Classify the following systems, and sketch their phase...

Classify the following systems, and sketch their phase portraits. (a) $\frac{d u}{d t}=-u+4 v$, (b) $\frac{d u}{d t}=-2 u+v$, (c) $\frac{d u}{d t}=5 u+4 v$, (d) $\frac{d u}{d t}=-3 u-2 v$, $\frac{d v}{d t}=u-2 v$. $$ \frac{d v}{d t}=u-4 v \text {, } $$ $$ \frac{d v}{d t}=u+2 v . \quad \frac{d v}{d t}=3 u+2 v . $$

Classify the following systems, and sketch their phase portraits.
(a) $\frac{d u}{d t}=-u+4 v$,
(b) $\frac{d u}{d t}=-2 u+v$,
(c) $\frac{d u}{d t}=5 u+4 v$,
(d) $\frac{d u}{d t}=-3 u-2 v$,
$\frac{d v}{d t}=u-2 v$.
$$
\frac{d v}{d t}=u-4 v \text {, }
$$
$$
\frac{d v}{d t}=u+2 v . \quad \frac{d v}{d t}=3 u+2 v .
$$

Key Concepts

Phase Portraits

A phase portrait is a graphical depiction of the trajectories of a dynamical system in the phase plane, showing how different initial conditions evolve over time. For linear systems, phase portraits visually represent the direction, stability, and nature of the trajectories around equilibria such as nodes, saddles, and spirals. This qualitative analysis is essential for understanding the global behavior of the system even when analytic solutions are difficult to obtain.

Classification of Equilibria

The nature of an equilibrium point in a linear system can be classified based on the signs and types (real or complex) of the eigenvalues. For example, real eigenvalues of opposite signs indicate a saddle point, while real eigenvalues of the same sign indicate nodes (stable if negative, unstable if positive). Complex eigenvalues typically signal spiral (or focus) behavior, with stability determined by the sign of their real parts, or centers if the real parts vanish.

Eigenvalue and Eigenvector Analysis

Determining the eigenvalues and eigenvectors of the matrix associated with a linear system is central to classifying the system's behavior near critical points. The eigenvalues indicate the type of equilibrium – such as nodes, saddles, spirals, or centers – and whether the equilibrium is stable, unstable, or otherwise. Eigenvectors provide the directions in which the trajectories locally approach or recede from the equilibrium.

Linear Systems of Differential Equations

This concept refers to systems where the rates of change of the variables are expressed as linear combinations of the variables themselves. Such systems can be written in the form x' = Ax, where A is a matrix of constants. Because of their linearity, these systems are amenable to analysis via algebraic methods, including eigenvalue-eigenvector techniques, which directly determine the behavior of solutions over time.

Matrix Representation of Differential Equations

Many systems of linear differential equations are most naturally expressed in matrix form. This involves organizing the coefficients of the variables into a matrix, which encapsulates the entire system's structure. Not only does this format simplify computation, but it also allows for the use of linear algebra tools in analyzing the system’s dynamical properties.

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