Refining and Optimizing Analytical Models
This section gives an example of how the model developed earlier could be extended to account for additional knowledge of the system's structure. Again, if you have not yet been exposed to this particular type of mathematical analysis, simply note that all models can be refined as we gain more knowledge about the system. Think back to the model you identified in the last section. How could you extend this model to make it a better reflection of reality?
Long-term behaviour (Equilibrium, Phase space)
Calculation (saddle points and nodes)
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
