- (20 points) Consider the initial value problem X(t) = f (t), x(0) = xo E R, for some continuous f : R R. Write the result of the first 2021 steps of Picard’s iterations explicitly (in terms of f).
- (15 points) True or false: the following initial value problem has more then one local solution.
±(t) = Ix’ , x(3) = 0 .
Prove your answer. - (20 points) Plot the bifurcation diagram for i = 5 — re-X2. Find explicit formulas for the fixed points, possibly as a function of the parameter r E R.
- (20 points) Consider the ODE
X(t) = u — where u, r, h E R are parameters. Find a reparametrization of the ODE (possibly of the time t too) such that it depends only on one parameter. - (25 points) Consider ± = rx + axe — x3. • Find the fixed point which is independent of the parameters a, r. Find its stability as a function of possibly a and r using linear stability analysis. • Plot the bifurcation diagrams in r for the ODE. Separate the cases of a > 0 and a < 0. • Plot the stability diagram, i.e., the different regions in the r — a plane where the ODE has a different number of fixed points.
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