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1 Introduction 2 the Van Der Pol Oscillator Van der Pol Oscillator Celestial Mechanics Wesley Cao · 1 Introduction The van der Pol oscillator is a non-conservative oscillator with non-linear damping. Energy is dissipated at high amplitudes and generated at low amplitudes. As a result, there exists oscillations around a state at which energy generation and dissipation balance. The state towards which the oscillations converge is known as a limit cycle, which shall be formalized later in this paper. Balthazar van der Pol was a pioneer in the field of radio and telecommunications. While he was working at Phillips, van der Pol discovered these stable oscillations. Van der Pol himself came across the system as he was building electronic circuit models of the human heart. Due to the unique nature of the van der Pol oscillator, it has become the cornerstone for studying systems with limit cycle oscillations. In fact, the van der Pol equation has become a staple model for oscillatory processes in not only physics, but also biology, sociology and even economics. For instance, it has been used to model the electrical potential across the cell membranes of neurons in the gastric mill circuit of lobsters [5, 9]. Additionally, Fitzhugh and Nagumo used the model to describe spike generation in giant squid axons [4, 7]. The equation has also been extended to the Burridge–Knopoff model which characterizes earthquake faults with viscous friction [3]. Thus, it stands to reason that we should develop a deep understanding of the van der Pol oscillator due to its widespread applications. 2 The Van der Pol Oscillator The van der Pol oscillator is described by the equation x¨ ǫ(1 x2)x ˙ + x = 0 − − and equivalently the autonomous system, x3 x˙ = y F (x) := y ǫ x − − µ 3 − ¶ y˙ = x − It differs from the systems we have studied in that it is non-conservative. That is, it is not volume-preserving except when ǫ = 0. If ǫ = 0, then the equation becomes x¨ + x = 0, the simple harmonic oscillator, which is conservative. We formalize our analysis with some definitions. Definition 1. A flow is a mapping φ : U R U, for any set U that satisfies × → φ(z, 0) = z z U • ∀ ∈ φ(φ(z,s),t)= φ(z,s + t) z U and s,t R • ∀ ∈ ∈ Page 1 of 9 Van der Pol Oscillator Celestial Mechanics Wesley Cao · Solutions to an autonomous system of the formz ˙ = v(z), where z U and v : U U will be a flow, namely φ(z,t) such that ∈ → d φ(z,t)= v(φ(z,t)) dt Definition 2. Given z U, the trajectory through z is defined to be ∈ Γz φ(z,t) : t R ≡ { ∈ } A trajectory traces the motion of a single point through the flow. Thus, under- standing the properties of trajectories is essential to our analysis. Definition 3. A trajectory Γ is called closed if it contains more than one point and there exists T R+ such that φ(z, T )= z z Γ. ∈ ∀ ∈ Definition 4. A limit cycle is an isolated closed trajectory. That is, there exists a neighborhood N U around this trajectory such that Γz is not closed for all z N. ⊆ ∈ As a result of this definition of limit cycles, limit cycles can only occur in nonlinear systems. In a linear system, closed trajectories are neighboured by other closed trajectories. Consequently, neighboring trajectories of a limit cycle must spiral into or away from the limit cycle. We shall analyze limit cycles of this system when ǫ> 0. 3 Results We begin with some preliminary results before we start our analysis of the van der Pol oscillator. Lemma 5. (Gronwall’s Lemma [1]) Let α,β,u be nonnegative real functions defined on [0, M] where β, u are continuous, α is differentiable, and t [0, M] ∀ ∈ t u(t) α(t)+ β(s)u(s)ds ≤ Z0 then t t t u(t) α(0) exp β(s)ds + α′(s) exp β(r)dr ds , t [0, M] ≤ µ Z0 ¶ Z0 · µ Zs ¶¸ ∀ ∈ t t Proof. For t [0, M], let γ(t) = 0 β(s)ds and v(t) = 0 β(s)u(s)ds. By the Funda- ∈ d mental Theorem of Calculus, exp(R γ(t)) = β(t) exp(R γ(t)) and dt − − − v′(t)= β(t)u(t) β(t)(α(t)+ v(t)) ≤ Therefore, t t exp( γ(t))v(t)= exp( γ(s))(v′(s) β(s)v(s))ds exp( γ(s))α(s)β(s)ds − Z0 − − ≤ Z0 − Page 2 of 9 Van der Pol Oscillator Celestial Mechanics Wesley Cao · Furthermore, integration by parts yields t t exp( γ(s))α(s)β(s)ds = α(t) exp( γ(t)) + α(0) + α′(s) exp( γ(s))ds Z0 − − − Z0 − Thus, it follows that u(t) α(t)+ v(t) = exp(γ(t)) exp( γ(t))α(t) + exp( γ(t))v(t) ≤ · − − ¸ t exp(γ(t)) α(0) + α′(s) exp( γ(s))ds ≤ · Z0 − ¸ t t t = α(0) exp β(s)ds + α′(s) exp β(r)dr ds µ Z0 ¶ Z0 · µ Zs ¶¸ Lemma 6. (Averaging Lemma [10]) Let U Rn be a compact set, ǫ be a real such that 0 < ǫ 1. If x and z are solutions to ⊆ ≪ x˙ = ǫf(x, t, ǫ) ( ) † and 1 T z˙ = ǫf(z) := ǫ f(z,t, 0)dt ( ) T Z0 ‡ respectively, where f : Rn R R+ Rn is C2 and periodic in t with period T , × × → and satisfy initial conditions x(0) = x0 and z(0) = z0 where x0 z0 = O(ǫ), then x(t) z(t) = O(ǫ) on a time scale t 1 . | − | | − | ∼ ǫ Proof. First we shall show that there is a C2 change of coordinates x = y + ǫw(y, t, ǫ) under which ( ) becomes † y˙ = ǫf(y)+ ǫ2g(y, t, ǫ) ( ) ≀ where g is periodic in t with period T . Define f(x, t, ǫ)= f(x, t, ǫ) f(x) − and consider the change of coordinatese x = y + ǫw(y, t, ǫ) ∂w where w is the anti-derivative of f. That is, ∂t = f(y,t, 0). Differentiating yields e ∂w ∂we x˙ =y ˙ + ǫ(y ˙ + ) ∂y ∂t Page 3 of 9 Van der Pol Oscillator Celestial Mechanics Wesley Cao · Therefore, ∂w ∂w y˙ = (1+ ǫ )−1(x ˙ ǫ ) ∂y − ∂t ∂w = (1 ǫ + O(ǫ2))(ǫf(y + ǫw)+ ǫf(y + ǫw, t, ǫ) ǫf(y,t, 0)) − ∂y − ∂w ∂f e ∂f e ∂f = ǫ(1 ǫ + O(ǫ2))(f(y)+ ǫw (y)+ ǫw (y,t, 0) + ǫ (y,t, 0) + O(ǫ2)) ∂y ∂y ∂y ∂ǫ − e e ∂w ∂f ∂f = ǫ(1 ǫ )(f(y)+ ǫw (y,t, 0) + ǫ (y,t, 0)) + O(ǫ3) ∂y ∂y ∂ǫ − e ∂f ∂f ∂w = ǫf(y)+ ǫ2 w (y,t, 0) + (y,t, 0) f(y) + O(ǫ3) ∂y ∂ǫ ∂y µ e − ¶ := ǫf(y)+ ǫ2g(y, t, ǫ) Since U is compact, f is uniformly continuous and g is bounded. Let L be the Lipschitz constant of f and B be the maximum absolute value of g, and u = y z. Subtracting ( ) and ( ) and then integrating yields − ‡ ≀ t t u(t)= u(0) + ǫ [f(y(s)) f(z(s))]ds + ǫ2 g(y(s), s, ǫ)ds Z0 − Z0 Taking absolute values, and using the triangle inequality yields t u(t) = u(0) + ǫL u(s) ds + ǫ2Bt | | | | Z0 | | By Gronwall’s Lemma, t ǫB u(t) u(0) exp(ǫLt)+ ǫ2B exp(ǫL(t s))ds exp(ǫLt) u(0) + | | ≤ | | Z0 − ≤ µ| | L ¶ 1 Therefore, if y(0) z(0) = u(0) = O(ǫ), then for t [0, ǫL ], we have y(t) z(t) = u(t) = O(ǫ).| Recall,− x(|t) | y(t)| = ǫ w(y, t, ǫ) = O(∈ǫ), so by the triangle| inequality− | | | | − | | | x(t) z(t) x(t) y(t) + y(t) z(t) = O(ǫ) | − | ≤ | − | | − | as desired. Now we are ready to delve into our analysis of the van der Pol oscillator. First, we prove a fairly powerful result that characterizes the unique limit cycle of the van der Pol oscillator. Moreover, all other trajectories will approach this limit cycle. Then we seek to deepen our understanding by describing the behavior of this limit cycle for various values of ǫ. Page 4 of 9 Van der Pol Oscillator Celestial Mechanics Wesley Cao · Theorem 7. (Lienard’s Theorem [8]) The van der Pol oscillator has exactly one limit cycle and all other trajectories spiral into it. Proof. We refer to Figure 1 throughout this proof. Let Pi =(xi, yi) for i = 0, 1, 2, 3, 4 where P0 P4 traces the trajectory Γ that starts at P0. We know Γ follows this path since on the··· positive y-axis,x> ˙ 0, y˙ < 0, so Γ spirals clockwise until it intersects the graph of y = F (x) at P2. At this point,x ˙ = 0, y˙ < 0 so Γ travels below the graph of y = F (x) after whichx< ˙ 0, y˙ < 0 so Γ curves left and keeps spiraling until it hits the negative y-axis at P4. Notice the van der Pol equation is symmetric with respect to origin. That is, if (x(t), y(t)) describes a trajectory, so does ( x(t), y(t)). Therefore, − − it follows that Γ is closed if and only if y4 = y0 or equivalently, u(0, y0) = u(0, y4) 1 2 2 − where u(x, y)= 2 (x + y ). Let A be the arc P0P4 of Γ. A is uniquely determined by α = x2 since trajectories do not cross.
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