https://file.aiquickdraw.com/mathbot/file/1725532819899dos5x14h.jpg

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.

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

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

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

This page explores the global and asymptotic stability of several dynamical systems. It includes detailed analysis of Lyapunov functions, as well as the conditions for exponential stability. Perfect for advanced undergraduate or graduate-level students studying stability theory.

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