The image contains the following problems:

1. Prove global stability of the origin of the system: $\dot{x} = -\frac{x}{1+x^2}$

2. Consider the system: $\dot{x}_1 = x_2, \quad \dot{x}_2 = -x_1 - x_2$ Is the origin asymptotically stable?

3. Prove global asymptotic stability of the origin of the system: $\dot{x}_1 = x_2, \quad \dot{x}_2 = -x_1 - \frac{x_2}{1 + x_2^2}$ Is the origin locally exponentially stable?

Would you like a detailed breakdown of the solutions to these questions, or do you have a specific part you'd like me to explain further?

Here are 5 related questions to expand on this topic:

1. What is the difference between global and local stability in dynamical systems?
2. How does the Lyapunov function help in proving stability of a system?
3. What are the key properties of an asymptotically stable system?
4. Can a system be globally stable but not asymptotically stable?
5. How can one derive exponential stability from Lyapunov's second method?

Tip: When analyzing stability, always check if the system is autonomous or non-autonomous as it affects the behavior and techniques used in the stability analysis.