91Ó°ÊÓ

In the context of vector fields in the complex plane, critical points are locations where the vector field is zero. These points are essential as they represent equilibrium states in a dynamic system. To find critical points for the vector fields given by \(\dot{z} = z^k\) and \(\dot{z} = \bar{z}^k\), we need to set the real and imaginary components of the vector field to zero.

By expressing the functions in terms of their real and imaginary parts, we arrive at the system of equations \(\dot{x} = r^k \cos(k \theta)\) and \(\dot{y} = r^k \sin(k \theta)\) for \(\dot{z} = z^k\). Setting these to zero, we see that the only solution is when \(r = 0\), which corresponds to \(z = 0\).Hence, \(z = 0\) is a critical point.

For the vector field given by \(\dot{z} = \bar{z}^k\), the corresponding system of equations is \(\dot{x} = r^k \cos(k \theta)\) and \(\dot{y} = -r^k \sin(k \theta)\). Again, setting these to zero, we find the critical point is at \(z = 0\).

• Z = 0 is the critical point for \(\dot{z} = z^k\) with index k.
• Z = 0 is the critical point for \(\dot{z} = \bar{z}^k\) with index -k.

Using polar coordinates simplifies the analysis of vector fields in the complex plane. This coordinate system represents a complex number \(z\) as \(r e^{i \theta}\), where \(r\) is the magnitude and \(\theta\) is the angle (or argument) of \(z\). This form often makes it easier to handle complex arithmetic.

To compute \(z^k\) in polar coordinates, we get \(z = r e^{i \theta}\). Thus, \(z^k = (r e^{i \theta})^k = r^k e^{i k \theta}\). Similarly, for the conjugate \(\bar{z} = r e^{-i \theta}\), \(\bar{z}^k = (r e^{-i \theta})^k = r^k e^{-i k \theta}\).

• Using polar coordinates, complex multiplication becomes simpler.
• The expressions \(z^k\) and \(\bar{z}^k\) yield cosine and sine terms that represent real and imaginary components respectively.

Phase portraits graphically represent the trajectories of a dynamical system in the phase plane. For complex vector fields defined by \(\dot{z} = z^k\) and \(\dot{z} = \bar{z}^k\), phase portraits help illustrate the behavior near the critical points.

To sketch these, we look at the indices \(\pm k\) near the origin:

• For \(\dot{z} = z^k\) with indices \(\pm 2\), the trajectories exhibit a double rotation around the origin.
• For \(\pm 3\), a triple rotation pattern emerges.

Generally, the number of rotations (or loops) in the phase portrait correlates with the index value \(k\), indicating how the system evolves over time.

The index of a vector field at a critical point characterizes the topological structure of the field around that point. It essentially counts how many times the vector field rotates while making a closed loop around the critical point.

For \(\dot{z} = z^k\), the index at the critical point \(z = 0\) is \(k\). It indicates that as we trace a path around the origin, the vector field completes \(k\) full rotations.

Conversely, for \(\dot{z} = \bar{z}^k\), the index at \(z = 0\) is \(-k\). Here, the field undergoes \(-k\) rotations around the origin. This negative value shows that the rotation direction is opposite to that of \(\dot{z} = z^k\).

• The index gives insight into the stability and behavior of critical points.
• It helps in predicting patterns in the phase portraits.