MTH108: SUMMARY OF QUALITATIVE ANALYSIS TECHNIQUES
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MTH108: SUMMARY OF QUALITATIVE ANALYSIS TECHNIQUES
In the module we saw a variety of different model types: discrete/continuous, linear/nonlinear, dimension one/two . Different techniques are required for each type . Linear equations have explicit solutions, as do separable differential equations . For nonlinear equations, we can understand the behaviour of solutions using qualitative analysis .
One-dimensional models
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DISCRETE MODELS |
CONTINUOUS MODELS |
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1st order equation A solution is . . . A plot shows . . . |
xn+1 = F (xn ), x0 = c a sequence (xn )n>0 xn against n |
a smooth function x(t) : [0 , o) 3 R x(t) against t |
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Solution is . . . rising wherever falling wherever constant wherever |
F (x) > x F (x) < x F (x) = x |
f (x) > 0 f (x) < 0 f (x) = 0 |
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So the equilibria are all the points x that satisfy |
F (x) = x |
f (x) = 0 |
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An equilibrium x is . . . stable if unstable if |
lF/ (x)l < 1 lF/ (x)l > 1 |
f/ (x) < 0 f/ (x) > 0 |
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Graphical technique: |
Cobweb diagram |
(1-dimensional) phase portrait |
Two-dimensional models
In the two dimensional case, we only studied tools for qualitative analysis of continuous models of the general form:
dx
= f (x , y)
dt
dy
= g(x , y) .
dt
We use the following steps:
(1) Calculate the nullclines: the points in the (x , y)-plane where 
= 0 and 
= 0 . The x-nullcline is the set of points where f (x , y) = 0 and the y-nullcline is the set of points where g(x , y) = 0 .
(2) Draw the nullcline diagram . Distinguish the two nullclines by labels using different colours . Add pairs of perpendicular arrows to show whether x and y are each increasing or decreasing in the regions between nullclines .
(3) Identify the equilibria: they are the points where the x-nullcline and y-nullcline intersect .
(4) Calculate the Jacobian matrix J(x,y) and evaluate it at each equilibrium . Find the eigenvalues of this linearisation to classify the equilibrium as either a source, saddle, sink or stable or unstable focus .
(5) Summarise all of the above information by sketching the phase portrait .
2022-07-30