(Gradable). (Suggested reading: Strogatz chapter 2.4). Let y, be a fixed point of the ODE y...

 

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(Gradable). (Suggested reading: Strogatz chapter 2.4). Let y, be a fixed point of the ODE y = f(y), that is f(y) = 0. Suppose that f is continuous and at least differentiable. We want to investigate how solutions y(t) behave with initial data starting "close" to y.. Consider the solution y(t) = y + n(t), we say that en(t) is a small perturbation from the fixed point y, and we are assuming that & is quite small. We would like to figure out how n behaves near the fixed point. (a) Find an equation for in terms of y. (b) Taylor expand f(y) to linear order around y, remembering that f(y+) = 0. Your answer will be of the form f(y) = a + bn + higher order terms (c) When b is non-zero, the bn term will determine the main dynamics of the solution. Write the differential equation satisfied by n. (d) Give a condition on f'(y.) to ensure that →y, as too. In this case we say that the fixed point y. is linearly stable. (e) Give a condition on f'(y) to ensure that 7 diverges away from y, as t→∞. In this case we say that the fixed point y, is linearly unstable. Note that this is not sufficient to conclude that the solutions diverge to infinity, only that as t→∞o, solutions with yo = y + do not converge to y.. (f) Suppose that f, f', f", f"" and f(4) are zero at y., but f(5) ‡0. Propose a modification of the above analysis to determine if y, is stable or unstable. Why does your modification work? Note that in this case y, is neither linearly stable or unstable, but we may still speak of the stability of the fixed point. (Gradable). (Suggested reading: Strogatz chapter 2.4). Let y, be a fixed point of the ODE y = f(y), that is f(y) = 0. Suppose that f is continuous and at least differentiable. We want to investigate how solutions y(t) behave with initial data starting "close" to y.. Consider the solution y(t) = y + n(t), we say that en(t) is a small perturbation from the fixed point y, and we are assuming that & is quite small. We would like to figure out how n behaves near the fixed point. (a) Find an equation for in terms of y. (b) Taylor expand f(y) to linear order around y, remembering that f(y+) = 0. Your answer will be of the form f(y) = a + bn + higher order terms (c) When b is non-zero, the bn term will determine the main dynamics of the solution. Write the differential equation satisfied by n. (d) Give a condition on f'(y.) to ensure that →y, as too. In this case we say that the fixed point y. is linearly stable. (e) Give a condition on f'(y) to ensure that 7 diverges away from y, as t→∞. In this case we say that the fixed point y, is linearly unstable. Note that this is not sufficient to conclude that the solutions diverge to infinity, only that as t→∞o, solutions with yo = y + do not converge to y.. (f) Suppose that f, f', f", f"" and f(4) are zero at y., but f(5) ‡0. Propose a modification of the above analysis to determine if y, is stable or unstable. Why does your modification work? Note that in this case y, is neither linearly stable or unstable, but we may still speak of the stability of the fixed point.

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a ANS WER The O DE for y t is y f y If we plug in y t y n t we get y f y n t We can expand f y n t using a Taylor series f y n t f y f y n t o n t Plu... View the full answer