Let H be a separable Hilbert space, and (e,) c H a total orthonormal sequence in...

 

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Let H be a separable Hilbert space, and (e,) c H a total orthonormal sequence in H. Let T: H → H be the right shift operator, defined as Tx = (x, en-1)en for every x = (x, enden. n=2 n=1 Note that, in particular, we have Ten = en+l> for every n e N. (c) Determine the steady states of system (2). (20 Marks) (d) Perform the linear stability analysis of system (2) and show that there is a bifurcation (i.e. change in the stability properties of the steady state) at a point b = b. = 1+ a. (20 Marks) Let H be a separable Hilbert space, and (e,) c H a total orthonormal sequence in H. Let T: H → H be the right shift operator, defined as Tx = (x, en-1)en for every x = (x, enden. n=2 n=1 Note that, in particular, we have Ten = en+l> for every n e N. (c) Determine the steady states of system (2). (20 Marks) (d) Perform the linear stability analysis of system (2) and show that there is a bifurcation (i.e. change in the stability properties of the steady state) at a point b = b. = 1+ a. (20 Marks)

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