From the image provided, we need to evaluate the statements regarding the discrete-time dynamical system $m_{t+1} = f(m_t)$, based on the graph given. The graph shows a blue curve representing $f(m)$ and a red line $m_{t+1} = m_t$ (the identity line). Let’s analyze the statements:

Analysis

1. Statement (i): The solution starting at $m_0 = 0.5$ is increasing.

   • From the graph, $f(0.5) > 0.5$, indicating that $m_{t+1} > m_t$. Therefore, the solution starting at $m_0 = 0.5$ increases.
   • This statement is True (1).

2. Statement (ii): The solution starting at $m_0 = 1.25$ is increasing.

   • From the graph, $f(1.25) < 1.25$, indicating that $m_{t+1} < m_t$. Therefore, the solution starting at $m_0 = 1.25$ decreases.
   • This statement is False (2).

3. Statement (iii): $m^* = 0$ is a stable equilibrium.

   • At $m^* = 0$, the graph intersects the line $m_{t+1} = m_t$. However, the slope of $f(m)$ near $m^* = 0$ appears to be less steep than 1, indicating stability.
   • This statement is True (1).

4. Statement (iv): The non-zero equilibrium is unstable.

   • There is another equilibrium where the blue curve intersects the red line (non-zero equilibrium). Observing the graph, the slope of $f(m)$ near this equilibrium is greater than 1, indicating instability.
   • This statement is True (1).

Final Answer

The correct sequence is 1, 2, 1, 1.

Would you like me to explain any step in more detail or analyze additional scenarios?

Related Questions:

1. How can you determine stability for equilibrium points in a discrete dynamical system?
2. What role does the slope of $f(m)$ play in determining stability?
3. How would the analysis change if $f(m)$ had a different shape?
4. What are the implications of unstable equilibria in real-world systems?
5. How does the graph illustrate the long-term behavior of solutions for varying $m_0$?

Tip:

Always evaluate the slope of $f(m)$ at equilibrium points to determine whether they are stable ($|f'(m^*)| < 1$) or unstable ($|f'(m^*)| > 1$).