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The trace-determinant plane is a helpful visual tool. It aids in understanding the nature of equilibrium points in a two-dimensional linear system. The plane consists of axes where the x-axis represents the trace (denoted as \( T \)) and the y-axis represents the determinant (denoted as \( D \)). This plane allows us to determine the type of system in question by using the pair \((T, D)\). Depending on their values, different regions of the plane correlate with specific types of equilibrium points (e.g., nodes, saddles, spirals). Locating the point \((-4, 13)\) in our example helps us identify important properties of the system’s eigenvalues, which tell us about the system's dynamic behavior.

A phase portrait is a graphical representation of the trajectories of a dynamical system. It shows the behavior of systems of differential equations over time. In the context of a two-dimensional system, the phase portrait is typically drawn in the plane defined by the system's variables. For the given system, the phase portrait will visualize the paths taken by all possible initial points in the system. When the equilibrium point is a stable spiral, as seen from our point \((-4, 13)\), the portrait will feature inward spiraling trajectories. This indicates that no matter where we start, the system tends to stabilize back at the equilibrium point. Credible phase portraits effectively describe the system's long-term behavior without requiring complex calculations.

Eigenvalues are critical in determining the stability and type of equilibrium points in a system of linear differential equations. For a matrix \( A \), the eigenvalues are roots of its characteristic equation, which generally take the form \( \lambda^2 - T\lambda + D = 0 \). In our case, the eigenvalues are derived from the trace \( T = -4 \) and the determinant \( D = 13 \). The condition \( T^2 < 4D \) suggests that the eigenvalues are complex. This indicates a rotating or spiral behavior in the associated phase portrait. If the real part (related to the trace \( T \)) of these complex eigenvalues is negative, the system spirals inward, indicating stability.

A stable spiral is a type of equilibrium characterized by inward spiraling trajectories. This indicates that as time progresses, all motion within the system will tend to settle towards the equilibrium point. In stability analysis, whether a spiral is stable or not depends on the sign of the real part of the complex eigenvalues.The criteria to determine the presence of a stable spiral are:

• The determinant \( D \) is positive.
• The trace \( T \) is negative.
• \( T^2 < 4D \), signaling that the eigenvalues are complex with negative real parts.

Understanding the concept of a stable spiral is crucial since, in real-world applications, systems often gravitate towards stability. This behavior makes the notion of stable spirals highly relevant in engineering, physics, etc.