# What Is the Next Step in Simplifying the Van der Pol Oscillator Equation?

• kornelthefirst
In summary, the problem involves a Van der Pol oscillator with small epsilon, which has a circular limit cycle in a phase space trajectory. The equation of motion and Fourier-series for the limit cycle are given. The goal is to simplify the equation to a simpler form, which involves using trigonometric identities and solving for x.
kornelthefirst
Homework Statement
We have a Van der Pol oscillator with small ##\epsilon## and after writing up a Fourier-series we have to bring it to a simpler form.
Relevant Equations
Equation of motion for the Van der Pol oscillator$$\ddot{x} + \epsilon(x^{2} - 1)\dot{x} + x = 0$$ Fourier-series for the limit cycle(already given) $$x_\epsilon^p(t) = \frac{a_0}{2} + \sum\limits_{k=1}^{\infty } [a_k \cos(k \omega t) + b_k \sin(k \omega t)]$$ Equation we need to arrive to$$\epsilon (x_p^2 - 1)\dot{x_p} = \epsilon a_1\omega[(1-\frac{a_1^2}{4})\sin(\omega t)-\frac{a_1^2}{4}\sin(3 \omega t)]$$
First i looked at the case of ## \epsilon = 0## and came to the conclusion, that this oscillator has a circular limit cycle in a phase space trajectory, when plotted with the axes x and ##\dot{x}##.
I have found that ##x_0^p (t) = a_1 \cos(t)## which implies that all other Fourier- coefficients have ##\epsilon## of at least power of 1
The limit cycle is independent of the starting conditions unless ##x = 0## and ##\dot{x} = 0##, so we can choose ##a_1## to be > 0 and ##b_1 > 0##.
If we put the equation of the Fourier-series back to the equation of motion we get$$\sum\limits_{k=1}^{\infty } [ - a_k \cos(k \omega t) - b_k sin( k \omega t)] + \epsilon (x^2-1)\dot{x} + \frac{a_0}{2}+\sum\limits_{k=1}^{\infty } [a_k \cos(k \omega t) + b_k sin( k \omega t)]$$ so simplified $$\epsilon (x^2-1)\dot{x} + \frac{a_0}{2} = 0$$ I am currently stuck here and can't find the next step. I can only assume it will include trigonometric identities, because i can see some patterns for some.

kornelthefirst said:
Homework Statement: We have a Van der Pol oscillator with small ##\epsilon## and after writing up a Fourier-series we have to bring it to a simpler form.

I'm not sure this is the best approach to this problem; could you please post the exact problem statement?

Relevant Equations: Equation of motion for the Van der Pol oscillator$$\ddot{x} + \epsilon(x^{2} - 1)\dot{x} + x = 0$$ Fourier-series for the limit cycle(already given) $$x_\epsilon^p(t) = \frac{a_0}{2} + \sum\limits_{k=1}^{\infty } [a_k \cos(k \omega t) + b_k \sin(k \omega t)]$$ Equation we need to arrive to$$\epsilon (x_p^2 - 1)\dot{x_p} = \epsilon a_1\omega[(1-\frac{a_1^2}{4})\sin(\omega t)-\frac{a_1^2}{4}\sin(3 \omega t)]$$

First i looked at the case of ## \epsilon = 0## and came to the conclusion, that this oscillator has a circular limit cycle in a phase space trajectory, when plotted with the axes x and ##\dot{x}##.
I have found that ##x_0^p (t) = a_1 \cos(t)## which implies that all other Fourier- coefficients have ##\epsilon## of at least power of 1
The limit cycle is independent of the starting conditions unless ##x = 0## and ##\dot{x} = 0##, so we can choose ##a_1## to be > 0 and ##b_1 > 0##.
If we put the equation of the Fourier-series back to the equation of motion we get$$\sum\limits_{k=1}^{\infty } [ - a_k \cos(k \omega t) - b_k sin( k \omega t)] + \epsilon (x^2-1)\dot{x} + \frac{a_0}{2}+\sum\limits_{k=1}^{\infty } [a_k \cos(k \omega t) + b_k sin( k \omega t)]$$

You should have $$\ddot x = -\sum_{n=1}^\infty n^2\omega^2 (a_n \cos (n\omega t) + b_n \sin (n\omega t)).$$

I am currently stuck here and can't find the next step. I can only assume it will include trigonometric identities, because i can see some patterns for some.

I think the idea is that $x(t) = a_1 \cos \omega t + \epsilon x_p(t)$ so that $$\begin{split} \ddot x + x &= -\epsilon(x^2 - 1)\dot x \\ (1 - \omega^2) \cos \omega t + \epsilon (\ddot x_p + x_p) &= \epsilon a_1 \omega (a_1^2 \cos^2\omega t - 1)\sin \omega t + O(\epsilon^2)\end{split}$$ subject to $\dot x_p(0) = x_p(0) = 0$. We do not need a sine term in the leading order solution since that just amounts to a shift in the origin of time, which merely moves us to a different point on the limit cycle. It is not necessary to expand $x_p$ as a fourier series in order to solve this, although expressing the right hand side as a series in $\sin n\omega t$ and knowing $\ddot y + y$ where $y = \sin n\omega t$ or $t\cos n\omega t$ will assist.

## What is a Van der Pol Oscillator?

The Van der Pol Oscillator is a type of non-conservative oscillator with non-linear damping. It was originally proposed by the Dutch physicist Balthasar van der Pol in the 1920s while studying electrical circuits. The system is described by a second-order differential equation that includes a non-linear damping term, which leads to self-sustained oscillations.

## What is a limit cycle in the context of the Van der Pol Oscillator?

A limit cycle in the context of the Van der Pol Oscillator refers to a closed trajectory in phase space that represents a stable, periodic solution to the oscillator's differential equation. Regardless of the initial conditions, the system's behavior eventually converges to this limit cycle, making it a key feature of the oscillator's dynamics.

## How does the parameter μ (mu) affect the behavior of the Van der Pol Oscillator?

The parameter μ (mu) in the Van der Pol Oscillator equation controls the strength of the non-linearity and damping. For small values of μ, the system behaves almost like a simple harmonic oscillator. As μ increases, the non-linearity becomes more pronounced, leading to more complex oscillatory behavior. For large values of μ, the system exhibits relaxation oscillations characterized by rapid changes in amplitude and slow variations in phase.

## What are the applications of the Van der Pol Oscillator?

The Van der Pol Oscillator has applications in various fields including electrical engineering, biology, and neuroscience. In electrical engineering, it was initially used to describe oscillations in vacuum tube circuits. In biology, it models heartbeats and other rhythmic physiological processes. In neuroscience, it helps to understand neuronal firing patterns and other brain oscillations.

## How can the Van der Pol Oscillator be simulated numerically?

The Van der Pol Oscillator can be simulated numerically using methods for solving differential equations, such as the Runge-Kutta method. By discretizing the time variable and iteratively solving the differential equation, one can generate time series data that represents the oscillator's behavior. Software tools like MATLAB, Python (with libraries such as SciPy), and Mathematica are commonly used for such simulations.

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