# Conservative system ODE

̇x = y

̇y = −x + x3 + ay

a) (1 pts) Find the fixed points of the system.

b) (1.25 pts) For which values of a is the system not conservative? Why?

c) (1 pts) Show that for the other values of a, E(x, y) = (-1/2)y^2-(1/2)x^2+(1/4)x^4 is a conserved quantity.

d) (1.25 pts) Study the stability of the fixed points.

(Hint: Make sure that you study the non-linear stability of each point.)

e) (1 pts) Is the system reversible? Why?

Eva May

28

## Answer

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Mathe

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The answer is accepted.

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The offered bounty is low for a five-part question.

I increased it

Still low. Answering this question will take over an hour. Think about how much the time an individual skilled to answer this question is worth. That would give you an idea for a fair bounty.

What do they mean by non-linear stability? Would checking the eigenvalues of the linearized system be enough?

Im not entirely sure but I think that you are correct about checking the eigenvalues