van der pol Definition and 12 Threads

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...

The van der pol non-linear equation is given as-: y''=A(1-y²)y'-By y=amplitude The analysis given by the book is this-: When y²<1 i.e when y is small A(1-y²) is negative. A(1-y²) is called damping term. I don't understand how is it negative? It obviously becomes positive in this case...

Homework Statement Given characteristic equation for a circuit containing a diode, I must figure out how to fit a polynomial to the curve so that the van der pol equation is obtained. The paper I am reading is here: http://www.unige.ch/math/hairer60/pres/pres_rentrop.pdf My doubts are...

Homework Statement As the title says, I am trying to implement two differential equations into Simulink. I don't know how to start. How do I know what to start with, which block goes where? when do I add something and when do I know to subtract something using the sum block. When I got my...

$$ x'' + \varepsilon(x^2-1)x'+x=0\quad \varepsilon\ll 1 $$ Using multiple scale perturbation and letting $f(x,x') = (x^2-1)x'$, we have $$ x_{0tt} + \varepsilon x_{1tt} + 2\varepsilon x_{0tT} + \cdots + \varepsilon f(x_0,x_{0t}) + \cdots + x_0 + \varepsilon x_1 + \cdots = 0 $$ where $T$ is our...

Hi All, for a Van Der Pol dyanmical sysetm , governed by the equations x' = a * (x - 0.3 x ^3 ) - y y' = x i read at http://www.scholarpedia.org/article/Van_der_Pol_oscillator that when the system is away from the curve y=x−x3/3 , "a relation |x˙| >> |y˙|=O(1/ϵ) is obtained from...

Hello All, I am struggling to understand the following passage (all equations can be found in a much better format at http://www.scholarpedia.org/article/Van_der_Pol_oscillator, eq(8) especially). Starting from (C being a constant) u' = C * u [4 - (u^2*v^2)] v' = C * v [4 - (u^2*v^2)]...

Homework Statement I have two separate problems, but I get the same answer for each. I feel like this must be wrong. Question 1: Find the leading term of a uniformly valid (for t > 0) asymptotic expansion of the solution of the IVP \ddot{x} + \epsilon\dot{x}(x^2-1)+x = 0 \mbox{ } x(0) =...

Homework Statement "Discuss the fixed points of the Van Der Pol equation depending on the perturbation parameter µ when µ >= 0" Homework Equations x'' - \mu(1-x^2)x' + x = 0 is the Van Der Pol Equation. The Attempt at a Solution Well, this question is really easy for most values of µ...

Hi, wondering if anyone can help with this; we have the Hamiltonian of a linear harmonic oscillator H=(p2+w2q2)/2 now we apply the frictional force F=ap(bq-1) where a and b are constants. how do we alter the Hamiltonian to take that into account? if it helps, the total time derivative of H...

Homework Statement Consider the biased van der Pol oscillator: \frac{d^2x}{dt^2}=u (x^2-1)\frac{dx}{dt} + x = a. Find the curves in (u,a) space at which Hopf bifurcations occur. (Strogatz 8.21) Homework Equations The Attempt at a Solution Not even sure where to start with this...

Hello guys, I am looking for the original article of Van der Pol. B. van der Pol, "The nonlinear theory of electric oscillations," Proc. IRE, vol. 22, pp. 1051--1086, Sept. 1934. do you have an idea of where i can find it? (online :blushing:, or what does proc. IRE means?) The main...