Dynamical systems and chaos

The map $f(x) = x^3$ has three fixed points. How many of them are attracting?

Classify the equilibrium at the origin of $\dot x = x + 2y$, $\dot y = 3x + 2y$.

For the logistic map with $r = 2.5$, how do orbits starting near the fixed point $x^* = 1 - 1/r = 0.6$ behave?

How many points of the circle have minimal period exactly $3$ under the doubling map $D(x) = 2x \bmod 1$?

On the torus $\R^2/\Z^2$, the flow $\dot x = 1$, $\dot y = \sqrt2$ has no equilibria, every orbit is bounded, and every orbit is dense — so no orbit is periodic or approaches a periodic orbit. Which hypothesis of the Poincaré–Bendixson theorem fails?

In Mathlib, which statement says that p lies on an orbit of f of period exactly 2 (periodic with period 2, but not a fixed point)?

Consider $\dot x = -y - x(x^2 + y^2)$, $\dot y = x - y(x^2 + y^2)$. Which statement about the origin is correct?

Write lyapunov(r, x0, n, transient) that estimates the Lyapunov exponent of the logistic map $f_r(x) = rx(1 - x)$: iterate transient times from x0 without recording, then return the average of $\ln|f_r'(x_k)| = \ln|r(1 - 2x_k)|$ over the next n iterates $x_k$.

Which statements are true? Choose all that apply.

For which values of $r$ is the period-2 orbit $\{p_+, p_-\}$ of the logistic map attracting?