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Stability Analysis for Two Variable Systems
Stability Analysis for Two Variable Systems :
Consider a two variable model of the form
and suppose there is a point (X_{0}, Y_{0}) that satisfies
f(X_{0}, Y_{0}) = g(X_{0}, Y_{0}) = 0
To check the stability of this point we write
X = X_{0} + x and Y = Y_{0} + y
assumming that x and y are small. Because of this, we can expand the functions f and g as
f(X, Y) = f(X_{0}, Y_{0}) + Ax + By and g(X, Y) = g(X_{0}, Y_{0}) + Cx + Dy
where
with all derivatives evalutated at the point X = X_{0}, Y = Y_{0}.
After this expansion the equations are of the form :
To solve these equations, we try to find a new variable
that satisfies a simpler equation
The new paramter λ is called an eigenvalue. Our job is to express α and λ in
terms of A , B, C and D.
From the definition of z, (2), and equation (3) we get
Putting in the expressions (1) for dx/dt and dy/dt this gives
If this is going to be true for all x and all y the coefficients of x and of y must match on both sides
Solving the first equation for α gives
and putting this into the second equation, after multiplying by C , gives
From the quadratic formula this gives
This can be simplified to
This is the main result. Note that the solution of equation (4) is
There are four possible cases: If the expression inside the square root of
(10) is positive, the λ's are
real numbers . If both λ values are negative , z (and so x and y) will decay
exponentially to zero. For
this case we must require that the larger of the two λ' s is negative (case a).
If either or both of the λ's
are positive , z (and so x and y) will grow exponentially (case b). If the factor
inside the square root
is negative, the λ's are imaginary
If we write λ = R + i I this means that
z(t) = z(0)exp(λt) = z(0)exp((R + i I)t) = z(0)exp(Rt)exp(iIt)
The relation
exp(iIt) = cos(It) + i sin (It)
then tells us that this produces an oscillating z (and so oscillating x and
y) with the amplitude of the
oscillations going like exp(Rt). If R = (A + D) / 2 < 0 (case c) the
oscillations will decrease to zero
while if R = (A + D) / 2 > 0 (case d) they will increase exponentially .
Summary  For
a) STABLE EXPONENTIAL
b) UNSTABLE EXPONENTIAL
c) STABLE OSCILLATORY
A + D < 0
d) UNSTABLE OSCILLATORY
A + D > 0
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