Blue-sky bifurcation

Let us consider the following dynamical system [Gavrilov and Shilnikov (2000), Kuznetsov (2004)]:

\begin{equation*} \begin{cases} \dot{x} = x \big( 2 + \mu - b(x^2+y^2) \big) + z^2 + y^2 + 2y \\ \dot{y} = -z^3 - (y+1)(z^2+y^2+2y) - 4x + \mu y \\ \dot{z} = z^2 (y+1) + x^2 - \varepsilon \end{cases}, \end{equation*}
due to Gavrilov and Shilnikov. For \(\varepsilon = 0.02\) and \(b = 10\), the system exhibits a blue-sky bifurcation at \(\mu_0 \simeq 0.275\). You can visualize the orbit passing through \((1,~-1,~2)\) at \(t = 0\): choose a value of \(\mu\) (decimal number between \(0.1\) and \(0.5\)). For \(\mu > \mu_0\), the orbit is attracted to a stable limit cycle.


Simulation

\(\mu = \) You must enter a decimal number between $0.1$ and $0.5$.

Orbit of the Gavrilov-Shilnikov model


Bibliography

N. Gavrilov and A. Shilnikov. Example of a blue sky catastrophe. Translations of the American Mathematical Society-Series 2, 200:99–106, 2000.

Y. Kuznetsov. Elements of Applied Bifurcation Theory. Applied Mathematical Sciences. Springer New York, 2004.