Liénard system

Let us consider the following Liénard system corresponding to a perturbation of a weak focus [Perko (2001)]:

\begin{equation*} \begin{cases} \dot{x} = y + \varepsilon p(x) \\ \dot{y} = -x, \end{cases} \end{equation*}
where the perturbation function \(p\) is given by:
\begin{equation*} p(x) = 72 x - \dfrac{392}{3} x^3 + \dfrac{224}{5} x^5 - \dfrac{128}{35} x^7 . \end{equation*}
You can visualize the phase portrait of the perturbed system: choose a value of \(10\varepsilon\) (decimal number between \(0\) and \(1\)). For \(\varepsilon > 0 \) sufficiently small, the system presents 2 stable limit cycles and 1 unstable limit cycle.


Simulation

\(10 \varepsilon = \) You must enter a decimal number between $0$ and $1$.

Phase portrait of the Liénard system


Bibliography

L. Perko. Differential equations and dynamical systems. Springer, New York, 3rd edition, 2001.