Coupled Differential Equations

Coupled Differential Equations #

Two rates that each need the other quantity.

Two rates that each depend on both quantities form one matrix equation, $\frac{dX}{dt}=MX$dX/dt = M X

Two quantities whose rates each depend on the other are coupled, so they must be solved together.

Stack them and the pair is a single matrix equation: $\frac{dX}{dt}=MX$dX/dt = M X.

Divide one rate by the other to get $\frac{dy}{dx}$dy/dx, and the pair prescribes a direction at every point.

Both rates vanish at the origin, so a pair starting there never moves at all.

Population models take the same form: each quantity appears in the rate of the other.

Euler’s method still works: step both together, each from the current pair of values.

Solving a Coupled System with Eigenvalues #

One exponential per eigenvalue, added together.

With two real eigenvalues the solution is $A{e}^{{\lambda }_{1}t}{v}_1$A e^(λ₁t) v₁ plus $B{e}^{{\lambda }_{2}t}{v}_2$B e^(λ₂t) v₂

Guess a solution that keeps one direction and only changes length: $X=e^{\lambda t}v$X = e^(λt) v.

Cancel $e^{\lambda t}$e^(λt) and the guess works exactly when v is an eigenvector of M.

For this M the characteristic polynomial is ${\lambda }^2-2\lambda -3$λ² − 2λ − 3, so $\lambda =3$λ = 3 or $\lambda =-1$λ = −1.

Solving $\left(M-\lambda I\right)v=0$(M − λI)v = 0 twice gives the directions (1, 2) and (1, −2).

Add the two, each with its own constant, and that is the general solution.

The initial values fix the constants: from X(0) = (2, 0), both A and B come out as 1.

In the long run the larger eigenvalue dominates, and the pair lines up along (1, 2).

Phase Portraits and Equilibrium Types #

Saddle, stable node, unstable node, spiral.

The signs of the two eigenvalues decide whether paths run into the origin, away from it, or past it

Plot y against x rather than against t, and each solution becomes one path.

The two eigenvector directions are straight paths: a path that starts on one never leaves it.

With one eigenvalue positive and one negative, paths come in, turn, and leave again. This is a saddle point.

With both eigenvalues negative every path runs into the origin. This is a stable node.

With both eigenvalues positive every path runs away from the origin instead. This is an unstable node.

Complex eigenvalues turn every path; with a negative real part, each path spirals in to the origin.

So the eigenvalues predict the shape of the portrait before a single path is drawn.

A Second-Order Equation as a Coupled System #

Name the derivative and the order drops.

Naming the first derivative turns one second-order equation into two first-order ones

Name the derivative v and one second-order equation becomes two first-order ones.

The first line is the definition $v=\frac{dx}{dt}$v = dx/dt, and the second line is the original equation solved for $\frac{dv}{dt}$dv/dt.

So $\frac{dX}{dt}=MX$dX/dt = M X with X = (x, v), and M reads straight off the coefficients.

Its characteristic polynomial is ${\lambda }^2+3\lambda +2$λ² + 3λ + 2, which is the auxiliary equation.

The roots −1 and −2 give the same solution the auxiliary equation gives.

Drawn as v against x, every path tends to the origin: the motion dies away.

Euler’s method applies too: one step of the pair is one step of the original equation.