Plot the phase portraits for the following systems as parameter $\alpha$ varies:
(i) $\dot{x} = \alpha x - x^2$, $\dot{y} = - y$.
(ii) $\dot{x} = \alpha x + x^3$, $\dot{y} = - y$.
Use numerical integration to plot phase portraits for the system:
\begin{align} \dot{x} &= -y + \mu x + x y^2,\\ \dot{y} &= x + \mu y - x^2. \end{align}Show that as parameter $\mu$ varies, the system undergoes a Hopf bifurcations at $\mu_c=0$. Classify the type of Hopf bifurcation that occurs.
Take the system: $\dot{x}=y-2x$, $\dot{y} = \mu + x^2 - y$.
(i) Scetch the nullclines.
(ii) Find the birfurcations that occur as $\mu$ varies and classify them.
(iii) Sketch the phase portrait as a function of $\mu$.
Take the biased Van der Poll oscillator: $\ddot{x} + \mu (x^2-1)\dot{x} + x = \alpha$.
(i) Find the curves in $(\mu, \alpha)$-space at which Hopf bifurcations occur.
(ii) Take values for $\mu$ and $\alpha$ slightly above and below the Hopf-bifurcation curves and integrate the system numerically. Do you see the structurally different behaviour on either side of the $(\mu, \alpha)$ curves?
Consider the dynamical system $$ \dot{x} = -\frac{\mathrm{d}V}{\mathrm{d}x}, $$ where $V(x) = \tfrac1{6}x^6 + \tfrac1{4}\gamma x^4 - \tfrac1{2}\varepsilon x^2 + h x$. This system describes a phase change.
(i) Take $h=\varepsilon=0$ and compute the bifurcation diagram as a function of parameter $\gamma$.
(ii) Now take $h=0$ but $\varepsilon=\varepsilon_f$. Again, draw the bifurcation diagrams as functions of $\gamma$. Consider separately the case $\varepsilon_f>0$ and $\varepsilon_f<0$.
(iii) Repeat question (ii). with $h\ne 0$.
(iv) In experiments, parameter $\varepsilon$ was switched from a negative value at $t=0$ to some positive value $\varepsilon_f>0$. The figure shows various trajectories $x(t)$ for different final values of $\varepsilon_f$. Explain intuitively why the curves have this strange shape. Why for large values of $\varepsilon_f$ we observe that the trajectories go almost straight up to their steady state, whereas for small $\varepsilon_f$ the trajectories plateau first before increasing sharply to their final level?
[and, finally, some food for thought]
(v) Ponder on how would you go along repeating this exercise without using phase-space and bifurcation diagrams but rather from getting a closed-form solution of the dynamical system.