A Study of the Van der Pol Equation

Kai Zhe Tan, s1465711

September 16, 2016

Abstract

The Van der Pol equation is famous for modelling biological systemsas well as being a good model to study its multiple time scale behaviour.This report analyses the ability of it oscillating in its limit cycle. Ge-ometric singular perturbation theory is being introduced to analyse theslow and fast manifolds of the system. Fenichel’s theorem will be used toexplore and approximate the equation for the perturbed manifolds whichare locally invariant to the critical manifolds. Fold points, which are notdescribed by Fenichel’s theorem, are being studied using blow-up tech-niques.

Contents

1 Introduction 2

2 Van der Pol equation 2

3 Geometric Singular Perturbation Theory 5

4 Fenichel’s Theorem 6

5 Fold Points 7

6 Summary 11

7 Acknowledgement 11

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1 Introduction

The Van der Pol equation is well-known for playing the central role in thedevelopment of nonlinear dynamics. Proposed by the Dutch physicist Balthasarvan der Pol, the second order nonlinear autonomous differential equation isdeveloped from electrical circuit experiments using vacuum tubes. Van der Poldiscovered the signals produced stable periodic motion known as limit cycle.Due to its unique nature in producing limit cycles, it has often been used todescribe physical systems as well as biological systems. For example, FitzhHughand Nagumo have employed this equation in describing the theoretical modelsof nerve membrane. [6]

2 Van der Pol equation

The Van der Pol oscillator is given by the equation

x+ µ(x2 − 1)x+ x = 0, µ > 0 (1)

where µ > 0 is a parameter. The equation is a simple harmonic oscillator, with anonlinear damping term µ(x2−1)x. This nonlinear term causes large amplitudeto decay when |x| > 1, but increases when it reaches |x| < 1. As a result, thesystem will reach a self-sustained oscillation which will be shown later and ithas a unique, stable limit cycle.

Figure 1: Figure shows few limit cycles starting from some initial conditions.Reprinted from [3].

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Figure 2: Figure shows limit cycles with different values of µ. Reprinted from[5].

The following discussion is based on Strogatz [4]. To analyse this further, aphase plane analysis is introduced. Notice that

x+ µx(x2 − 1) =d

dt(x+ µ[

1

3x3 − x]). (2)

Let F (x) = 13x

3 − x and define a different phase plane variable w = x+ µF (x).We differentiate w with respect to t and compare w with the Van der Polequation (1) to obtain

w = x+ µx(x2 − 1) = −x. (3)

Now, (3) can be rewritten to

x = w − µF (x),

w = −x.(4)

With a change of variable y = wµ , the equation can be rewritten in terms of x

and yx = µ[y − F (x)],

y = − 1

µx.

(5)

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Figure 3: Plot of the autonomous system (5). Reprinted from [4]

Consider a typical tracjectory in the phase plane as plotted in Figure 1. Forany initial point except the origin, the trajectory will travel horizontally in afast pace until it reaches the cubic nullcline y = F (x). Then, it slowly movestowards the origin following the nullcline until it reaches the turning point of thenullcline. It then travels horizontally quickly to the opposite branch of the cubicnullcline and the motion continues periodically by crawling on the nullcline andjumping off from the nullcline to the opposite side of the nullcline.

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3 Geometric Singular Perturbation Theory

In this relaxation oscillation, we can see that the limit cycle consists of slowmanifolds and fast manifolds. Geometric singular perturbation theory would beused to analyse problems with different time scales. It uses invariant manifoldsto understand the global structure of the phase space. As biological models arefairly complicated and involve many time scales, geometric singular perturba-tion theory would be useful. The following discussion is based on Hek [1].

Given the autonomous system (5), the timescale could be rescaled to

x = y − F (x),

y = − 1

µ2x

(6)

and µ can be replaced by ε = 1µ2 becoming

x = y − F (x),

y = −εx.(7)

Now, eqn. (10) are basic equations of singularly perturbed systems of ODEswith two different time scales in the form of

u = f(u, v, ε),

v = εg(u, v, ε)(8)

with · = ddt , u, v ⊆ R. With a change of time scale, system (10) can be rewritten

asεx′ = y − F (x),

y′ = −x(9)

where ′ = ddτ and τ = εt. The timescale for t is assigned to be fast whereas the

one for τ is slow. Therefore, the equations (8) is said to be the fast system and(9) is the slow system. As µ is much greater than 1, ε, the inverse-square of µwould be much smaller than 1. When ε → 0. the limits are respectively givenby

x = y − F (x),

y = 0(10)

and0 = y − F (x),

y′ = −x.(11)

Equation (11) which is known as the reduced system shows that the cubic spline,y = F (x),- is the set of critical points for the slow variable. These two sets ofequation are the same with the condition that ε 6= 0.

As ε decreases, the trajectories jump ’faster’ to another end of the cubicspline. We can now continue to analyse the dynamics of the system with smallnonzero ε using Fenichel’s Theorem.

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Figure 4: Plot based on (10). Red line shows the cubic spline, black line showsthe plot when ε = 0.1 and yellow line indicates the plot when ε = 0.01.

4 Fenichel’s Theorem

The following discussion is based on Hek [1].

Theorem 1 (Fenichel) SupposeM0 ⊂ f(u, v, 0) = 0 is compact, possibly withboundary, and normally hyperbolic, that is, the eigenvalues λ of the Jacobianδfδu (u, v, 0)|M0 all satisfy Re(λ) 6= 0. Suppose f and g are smooth. Then forε > 0 and sufficiently small, there exists a manifold Mε,O(ε) close and diffeo-morphic to M0, that is locally invariant under the flow of (8).

The critical manifold for (6) is bounded in the range of (−√

3,√

3) which iscontained in the set f(u, v, 0) = 0. However, the eigenvalues of the Jacobian

λ =∂f

∂u(u, v, 0)|M0

= x2 − 1

will be non-zero except for x = ±1. These points are fold points, or known asjump points where the manifold is non-hyperbolic at these points. Theorem 1 isnot applicable at these points and thus these points need special attention later

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in this report.

Theorem 1 assures that for sufficiently small ε 6= 0 the critical manifoldM0

persist as perturbed manifold Mε that are invariant for the flow with ε 6= 0.The perturbed manifold Mε can be approximated by the asymtotic expansionusing its invariance. Assume that the perturbed manifold can be described bythe graph (x, y)|y = pε(x) , the manifold is invariant under the flow of

y′ =dpε(x)

dxx′.

By substituting in y′,

εx =dpε(x)

dxx′.

Now expandy = pε(x) = p0(x) + εp1(x) + ε2p2(x) + . . .

where p0 = 13x

3 − x describes the critical manifold M0. Then compare bothsides of the equation by the orders of ε :

O(1) : 0 = 0,

O(ε) : −x =dp0dx

p1(x),

O(ε2) : 0 =dp2dx

p0(x) +dp1dx

p1(x) +dp0dx

p2(x),

...

This yields the approximation

pε(x) =1

3x3 − x− ε x

x2 − 1+ ε2

x(x2 + 1)

2(x2 − 1)4+ . . .

for x 6= ±1. Interestingly, this approximation pε(x) is undefined for x = ±1,which is a good cross check as Fenichel’s theorem does not apply to these foldpoints.

5 Fold Points

The following section is based on Krupa and Szmolyan [2].

The blow-up techniques are one of the ways to analyse fold points. Thismethod is a coordinate transformation on the fold points where they are ’blown-up’ to a two-sphere. In certain trajectories on the sphere, one gains enough hy-perbolicity and Fenichel’s theorem and other standard techniques can be usedto describe the phenomena. The technique is a generalisation of the blow-upmethods of planar vector field.

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Before that, we need to make a few assumptions. First, the fold points haveto be shifted to the origin for simplicity in explanation (the fold points willbe refer as the origin here onwards). Let S = (x, y) : f(x, y, 0) = 0 be thecritical manifold. There exists a neighbourhood U of the origin such that (0,0)is the only point in U ∩ S, where the hyperbolicity vanishes and that S ∩ U isapproximately a parabola. Let Sa be the left branch of the parabola and Srbe the right branch of the parabola. Assuming that Sa is attracting towardsthe origin, Sr is repelling away from the origin and the origin is non-hyperbolic,weakly attracted towards the left and weakly repelled from the right.

Figure 5: Slow manifold S, its locally invariant perturbed manifold Sε and thefold point. Reprinted from [2].

There exists an arbitrarily small neighbourhood V of (0,0), the manifoldSa and Sr perturb smoothly to locally invariant manifolds Sa,ε and Sr,ε forsufficiently small ε 6= 0. We would need to rewrite the system (8) into itsextended system in R3. For sufficiently small ε 6= 0, we get

x′ = f(x, y, ε),

y′ = εg(x, y, ε),

ε′ = 0,

(12)

and its canonical form

x′ = −y + x2 + h(x, y, ε),

y′ = εg(x, y, ε),

ε′ = 0

(13)

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with ε =constant, h(x, y, ε) = O(ε, xy, y2, x3), g(x, y, ε) = −1 +O(x, y, z) afterrescaling. The blow-up transformation for (13) is

x = rx, y = r2y, ε = r3ε. (14)

We define B = S2×[0, ρ], where the constant ρ is related to ε0 by ε0 = p3. Then,the transformation can be written as a mapping φ : B → R3 with (x, y, ε) ∈ S2.

To avoid lengthy calculations, 3 different charts are introduced. Chart K2

describes the upper half-sphere defined by ε = 1 while chartsK1 andK3 describethe neighbourhoods of parts of the equator of S2. In this report, we wouldonly concentrate on K2, where the dynamics of the blown-up vector field in aneighbourhood of the upper half-sphere is studied. The blow-up transformationin chart K2 is given by

x = r2x2, y = r22y2, ε = r32, (15)

with coordinate (x2, y2, r2) ∈ R3 by setting ε = 1.

To study chart K2, (15) is inserted into the canonical form of the system(13). As ε is a constant and ε′ = 0 , we get

(r32)′ =0,

3r22r′2 =0,

(16)

thus r′2 = 0. On the other hand,

x′ = (r2x2)′ = r2x′2,

x′2 = −r2y2 + r2x22 +O(r22, r

22x2y2, r

32y

22 , r

22x

32),

(17)

andy′ = (r22y2)′ = r22y

′2,

y′2 = r2(−1 +O(r2x2, r22y2, r

32)).

(18)

To eliminate the factor r2, we desingularise the equations by rescaling timet2 := r2t. Thus, we get

x2 = x22 − y2 +O(r2),

y2 = −1 +O(r2),

r2 = 0

(19)

where ˙ denotes differentiation with respect to t2.

The Riccati equation is obtained with the condition r2 = 0, where

x2 = x22 − y2,y2 = −1.

(20)

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Its solution can be expressed in terms of special functions. The assymptoticexpansions of its unique orbit is given by

s(x2) = x22 +1

2x2+O(

1

x42), x2 → −∞

s(x2) = −Ω0 +1

x2+O(

1

x32), x2 →∞

(21)

Figure 6: Solutions of the Riccati eqaution. Reprinted from [2].

Notice that the orbit leads the incoming attracting slow manifold across theupper half of the sphere S2 to the point qout then it moves towards the fast flowdirection.

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Figure 7: Phase portrait of the blown-up vector field Sε and the fold point.Reprinted from [2].

6 Summary

The main aim of this report is to study the Van der Pol equation in terms of itsslow-fast behaviour using geometric singular perturbation theory. First, I havedone a phase plane analysis of the equation to get a rough idea of the phase por-trait of the system. Fenichel’s theorem was used to find the perturbed manifoldthat diffeomorphic to the normally hyperbolic critical manifold. I have carriedout an asymptotic expansion to approximate the perturbed manifold using itsinvariance. For the fold points that do not satisfy the normally hyperbolicitycriterion, blow-up method of planar vector field is introduced. Due to timeconstraints, I have only focused on the second chart, which describes the upperhalf-sphere. The trajectory was weakly attracted towards the fold point, andwhile being weakly repelled by another branch of the parabola, it is being ab-sorbed by the fast flow which caused a smooth jump from the slow manifold tothe fast manifold. This report can be continued by analysing two other charts,which will give a deeper analysis of the fold points.

7 Acknowledgement

I would like to express my deepest gratitude towards my project supervisor, Dr.Nikola Popovic, who has guided me throughout this project. His full supportand encouragement has enabled me to finish this report. His commitment togive his time generously has been very much appreciated. Besides that, I wouldlike to thank the University of Edinburgh School of Mathematics for giving me

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an opportunity on this scholarship and supporting me financially throughoutthe duration of my project.

References

[1] Geertje Hek, Geometric Singular Pertubation Theory in Biological Practice,Springer, 2009.

[2] M. Krupa and P. Szmolyan, Extending Geometric Singular PertubationTheory to Nonhyperbolic Points - Fold and Canard Points in Two Dimen-sions, Society for Industrial and Applied Mathematics, 2001.

[3] Marios Tsatsos, Theoretical and Numerial Study of the Van der PolEquation, Retrieved from https://arxiv.org/ftp/arxiv/papers/0803/

0803.1658.pdf 2006, Online; accessed 29-August-2016.

[4] Steven H. Strogatz, Nonlinear Dynamics and Chaos with Applications toPhysics, Biology, Chemistry and Engineering, Westview Press, 2015.

[5] Van der Pol Oscillator, Van der Pol Oscillator — Wikipedia, the free ency-lopedia. Retrieved from https://en.wikipedia.org/wiki/Van_der_Pol_

oscillator. Online; accessed 26-August-2016.

[6] Wesley Cao, Van der Pol Oscillator, Retrieved from http:

//www.math.cornell.edu/~templier/junior/final_paper/Wesley_

Cao-vanderpol.pdf, 2013, Online; accessed 25-August-2016.

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