Differential Equations · Chapter 06

Linear systems and the phase plane

Two quantities that change together. A linear system is a matrix acting on a vector, its solutions are paths through the phase plane, and the eigenvalues of the matrix decide the whole picture: saddles, nodes, spirals and centres.

x^{'} = A x

det (A - λ I) = 0

e^{λ t} v

e^{A t}

λ_{1} λ_{2} < 0

Re λ

x (t_{0}) = x_{0}

A v = λ v

Lessons

Build the foundation

Look for the simplest possible motion: a solution that moves along a straight line through the origin, only stretching or shrinking as time goes on. Such a solution is a fixed vector times an exponential, and substituting it into the system gives a condition on the vector: the matrix must send it to a multiple of itself. That is exactly an eigenvector, and the multiple is its eigenvalue. Two independent eigenvectors give two straight-line solutions, and every solution is a combination of them.

Read lesson →

05SaddlesWhen the two eigenvalues have opposite signs, one straight line carries solutions out and the other carries them in. Every other solution does some of each: it comes in close to the incoming line, swings past the origin and leaves along the outgoing one, like a ball rolled over a mountain pass. The origin is then a saddle, and it is unstable: only the solutions that start exactly on the incoming line ever reach it.Read lesson →

06NodesWhen both eigenvalues are real and have the same sign, every solution moves the same way: in, if both are negative, or out, if both are positive. The origin is a node. The two parts of a solution fade at different rates, so the faster one disappears first, and every trajectory comes in along the direction of the slower eigenvector, tangent to the slow line. Reversing time turns a stable node into an unstable one: the same picture, with the arrows the other way.Read lesson →

07SpiralsWhen the eigenvalues are a complex pair there is no real eigenvector, so no solution moves in a straight line: every solution turns about the origin. The imaginary part of the eigenvalues sets how fast it turns, and the real part whether it winds in, towards the origin, or out, away from it. The origin is a spiral, stable when the real part is negative. The underdamped spring of chapter five is exactly this: its trajectories spiral in as the oscillation dies.Read lesson →

08CentresWhen the eigenvalues are purely imaginary there is turning but no winding: every solution goes round a closed loop for ever, and the origin is a centre. The frictionless spring is the model: its energy is conserved, and the loops are the curves of constant energy. A centre is stable, since nearby solutions stay near, but not asymptotically stable, since they never arrive. It is also fragile: the smallest friction turns it into a stable spiral, and the smallest push into an unstable one.Read lesson →

09Repeated eigenvaluesOn the border between nodes and spirals sits the case of one repeated eigenvalue. If the matrix has only one eigenvector direction, it gives only one straight-line solution, and the second needs a factor of time, just as a repeated root needed a factor of x in chapter four. The trajectories then all come in tangent to that single direction, turning part of the way round as they do: a degenerate node. If instead every vector is an eigenvector, the matrix is a multiple of the identity and every solution is a straight ray: a star.Read lesson →

10Chapter practiceBring the chapter together: write a system as a matrix, find its eigenvalues and straight-line solutions, fit a solution to its start, and tell a saddle, a node, a spiral and a centre apart from the trace and the determinant. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Linear systems and the phase plane

Build the foundation