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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 (X0, Y0) that satisfies

f(X0, Y0) = g(X0, Y0) = 0

To check the stability of this point we write

X = X0 + x and Y = Y0 + y

as summing that x and y are small . Because of this, we can expand the functions f and g as

f(X, Y) = f(X0, Y0) + Ax + By and g(X, Y) = g(X0, Y0) + Cx + Dy


with all derivatives evalutated at the point X = X0, Y = Y0.

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 ex pression 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 exp onentially 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 + D < 0


A + D > 0

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