1. Equilibrium points of ODE Find the equilibrium points of the following system of ODEs. (1) The function z : R → R 2 satisfies z˙(t) =  1 2 1 2 z(t). (2) The functions x, y : I ⊂ R → R satisfy x˙(t) = x(t) − y 2 (t) y˙(t) = x 3 (t) + y 2 (t) (3) The functions x, y, z : I ⊂ R → R satisfy x˙(t) = −y(t) − z(t) y˙(t) = x + ay(t) z˙(t) = b + z(t)(x(t) − c) where a, b, c > 0 and c 2 − 4ab > 0. This system is known as Roessler attractor in the literature. It is an example of chaotic dynamics. (4) The function u : I ⊂ R → R 3 satisfies u˙(t) = F(u(t)), where F : R 3 → R 3 is defined as F   x y z   =   x 2 + yz y 2 + xz z 2 + xy   This system is known as the Hamilton ODE associated to three-dimensional Ricci flow. 2. Equilibrium points of ODE II Please rewrite the following ODEs of second order into a system of first order ODE. Then find the equilibrium points of the system of first order ODE. (1) The function x : I ⊂ R → R satisfies x¨(t) + 2βx˙(t) + ω 2 0x(t) = 0. for constants β > 0 and ω0 > 0. (2) The function ϕ : I ⊂ R → R satisfies ϕ¨(t) + 2βϕ˙(t) + ω 2 0 sin(ϕ(t)) = 0 for constants β > 0 and ω0 > 0. 1 2 MATH 2030 (FALL 2019) ORDINARY DIFFERENTIAL EQUATIONS EXERCISE SHEET NR. 10 3. Review: Linear equations with constant coefficents Please solve the initial value problem    z˙(t) =  0 4 4 0 z(t) z(0) = z0 ∈ R 2 . for the unknown function z : R → R 2 . Moreover, please answer the following questions (1) What are the equilibrium points of the above system? (2) For which z0 ∈ R 2 do we have limt→∞ z(t) = 0 ∈ R 2 ? (3) Is the origin a stable equilibrium point? 4. Review: Integration Please decide, which of the following improper integrals exist and compute their value. (1) Consider for α > 0 the expression Z ∞ 1 1 x α dx. (2) Consider for s > 0 the expression Z ∞ 0 exp(−ts) dt. (3) Consider for s > 0 the expression Z ∞ 0 exp(−ts)t 2 dt. (4) Consider for s > 0 the expression Z ∞ 0 exp(−ts) sin(t) dt. Reminder: Let a ∈ R and f a continous function on R. We define the improper integral R ∞ a f(x) dx by Z ∞ a f(x) dx = lim b→∞ Z b a f(x) dx, if the limit on the right-hand side exists.