The Van der Pol oscillator

Chapter 3 The Van der Pol oscillator

The Van der Pol oscillator is named after Dutch physicist Balthazar van der Pol, who first found self excited periodic states of the system

(3.1) Van der Pol studied the system in the 1920s by implementing it on circuits using

¨ 2 x−ǫ α−x ˙x + x = 0.

a triode[ 5 ]. He also applied sinusoidal excitation to the system with different frequencies, because of which the system started to show irregular outputs. This was one of the first encounters of chaotic behavior.

3.1 Fixed points

Let us start the investigation of 3.1 by rewriting it in the following from:

To find the fixed points, we have to solve

0=x 2 ,

0=ǫ α−x 1 x 2 −x 1 , which has only one – trivial – solution (x 1 ,x 2 ) = (0, 0). To examine the stability

we have to write the Jacobian matrix at the fixed point:

0  1 0 1 J (0, 0) = 

−2ǫx 1 x 2 − 1 ǫ (α − x 2 1 )

(x

1 ,x

Therefore the characteristic equation becomes

−λ

det(J(0, 0) − λI) = =λ − ǫαλ + 1 = 0. (3.5)

−1 ǫα − λ

The roots of the characteristic equation are

for which the cases (excluding multiple roots) are:

A) ǫα > 4: λ + 1,2 ∈R , unstable node (repeller).

B) 4 > ǫα > 0: λ 1,2 ∈ C, Re{λ 1,2 } > 0, unstable focus (spiral repeller).

C) 0 > ǫα > −4: λ 1,2 ∈ C, Re{λ 1,2 } < 0, stable focus (spiral attractor).

D) ǫα < −4: λ − 1,2 ∈R , stable node (attractor).

(a) ǫ = 2, α = 3 (b) ǫ = 1/2, α = 5/2

(c) ǫ = 1/2, α = −2 (d) ǫ = 2, α = −3

Table 3.1. The fixed point of the Van der Pol oscillator with different param- eter values

(a) ǫ = 0.1 (green) to ǫ = 0.4 (blue), in (b) ǫ = 0.5 (blue) to ǫ = 5 (red), in in- increments of 0.1

crements of 0.5

Figure 3.1. Limit cycle of the Van der Pol oscillator with different values of ǫ(α = 1)

3.2 Limit cycle

When ǫ = 0 the system becomes ¨ x + x = 0 for which the solution is a trivial harmonic motion. It is shown in [ 18 ] that if α = 1 and ǫ is small, the system

can be written in the polar form ˙r = (4 − r 2 ) which has a limit cycle at r = 2.

To prove the existence of a (stable) limit cycle for ǫ, α > 0 we have to apply the Poincaré-Bendixson theorem.

In order to succeed we have to find a domain which does not contain any fixed points, and the trajectories do not leave it. Near the origin, when x 1 is small, x 2 1 is negligible, thus the equation becomes linear with negative damping. Therefore the solutions diverge from the vicinity of the origin. When x 1 is large the quadratic term takes over and the damping becomes positive, causing the system to loose

energy and the trajectories to get closer to the origin [ 18 ].

If ǫ is not small, increasing it results in an increase in the amplitude of x 2 (t), while the amplitude of x 1 (t) stays the same. ǫ also has an effect on the oscillation period T , which increases with ǫ too. In fact for ǫ → ∞ x 1 (t) converges to a square signal.

Figure 3.2. Solutions of the Van der Pol oscillator with different values of ǫ

On the other hand, an increase in α cause both amplitudes to grow and the shape of the limit cycle to distort more and more from the circular shape attained at small values of α. Plotting the amplitudes against α yields a tool that can help

visualize the qualitative changes in the system’s behavior, similarly to Figure 2.20 .

), red: max(x 1 4

blue: max(x

Figure 3.3. Bifurcation diagram of the Van der Pol oscillator with the bifur- cation parameter α

The phenomenon illustrated by Figure 3.3 is called Hopf bifurcation[ 7 ], which happens when a pair of complex conjugate roots in the stability spectrum gets shifted from the stable half (Re{λ 1,2 } < 0) of the complex plane to the unstable half (Re{λ 1,2 } > 0). In this particular case this happens when α becomes positive, as ǫ > 0. If the damping is positive (and stays so while changing the bifurcation The phenomenon illustrated by Figure 3.3 is called Hopf bifurcation[ 7 ], which happens when a pair of complex conjugate roots in the stability spectrum gets shifted from the stable half (Re{λ 1,2 } < 0) of the complex plane to the unstable half (Re{λ 1,2 } > 0). In this particular case this happens when α becomes positive, as ǫ > 0. If the damping is positive (and stays so while changing the bifurcation

1 Figure 3.4. Phase portrait of the Van der Por oscillator at α = 1, ǫ = 2 (for

larger see Figure C.4 )

3.3 External forcing

The autonomous Van der Pol equation shows interesting phenomena exclusive to nonlinear systems. However, introducing a forcing term allows the system to show chaotic behavior. In case of periodic excitation, the system is defined as

At the parameter values α = 1, ω = , A = 1.2 the system can exhibit chaotic

behavior at certain values of ǫ. In order to find these, we can plot a bifurcation diagram of the x 1 coordinate of the Poincaré section of the non-autonomous system versus the bifurcation parameter ǫ. The sampling by the plane must be done at time intervals of T = 2π/ω = 10. Figure 3.5 also shows that the attractor tries to “lock” itself to the excitation frequency resulting in sections of the bifurcation diagram where the solutions are frequency locked (e.g. between ǫ=8.6 and 9.1 or ǫ=

9.8 and 10.8 ). These frequency locked regions are separated by the chaotic regions.

Note that even after a long transient (t trans = 1000) the frequency-locked states are still not completely settled down (the simulation run time was t end −t trans = 1000 after the transient, with a time step of t step = 0.001), the lines do have some

spreading along the x 1 axis.

(k*T), (T=2 x 1 −1

Figure 3.5. Bifurcation diagram of the forced Van der Pol oscillator with the

bifurcation parameter ǫ (for larger see Figure C.5 )

After analyzing the bifurcation diagram I found points, which resulted in an exceptionally narrow spreading, e.g. at ǫ = 6.72, 9.06, 10.1, 13.92. I thought it would be interesting to analyze the spectra of these signals, and compare it to

others, where the system is chaotic[ 18 ], e.g. at ǫ = 8.53. In order to do this, I calculated the normalized FFT spectrum (N = 2 20 ,f s = 1/t step = 1000) of the x 1 and x 2 signals after the transient. Because of the large value of ǫ, the spectrum contained relatively low frequency components only (it is practically zero after over f=10). For a better picture, I windowed the signal with Hanning window.