Consider the predator/prey model x^1 = 9x - x^2 -xy y^1 = -5xy + xy. Find all li

Question

Consider the predator/prey model x^1 = 9x - x^2 -xy y^1 = -5xy + xy. Find all linearization of this system at the critical points (0,) [x' y']=A[x y], where A= Then solve this to find the eigenvalues of the literalized system (enter any complex numbers you may obtain by using "i" for root of -1. For real answers, enter them in ascending order, for complex, enter the a - ib root before a +ib): lambda_1 = , lambda_2 = ,The critical point is stable unstable asymptotically stable and is a(n) center improper node saddle point node spiral source spiral sink

Explanation / Answer

The angular velocity of the minute hand is:

Vm = 2*pi rad per hour = 2*pi/3600 rad/sec

The rotational kinetic energy is

Ekr = (1/2)*I*w^2

The rotational inertia of a rod moving about it's end is:

I = (1/3)*M*L^2

For the minute hand, I = (1/3)*100*(4.2)^2 = 588 kg*m^2

Em = (1/2)*(588)*(2*pi/3600)^2 = 8.956x10^-4 J

The hour hand moves 1/12 of a revolution in an hour, so the angular velocity is:

Vh = 2*pi/12 rad per hour = pi/21600 rad/sec

The moment of inertia is (1/3)*300*2.7^2 = 729 kg*m^2

Eh = (1/2)*(729)*(pi/21600)^2 = 7.7106x10^-6 J

The total rotational energy of the hands is Em + Eh, obviously Em dominates.

Ek = 8.956x10^-4 + 7.7106x10^-6 = 9.033x10^-4 J

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