The image shows a system of differential equations that represents a simplified aircraft landing gear model. The system is defined as follows:

$\dot{u} = G(u, \lambda) \iff \begin{bmatrix} \dot{x} \\ \dot{y} \end{bmatrix} = \begin{bmatrix} -y + (a - x) e^{x/b} \\ \varepsilon \left( x - \alpha + \frac{y}{\beta} \right) \end{bmatrix}$

Where the parameters are given by:

• $a = -3$
• $b = 10$
• $\alpha = -15$
• $\beta = 3$
• $\varepsilon = 0.001$

The text explains that the steering angle $\alpha$ has been changed from 0 to $-15$ compared to a previous assignment, and as a consequence, the system can exhibit a periodic orbit (denoted by $\Gamma$) when the steering angle is non-zero. This periodic orbit arises because the lateral forces on the landing gear may cause the aircraft to slip out of control, leading to this periodic motion.

Would you like help analyzing the behavior of this system, such as finding fixed points or discussing the periodic orbits?

Here are some questions that could deepen your understanding:

1. What is the significance of each parameter ($a$, $b$, $\alpha$, $\beta$, $\varepsilon$) in the system?
2. How would changing $\varepsilon$ affect the system's behavior?
3. What does it mean for the system to have a periodic orbit, and how could you find it analytically or numerically?
4. How is stability related to the steering angle $\alpha$, and what is the implication of $\alpha = -15$?
5. What numerical methods could be used to simulate this system and identify its dynamic behavior?

Tip: To analyze the stability of the system, you can linearize the differential equations around fixed points and study the eigenvalues of the Jacobian matrix.