Linear Algebra with Differential Equations/Non-Linear Differential Equations

=Some Graphical Analysis=

So far we've dealt with $$\mathbf{A}$$ being a constant matrix, and other niceties; but when it is otherwise, and thus a non-linear differential equation, the best way to find a solution is by graphical means. By taking the independent variables on the axis of a graph, we can note several types of behavior that suggest the form of a solution.

So without adue, here are the main types of behaviors, and their suggested causes:

A nodal source (the graph tends away from a point): real, distinct positive eigenvalues.

A nodal sink (the graph approaches in towards a point): real, distinct negative eigenvalues.

A saddle point (the graph approaches from one end and deviates away at another): real, distinct, opposite eigenvalues.

A spiral point (spirals in or away from a point): a complex eigenvalues.

A series of ellipses around a point: a purely imaginary eigenvalue.

A star point (straight lines deviating or coming towards a point): repeated eigenvalues.