Consider the system d/dt[x/y]=[x(1x^23y^2), y(3x^23y^2)] Compute the Jacobian ma
ID: 3144837 • Letter: C
Question
Consider the system d/dt[x/y]=[x(1x^23y^2), y(3x^23y^2)]
Compute the Jacobian matrix for the above system.
Find the critical points (x,y)(x,y). Enter your answers according to the lexicographical order of the critical points, i.e.,
(a,b)(c,d) if and only if a
For example (1,0),(2,5),(2,6)(1,0),(2,5),(2,6) are in lexicographical order.
Critical Points (x,y)(x,y) :
Next we will classify each critical point as (1) Stable node, (2) Stable spiral, (3) Unstable spiral, (4) Unstable node, (5) Saddle point, (6) Stable Borderline case, (7) Unstable Borderline case, or (8) a Center. Now, in exactly the same order as you entered the critical points enter a comma separated list triples in parentheses (,,k) where and are the trace and determinant of the Jacobian matrix evaluated at the corresponding critical point and kk is the integer 1, 2, 3, 4, 5, 6, 7, 8 describing the stability property of the critical point.
Stability (,,k) :
Explanation / Answer
d/dt[x/y]=[x(1x^23y^2), y(3x^23y^2)]
f = x(1x^23y^2)
g = y(3x^23y^2)
df/dx = 1 3x^23y^2 df/dy = 6xy
dg/dx = 2xy dg/dy = 3x^29y^2
Jacobian Matrix is given by 1 3x^23y^2 2xy
6xy 3x^29y^2
Critical Points:
(0, 0) is unstable as real part of eigen vale is positive
(0, 1) is stable as real part of eigen vale is negavtive
(0, 1) is stable as real part of eigen vale is negavtive
(1, 0) is stable as real part of eigen vale is negavtive
(1, 0) is stable as real part of eigen vale is negavtive