Define the Poincaré index of a closed curve $\mathcal{C}$ for a vector field $\mathbf{f}(\mathbf{x}), \mathbf{x} \in \mathbb{R}^{2}$.

Explain carefully why the index of $\mathcal{C}$ is fully determined by the fixed points of the dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ that lie within $\mathcal{C}$.

What is the Poincaré index for a closed curve $\mathcal{C}$ if it (a) encloses only a saddle point, (b) encloses only a focus and (c) encloses only a node?

What is the Poincaré index for a closed curve $\mathcal{C}$ that is a periodic trajectory of the dynamical system?

A dynamical system in $\mathbb{R}^{2}$ has 2 saddle points, 1 focus and 1 node. What is the maximum number of different periodic orbits? [For the purposes of this question, two orbits are said to be different if they enclose different sets of fixed points.]