ω(γ) =γ0.
Proof. Suppose γ1 =ω(γ)\γ0 is not empty.
First we show there exists a point x0 ∈ γ0 which is an accumulation point of γ1
(i.e. every open neighborhood Ux0 of x0 contains a point of γ1.) Suppose there exists nox0. Then every point x∈γ0 has a neighborhood Ux disjoint ofγ1. These open sets make a cover of γ0. γ0 is bounded and compact. So we can choose a finite subcover {Uxj|j = 1,2, . . . , n}of γ0. The set jUxj is an open neighborhood of γ0 and disjoint of γ1. This is a contradiction because ω(γ) is connected.
So let x0 ∈ γ0 be an accumulation point of γ1 and τ0 an open transversal which contains x0. For x1 ∈γ1 sufficiently near x0 the orbit γ(x1) intersects the transversal τ0. According to the theorem (2.3.3) this can happen only at one point x0 and so x1 ∈γ0. From this contradiction we conclude γ1 =ω(γ)\γ0 is empty. 2
Theorem 2.3.5 Ifx0 ∈γ∩ω(γ)is not critical thenγ is a periodic orbit andω(γ) =γ.
Proof. Follows from the two previous theorems. 2
Definition 2.3.3 A set M is called a minimal set of the system(X,ϕ) ifM is closed, invariant, not empty and if M has no proper subset with these three properties.
Theorem 2.3.6 Suppose thatF is a nonempty, compact, invariant set of the dynamics (X,ϕ), then there exists a minimal set in F.
Proof. Let F be the set of closed invariant subsets of F. The subset relation is a partial order in F. Let {Ui} be a totally ordered family in (F,⊂). By the Bolzano- Weierstrass Theorem iUi is nonempty. As iUi is closed and invariant, it belongs to F and is thus a lower bound for the totally ordered familyF. By Zorn’s Lemma there
exists a minimal element ofF. 2
Theorem 2.3.7 If M is a bounded, minimal set of the system (IR2,ϕ), then M is a critical point or a periodic orbit.
Proof. The set M is not empty and invariant. So it contains at least one orbit γ. As a minimal set M is closed and so it contains limit sets α(γ) and ω(γ). If M contains a critical point then as a minimal set M contains this point only.
If M contains no critical point, γ and ω(γ) consist of ordinary points. Because M is minimal, γ and ω(γ) are not disjoint. Thus γ ⊂ω(γ). From theorem (2.3.5) we see γ
periodic. 2
Theorem 2.3.8 (Poincar´e-Bendixson) Let X ∈IR2. Consider the continuous flow (X,ϕ) and assume thatγ+ is a bounded, positive orbit and thatω(γ) contains ordinary points only. Then ω(γ) is a periodic orbit (i.e. a limit cycle).
Proof. Because γ+contains infinitely many points in a bounded region, it has at least one limit point and so ω(γ) is nonempty. Let p be a point of the ω-limit set. As an invariant set ω(γ) contains the orbit γ(p) starting from point p. The orbit γ(p) is a minimal invariant set containing ordinary points only. Then it follows from the theo- rem (2.3.7) that γ(p) is periodic. From the theorem (2.3.4) we see ω(γ) =γ(p). 2
Theorem 2.3.9 (Poincar´e-Bendixson) Let X ∈IR2. Consider the continuous flow (X,ϕ) and assume A is a bounded positively invariant set with no critical point. Then A contains a limit cycle.
Proof. Follows immediately from the previous theorem. 2
2.4 Linear systems
Let us first consider the linear system
X˙ = a11X+a12Y +b1
Y˙ = a21X+a22Y +b2
(2.6)
where A= a11 a12
a21 a22
is a non singular matrix. The change of the variables x
y = X
Y +A−1 b1
b2
(2.7)
leads to an equivalent system
˙ x
˙
y =A x
y . (2.8)
So we can always suppose that the critical point is in the origin. The eigenvalues λ1 andλ2 are the roots of the characteristic equation
det(A−λI) = 0 (2.9)
λ2−trace(A)λ+ det(A) = 0 (2.10)
Because A is a non-singular matrix, the rootsλ1 and λ2 of the equation (2.10) are non-zero. There exists a real non-singular matrixT such thatT−1AT is in Jordan form
T−1AT = λ1 a 0 λ2
(2.11)
where a= 0 ifλ1=λ2, and a = 0 or a= 1 ifλ1 =λ2.
If the eigenvectors ofA form a linearly independent set, thenT can be constructed by putting the eigenvectors into the columns of T. If (u v) is a solution of the equation
˙ u
˙
v =T−1AT u v . (2.12)
then (x y) = T(u v) is a solution of (2.8). So in a qualitative classification we suppose A to be of the form (2.11). If the laws of the motion are
˙ x
˙
y = λ1 a
0 λ2
x y . (2.13)
then the corresponding flow is
ϕ(x, y, t) = (xeλ1t+ayteλ2t, yeλ2t) (2.14)
Let ψ(x, y, t) be the flow of the system with eigenvalues λ3 and λ4. In the first quarter of the state space (i.e. {(x, y)∈IR2|x >0, y >0) we easily find the topological conjugaciesh1 andh2 with
h1(x, y) = (xλ3/λ1, yλ4/λ2) (2.15)
h2(x, y) = (yλ3/λ2, xλ4/λ1) (2.16)
There are six equivalence classes of the flows on a plane generated by a linear model.
We collect these classes into four groups described below.
(A) The node. If the eigenvalues are real and have the same sign, then we call the critical point the node. If λ1 =λ2, then by eliminating t from (2.14) we see that
|y| = c|x|λ2/λ1. If the eigenvalues are negative, then the origin is an attracting node and if the eigenvalues are positive, then the origin is a repelling node. In either case there are no periodic orbits.
Ifλ1 =λ2 =λ, then the solution is either (x =c1eλt,y=c2eλt) or (x=c1eλt+c2teλt, y=c2eλt).
y
x
y
x
(a) λ1 =−1, λ2 =−4. (b) λ1 = 4, λ2 = 1.
Figure 10. (a) Attracting and (b) repelling nodes.
(B) The saddle point. If the eigenvalues are real and have different signs, then we call the critical point the saddle point. In the phase space the orbits are given by
|y|=c|x|−|λ2/λ1|. Two orbits arrive in the origin (stable manifold) and two orbits leave the origin (unstable manifold). The saddle point is not an attractor (or a repellor) and there are no periodic orbits. (See Figure 11 (a).)
(C) The focus. If the eigenvalues are complex conjugates, λ1,2 = α +βi, with αβ = 0, then we call the critical point the focus. The solution of the complex equation (2.13) is complex and the solution of the original equation with real coefficients
x
y =T c1eαteiβt c2eαte−iβt . (2.17)
is a linear combination of the terms eαtcosβt and eαtsinβt. The orbit is an elliptic spiral spiraling in to the origin if α <0 and spiraling out from the origin if α >0. So the focus is attracting if α<0 and repelling if α>0. (α<0 in Figure (11(b)).)
(D) The center. If the eigenvalues are purely imaginary, λ1,2 = ±βi, we call the critical point the center. The solution of the complex equation (2.13) is complex and the solution of the original equation with real coefficients is a linear combination of the terms cosβt and sinβt. The orbits are ellipses. All orbits are closed but not
(a)
y
x
(b)
y
x
Figure 11. (a) A saddle point, (b) an attracting focus.
asymptotically stable, and so the system has no attractor. Because there is no attractor, the model is not robust.
Let usfinally consider the set of equations (2.6) with a singular matrix of coefficients.
If det(A) = 0, we can denote
A= α β
kα kβ (2.18)
Thus the model is
X˙ = αX+ βY +b1
Y˙ =kαX+kβY +b2
. (2.19)
If b2 = kb1, then the set of critical points is equal to the line αX +βY +b1 = 0. If b2=kb1, then the system has no critical points because the pair of equations
αX+ βY +b1= 0 kαX+kβY +b2= 0 (2.20)
has no solution. In both cases we can conclude using the theorem (2.1.1) that the model (2.19) has no periodic orbits.
Combining all the results, we see that the only case in which the model has a periodic solution is the center. For a system with a center there is no attracting orbit.
So the model has no limit behavior to tend to.
Example 2.4.1 (The second order differential equation with constant coefficients):
Let us consider the model
Y¨ +aY˙ +bY =c (2.21)
where the parameters a, b and c are all real constants, and b = 0. By the change of the variable y=Y −c/bthe system can be written as
˙ x
˙
y = −a −b
1 0
x (2.22) y
The only critical point is (0,0). The eigenvalues are λ1,2 = −a±√
a2−4b (2.23) 2
If b < 0, then both eigenvalues are real and they have different signs. So the origin is a saddle point. If 0< b≤a2/4, then the eigenvalues are real and they either have the same sign or one of them is zero. Thus the model has a node at the origin. Ifb > a2/4, then the eigenvalues are complex. If a = 0, then the model has a center at the origin and if a= 0 then the model has a focus at the origin.
In the case of focus we conclude from the equation y˙ =x that the orbit circulates counter-clockwise. Otherwise the orbit of focus is rather clear. Next we sketch the orbit in the case of node. If the eigenvalues are not equal, then the eigenvectors are (λ1 1) and (λ2 1) . So the solution of (2.22) is
x
y = λ1 λ2
1 1
c1eλ1t c2eλ2t
= λ1c1eλ1t+λ2c2eλ2t c1eλ1t+c2eλ2t (2.24)
If the eigenvalues are real and |λ1| > |λ2|, then the orbit has two asymptotic directions
x≈λ2y near the origin, x≈λ1y far from the origin.
(2.25)
Eliminating t leads to the equation of orbit |x−λ1y|=c|x−λ2y|λ2/λ1. Some of these orbits are drawn in Figure (12).
One remark we may make is that although the linear autonomous system can have closed orbits in the phase space, they are not attractors. A closed orbit is a limit set for no other orbit than itself. So a linear autonomous model cannot exhibit stable periodic behavior.
To model a robust periodic behavior we must use either nonlinear or time-dependent models. We will choose the nonlinear autonomous way.
(a) x y x = yλ2
x = yλ1
(b) x
y
x = yλ1
λ2 x = y
Figure 12. (a) a repelling nodeλ1 >λ2 >0, (b) an attracting nodeλ1 <λ2 <0.
2.5 The critical points of a nonlinear system
Smooth nonlinear dynamics is locally linear. Let g : IR2 → IR2, z → (g1(z), g2(z)) be a nonlinear vector function and let z0 = (x0, y0) be a critical point for the dynamical system
˙
z =g(z) (2.26)
(i.e. g(z0) = (0,0)) The derivative of the function g at the point z0 is a linear trans- formation g(z0) =A such that
g(z0+h) = Ah+r(h), and (2.27)
||r(h)||
||h|| → 0, as||h||→0.
(2.28)
So if the state point is near the critical point, then h is small and we can regard r(h) as insignificant. Then the dynamics in local co-ordinates is approximately linear and determined by the derivative
h˙ ≈Ah (2.29)
The derivative at the critical point gives us only a local presentation of the small dynamics. The large-scale dynamics is much more difficult to analyze.
Example 2.5.1 (The Lotka-Volterra equation): Consider the system
˙
x=g1(x, y) =ax−bxy
˙
y=g2(x, y) =bxy−cy (2.30)
y
x .
Figure 13. Phase portrait of the predator-pray model with a = 2, b = 1 and c= 3.
This system is formulated by Lotka and Volterra to describe the interaction of two species where x denotes the population density of prey, y the population density of predator. The critical points (0,0) and (c/a, a/b) give the equilibrium solutions for the population model. We shall study the dynamics in the neighborhood of the non-trivial equilibrium.
The derivative of (2.30) is
g x
y = ∂g1/∂x ∂g1/∂y
∂g2/∂x ∂g2/∂y = a−by −bx by bx−c . (2.31)
In the non-trivial critical point
A=g c/b
a/b = 0 −c
a 0
(2.32)
and the eigenvaluesλ1,2 =±i√
acof Aare purely imaginary and so the dynamics have a center at (c/a, a/b). See Figure (13) for a phase portrait of the predator-pray model with parameter values a= 2, b= 1,c= 3.
3. Limit cycles of the Li´ enard equation
3.1 The Li´ enard equation
The basic form of theLi´enard’s differential equationstudied originally by Li´enard [1928]
is
¨
x+f(x)x˙ +x= 0 (3.1)
wheref ∈C0(IR). In this thesis we will consider the properties of this class of nonlinear ordinary differential equations. Most of the results will be formulated for the basic form.
If f(x) =m(x2−1) (m∈IR is a constant), then (3.1) is equal to theVan der Pol’s differential equation modeling the electric circuit with a triode tube and inductive feedback. Van der Pol’s differential equation is one of the most widely used models for nonlinear oscillations.
The Li´enard equation (3.1) can be written in an equivalent form on the Li´enard plane (x, y) by
˙
x=y−F(x)
˙ y=−x (3.2)
whereF(x) = 0xf(s)dsis the generating function of the systems (3.1) and (3.2). The use of F rather than f is a matter of convenience, because we usually use the Li´enard plane presentation. F ∈ C1(IR), F(0) = 0 and the only critical point on the Li´enard
y = F(x)
x y
y' x'
> 0
< 0
y' x'
< 0
< 0 y' x'
< 0
> 0 y'
x'
> 0
> 0
Figure 14. Direction of the motion on the Li´enard plane.
plane is (0,0). The eigenvalues of the derivative matrix at the origin are λ1,2 = −f(0)± f(0)2−4
2 .
(3.3)
So if f(0)>0 (f(0)<0), then the origin is an attracting (repelling) critical point. If there is a closed orbit in the phase space (i.e. in the Li´enard plane), it is a clockwise orbit around the origin (see Figure (14)). The number of closed orbits on the phase portrait depends on the generating function F.
An equivalent formulation of the Li´enard equation is the variant of the basic form
¨
x+F(x) +˙ x= 0.
(3.4)
Differentiating the equation (3.4) and changing the variable toξ=x˙ we get ¨ξ+F (ξ)ξ+˙ ξ = 0, which clearly is a Li´enard equation.
In equation (3.1), f(0) may be non-zero. In (3.4) we normally assume F(0) = 0 in which case the equation (3.4) is equivalent to
ξ˙=y−F(ξ)
˙ y=−ξ (3.5)
Further we usually write the third term of equation (3.1) or (3.4) without any coeffi- cient. This simplifies the reasoning in the following sections.
Often some application leads to an equation in which the assumption above is not fulfilled. Let the equation be similar to the basic form, except the third term containing
a coefficient, a= 0, a= 1.
¨
x+F(x)x˙+ax= 0, a= 0, a= 1.
(3.6)
In this case we rescale the time variable and change the variable τ = t√
a (3.7)
ξ(τ) = x(t).
(3.8) Sox˙ =ξ√
a and ¨x=ξ a. Substituting these in (3.6) leads to the equation ξ¨+F(ξ)
√a ξ˙+ξ = 0 (3.9)
which is of the basic form.
Let us consider an equation which is similar to the variant of the basic form, except the third term containing a coefficient a = 0, a = 1, and the generating function F does not fulfil the assumption F(0) = 0.
¨
x+f(x) +˙ ax= 0, a = 0, a= 1, f(0) = 0.
(3.10)
Again we rescale the time variable and change the variable and the generating function τ = t√
a, (3.11)
ξ(τ) = x(τ/√
a)−f(0)/a, (3.12)
g(s) = [f(s√
a)−f(0)]/a.
(3.13)
Substitution of these to (3.10) gives us
ξ¨+g(ξ) +ξ= 0 (3.14)
which is a variant of the basic form and g(0) = 0.
3.2 The critical point of the model
In the previous section we considered several forms of the Li´enard equation. Now we continue to consider the equation
¨
x+f(x)x˙ +x= 0 (3.15)
where f ∈C0(IR). In Li´enard plane the equation has the equivalent form
˙
x=y−F(x)
˙ y=−x (3.16)
whereF(x) = 0xf(s)dsis the generating function of the system. The only critical point on the Li´enard plane is (0,0). The eigenvalues (2.9) are either complex or they have the same sign. So if f(0)> 0 (f(0) < 0), then the origin is an attracting (repelling) critical point. It is a focus if|f(0)|<2, a center if |f(0)|= 2 and a node if |f(0)|>2.
We can prove more than just this local proposition. The first theorem to prove appears to be a trivial one, but it will be needed later. The other theorem describes the behaviour of the system in the neighborhood of the critical point.
Letf be aC0(IR)-function andF = 0xf(s)ds. Letrbe a positive real number such that |F(x)|>0 if 0 <|x|≤r and let Dr be the open disk Dr ={(x, y)|x2+y2 < r2}.
Theorem 3.2.1 If f(0) > 0 then the disk Dr is a positively invariant set of the flow (3.16) and iff(0)<0 then the complement IR2\Dr of the disk is a positively invariant set of the flow (3.16).
Proof. If f(0) > 0 (f(0) < 0) then xF(x) ≥ 0 (≤ 0) when −r ≤ x ≤ r. Let ρ = √
x2+y2 be the distance of the state point (x, y) from the origin. If x ∈ [−r, r]
then
d
dt(ρ2) = 2xx˙ + 2yy˙ =−2xF(x) (3.17)
Soρ2 does decrease (increase) on the boundary ∂Dr if f(0) >0 (f(0)<0). 2
Theorem 3.2.2 If f(0) > 0, then the critical point (0,0) of the flow (3.16) is an attractor and the disk Dr is contained in its basin of attraction. If f(0) <0 then the critical point(0,0) is a repellor and every orbit with the initial point inDr\(0,0)leaves the disk Dr in finite time.
Proof. Let f(0) > 0. Then according to the previous theorem, the open disk Dr is an open positively invariant neighborhood of the origin. LetV ={(x, y)|x2+y2 < r20}
be another open neighborhood. The origin is an attractor if there exists a T > 0 for which ϕt(Dr)⊂V for all t≥ T.
Letηbe an arc of an orbit with initial pointxη inDr\V. It turns clockwise around the origin. Next we estimate the change of ρ2 during one cycle around the origin. Let x1 =r0/2 and y1 = r20−x21. Then (x1, y1) and (x1,−y1) are two points on the circle
∂V. There exists a number ε > 0 such that F(x) >ε if x1 < x < r. Now the change during one cycleC is
∆(ρ2) =
C
dρ2 dt dt=
C−2xF(x)dt (3.18)
= 2
CF(x)dy <2 −
y1 y1
εdy=−4εy1. (3.19)
So infinite numbernof roundsηwill go insideV. The number nis independent of the initial point ofη. It is easy to see that the length of one turn around the origin is less than 4r and the speed of the continuous motion is strictly positive during these turns in Dr\V. So the speed has a positive lower bound in Dr\V and thus there exists T so that ϕT(xη) ∈ V. T is independent of the initial point of η andV is positively invariant. So ϕt(Dr)⊂V for allt≥T.
The case of the repellor is handled in a similar way. 2
3.3 The existence of the limit cycle
The basic theme of this paper is the search for a simple robust model of periodic behaviour. The last two theorems show us how the system explodes on one condition and converges on another condition. It would be a nice procedure to show that the system tends to explode near the origin and tends to converge far from the origin. In this case the Poincar´e-Bendixson theorem would trivially affirm the existence of the limit cycle. Unfortunately the set of divergence
E = (x, y)|d
dt(x2+y2)≥0 ={(x, y)|xF(x)≤0} (3.20)
is unbounded. So we must do some extra work to confirm the necessary conditions for the Poincar´e-Bendixson theorem.
