Refining and Optimizing Analytical Models

On the last page, some new notation was introduced:

\( \vec M (t) = \vec X (t) - \vec X_0 =\left[\begin{array}{l}P(t) \\G(t)\end{array}\right] -\left[\begin{array}{l}P_0 \\G_0\end{array}\right] \),

where (P0,G0) is an equilibrium point. You have learned that a system of differential equations that is linearised around an equilibrium point, can be written as:

\( \dfrac{d \vec M}{dt} = J (\vec X_0) \vec M (t) \).

When our rainbowfish/gourami system is linearised around (P0,G0)=(100,0), the result is:

\( \dfrac{d \vec M}{dt} \left[\begin{array}{l}-0.7 & -4) \\0 & 0.55\end{array}\right] \vec M(t) \).

In the next video, Dennis explains what this new differential equation can tell you about the original differential equations.

Equilibrium points in a system

In the video you have seen that (100,0) is a saddle point by looking at the solutions of the linearised differential equation. The behaviour of solutions near a saddle point is explained by the eigenvalues of the Jacobian matrix: one is positive, and one is negative.

Of course, the eigenvalues of a 2×2-matrix can also be both negative or both positive.  Then both factors eλ1t and eλ2t will either both decrease in time (when λ1<0 and λ2<0) or both increase in time, (when λ1>0 and λ2>0). The equilibrium points are then called nodes.

An equilibrium point X→0 is called a saddle point if the Jacobian matrix J(X→0) has one negative and one positive eigenvalue. A saddle point is unstable because some of the solutions that start near the equilibrium point (here the origin) leave the neighborhood of the origin. A typical sketch of the solutions near a saddle point in the phase plane is given by

An equilibrium point X→0 is called a stable node if the Jacobian matrix J(X→0) has two negative eigenvalues: all solutions that start near the equilibrium point stay near the equilirium point. An example of a phase portrait is

An equilibrium point X→0 is called an unstable node if the Jacobian matrix J(X→0) has two positive eigenvalues. A typical sketch of the solutions near an unstable node in the phase plane is given by