LS 30B Homework 7 Due Friday, June 3, 2016 1. Consider the following slight vari

Question

LS 30B Homework 7 Due Friday, June 3, 2016 1. Consider the following slight variation on the Rayleigh clarinet reed model, with a parameter a for the "friction" a V (Here X is the height of the tip of the clarinet reed, and V is its velocity.) (a) Regardless of the value of a, this system has only one equilibrium point. Where is it? (b) Compute the Jacobian matrix of the system (as functions of X and V). For this part, leave a as an unknown parameter (which means treat it as a constant when taking derivatives (c) Assume a 1. Use the Jacobian to classify what type of equilibrium point the system has. 1. Once again use the Jacobian to classify the equilibrium (d) Now assume a point (e) Based on your answers to parts (c) and (d), together with what you know about this model, what phenomenon occurred when a changed from 1 to -1? Find the exact value of a where it occurred. (Hint: Remember that the characteristic polynomial of a 2 x 2 matriz I is A2 (a d) (ad-bc), so that (using the quadratic formula) the eigenvalues are a d a d) 2-4 (ad -bc) In particular, if the eigenvalues are compler, it's because the big mess inside the square root here is negative. But the real part of the eigenvalues will just be the other piece of the above erpression at d This will be very useful for part e))

Explanation / Answer

(a) If we donot cosider the value of a then the system has an equilibrium point at origin which is (0,0) i.e, both the height and velocity becomes 0.

(b) Jacobian matrix of the system will be

I v       1I

I -1-v3 -av         -x-3v2-a I                     

(c)                  

when a=1 then (-a, 0) is a sink and (a, 0) is a saddle point equilibrium.

(d)

when a= -1 there will be line equilibrium since the eigen value becomes zero.

(e) The given system is a saddle-node bifurcation or tangent bifurcation which is collision and disappearance of two equilibria.

As value of a goes from 1 to -1 the exact value will be 1.

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