A Study of the Van der Pol Equation
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1 A Study of the Van der Pol Equation Kai Zhe Tan, s September 16, 2016 Abstract The Van der Pol equation is famous for modelling biological systems as 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. Geometric singular perturbation theory is being introduced to analyse the slow and fast manifolds of the system. Fenichel s theorem will be used to explore and approximate the equation for the perturbed manifolds which are locally invariant to the critical manifolds. Fold points, which are not described by Fenichel s theorem, are being studied using blow-up techniques. 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 1
2 1 Introduction The Van der Pol equation is well-known for playing the central role in the development of nonlinear dynamics. Proposed by the Dutch physicist Balthasar van der Pol, the second order nonlinear autonomous differential equation is developed from electrical circuit experiments using vacuum tubes. Van der Pol discovered the signals produced stable periodic motion known as limit cycle. Due to its unique nature in producing limit cycles, it has often been used to describe physical systems as well as biological systems. For example, FitzhHugh and Nagumo have employed this equation in describing the theoretical models of nerve membrane. [6] 2 Van der Pol equation The Van der Pol oscillator is given by the equation ẍ + µ(x 2 1)ẋ + x = 0, µ > 0 (1) where µ > 0 is a parameter. The equation is a simple harmonic oscillator, with a nonlinear damping term µ(x 2 1)ẋ. This nonlinear term causes large amplitude to decay when x > 1, but increases when it reaches x < 1. As a result, the system will reach a self-sustained oscillation which will be shown later and it has a unique, stable limit cycle. Figure 1: Figure shows few limit cycles starting from some initial conditions. Reprinted from [3]. 2
3 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, a phase plane analysis is introduced. Notice that ẍ + µẋ(x 2 1) = d dt (ẋ + µ[1 3 x3 x]). (2) Let F (x) = 1 3 x3 x and define a different phase plane variable w = ẋ + µf (x). We differentiate w with respect to t and compare ẇ with the Van der Pol equation (1) to obtain Now, (3) can be rewritten to ẇ = ẍ + µẋ(x 2 1) = x. (3) ẋ = w µf (x), ẇ = x. With a change of variable y = w µ, the equation can be rewritten in terms of x and y ẋ = µ[y F (x)], ẏ = 1 µ x. (5) (4) 3
4 Figure 3: Plot of the autonomous system (5). Reprinted from [4] Consider a typical tracjectory in the phase plane as plotted in Figure 1. For any initial point except the origin, the trajectory will travel horizontally in a fast pace until it reaches the cubic nullcline y = F (x). Then, it slowly moves towards the origin following the nullcline until it reaches the turning point of the nullcline. It then travels horizontally quickly to the opposite branch of the cubic nullcline and the motion continues periodically by crawling on the nullcline and jumping off from the nullcline to the opposite side of the nullcline. 4
5 3 Geometric Singular Perturbation Theory In this relaxation oscillation, we can see that the limit cycle consists of slow manifolds and fast manifolds. Geometric singular perturbation theory would be used to analyse problems with different time scales. It uses invariant manifolds to understand the global structure of the phase space. As biological models are fairly complicated and involve many time scales, geometric singular perturbation theory would be useful. The following discussion is based on Hek [1]. Given the autonomous system (5), the timescale could be rescaled to ẋ = y F (x), ẏ = 1 µ 2 x (6) and µ can be replaced by ε = 1 µ 2 becoming ẋ = y F (x), ẏ = εx. Now, eqn. (10) are basic equations of singularly perturbed systems of ODEs with two different time scales in the form of u = f(u, v, ε), v = εg(u, v, ε) with = d dt, u, v R. With a change of time scale, system (10) can be rewritten as εx = y F (x), y (9) = x where = d dτ 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 µ would be much smaller than 1. When ε 0. the limits are respectively given by ẋ = y F (x), (10) ẏ = 0 and 0 = y F (x), y = x. (7) (8) (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 of equation are the same with the condition that ε 0. As ε decreases, the trajectories jump faster to another end of the cubic spline. We can now continue to analyse the dynamics of the system with small nonzero ε using Fenichel s Theorem. 5
6 Figure 4: Plot based on (10). Red line shows the cubic spline, black line shows the plot when ε = 0.1 and yellow line indicates the plot when ε = Fenichel s Theorem The following discussion is based on Hek [1]. Theorem 1 (Fenichel) Suppose M 0 {f(u, v, 0) = 0} is compact, possibly with boundary, and normally hyperbolic, that is, the eigenvalues λ of the Jacobian δf δu (u, v, 0) M 0 all satisfy Re(λ) 0. Suppose f and g are smooth. Then for ε > 0 and sufficiently small, there exists a manifold M ε, O(ε) close and diffeomorphic to M 0, that is locally invariant under the flow of (8). The critical manifold for (6) is bounded in the range of ( 3, 3) which is contained in the set f(u, v, 0) = 0. However, the eigenvalues of the Jacobian λ = f u (u, v, 0) M 0 = x 2 1 will be non-zero except for x = ±1. These points are fold points, or known as jump points where the manifold is non-hyperbolic at these points. Theorem 1 is not applicable at these points and thus these points need special attention later 6
7 in this report. Theorem 1 assures that for sufficiently small ε 0 the critical manifold M 0 persist as perturbed manifold M ε that are invariant for the flow with ε 0. The perturbed manifold M ε can be approximated by the asymtotic expansion using its invariance. Assume that the perturbed manifold can be described by the graph {(x, y) y = p ε (x)}, the manifold is invariant under the flow of By substituting in y, Now expand y = dp ε(x) dx x. εx = dp ε(x) dx x. y = p ε (x) = p 0 (x) + εp 1 (x) + ε 2 p 2 (x) +... where p 0 = 1 3 x3 x describes the critical manifold M 0. Then compare both sides of the equation by the orders of ε : O(1) : 0 = 0, O(ε) : x = dp 0 dx p 1(x), This yields the approximation O(ε 2 ) : 0 = dp 2 dx p 0(x) + dp 1 dx p 1(x) + dp 0 dx p 2(x),. p ε (x) = 1 x 3 x3 x ε x ε2 x(x2 + 1) 2(x 2 1) for x ±1. Interestingly, this approximation p ε (x) is undefined for x = ±1, which is a good cross check as Fenichel s theorem does not apply to these fold points. 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. This method is a coordinate transformation on the fold points where they are blownup to a two-sphere. In certain trajectories on the sphere, one gains enough hyperbolicity and Fenichel s theorem and other standard techniques can be used to describe the phenomena. The technique is a generalisation of the blow-up methods of planar vector field. 7
8 Before that, we need to make a few assumptions. First, the fold points have to be shifted to the origin for simplicity in explanation (the fold points will be refer as the origin here onwards). Let S = {(x, y) : f(x, y, 0) = 0} be the critical 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 is approximately a parabola. Let S a be the left branch of the parabola and S r be the right branch of the parabola. Assuming that S a is attracting towards the origin, S r 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 S ε and the fold point. Reprinted from [2]. There exists an arbitrarily small neighbourhood V of (0,0), the manifold S a and S r perturb smoothly to locally invariant manifolds S a,ε and S r,ε for sufficiently small ε 0. We would need to rewrite the system (8) into its extended system in R 3. For sufficiently small ε 0, we get x = f(x, y, ε), y = εg(x, y, ε), ε = 0, (12) and its canonical form x = y + x 2 + h(x, y, ε), y = εg(x, y, ε), ε = 0 (13) 8
9 with ε =constant, h(x, y, ε) = O(ε, xy, y 2, x 3 ), g(x, y, ε) = 1 + O(x, y, z) after rescaling. The blow-up transformation for (13) is x = r x, y = r 2 ȳ, ε = r 3 ε. (14) We define B = S 2 [0, ρ], where the constant ρ is related to ε 0 by ε 0 = p 3. Then, the transformation can be written as a mapping φ : B R 3 with ( x, ȳ, ε) S 2. To avoid lengthy calculations, 3 different charts are introduced. Chart K 2 describes the upper half-sphere defined by ε = 1 while charts K 1 and K 3 describe the neighbourhoods of parts of the equator of S 2. In this report, we would only concentrate on K 2, where the dynamics of the blown-up vector field in a neighbourhood of the upper half-sphere is studied. The blow-up transformation in chart K 2 is given by with coordinate (x 2, y 2, r 2 ) R 3 by setting ε = 1. x = r 2 x 2, y = r 2 2y 2, ε = r 3 2, (15) To study chart K 2, (15) is inserted into the canonical form of the system (13). As ε is a constant and ε = 0, we get (r 3 2) =0, 3r 2 2r 2 =0, (16) thus r 2 = 0. On the other hand, x = (r 2 x 2 ) = r 2 x 2, x 2 = r 2 y 2 + r 2 x O(r 2 2, r 2 2x 2 y 2, r 3 2y 2 2, r 2 2x 3 2), (17) and y = (r 2 2y 2 ) = r 2 2y 2, y 2 = r 2 ( 1 + O(r 2 x 2, r 2 2y 2, r 3 2)). (18) To eliminate the factor r 2, we desingularise the equations by rescaling time t 2 := r 2 t. Thus, we get x 2 = x 2 2 y 2 + O(r 2 ), y 2 = 1 + O(r 2 ), r 2 = 0 where denotes differentiation with respect to t 2. The Riccati equation is obtained with the condition r 2 = 0, where (19) x 2 = x 2 2 y 2, y 2 = 1. (20) 9
10 Its solution can be expressed in terms of special functions. The assymptotic expansions of its unique orbit is given by s(x 2 ) = x O( 1 2x 2 x 4 ), x 2 2 s(x 2 ) = Ω x 2 + O( 1 x 3 2 ), x 2 (21) Figure 6: Solutions of the Riccati eqaution. Reprinted from [2]. Notice that the orbit leads the incoming attracting slow manifold across the upper half of the sphere S 2 to the point q out then it moves towards the fast flow direction. 10
11 Figure 7: Phase portrait of the blown-up vector field S ε 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 its slow-fast behaviour using geometric singular perturbation theory. First, I have done a phase plane analysis of the equation to get a rough idea of the phase portrait of the system. Fenichel s theorem was used to find the perturbed manifold that diffeomorphic to the normally hyperbolic critical manifold. I have carried out an asymptotic expansion to approximate the perturbed manifold using its invariance. For the fold points that do not satisfy the normally hyperbolicity criterion, blow-up method of planar vector field is introduced. Due to time constraints, I have only focused on the second chart, which describes the upper half-sphere. The trajectory was weakly attracted towards the fold point, and while being weakly repelled by another branch of the parabola, it is being absorbed by the fast flow which caused a smooth jump from the slow manifold to the 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 support and encouragement has enabled me to finish this report. His commitment to give his time generously has been very much appreciated. Besides that, I would like to thank the University of Edinburgh School of Mathematics for giving me 11
12 an opportunity on this scholarship and supporting me financially throughout the duration of my project. References [1] Geertje Hek, Geometric Singular Pertubation Theory in Biological Practice, Springer, [2] M. Krupa and P. Szmolyan, Extending Geometric Singular Pertubation Theory to Nonhyperbolic Points - Fold and Canard Points in Two Dimensions, Society for Industrial and Applied Mathematics, [3] Marios Tsatsos, Theoretical and Numerial Study of the Van der Pol Equation, Retrieved from pdf 2006, Online; accessed 29-August [4] Steven H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and Engineering, Westview Press, [5] Van der Pol Oscillator, Van der Pol Oscillator Wikipedia, the free encylopedia. Retrieved from oscillator. Online; accessed 26-August [6] Wesley Cao, Van der Pol Oscillator, Retrieved from http: // Cao-vanderpol.pdf, 2013, Online; accessed 25-August
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